Areas
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Curve Length
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Mean Value Theorem for Integrals
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Average Function Value
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Numerical Integration
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[{"Name":"Areas","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Areas","Duration":"15m 37s","ChapterTopicVideoID":4476,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this section, I\u0027ll be talking about 1 of the main applications of"},{"Start":"00:03.990 ","End":"00:08.100","Text":"definite integrals and that is for computing areas."},{"Start":"00:08.100 ","End":"00:11.160","Text":"I\u0027d like to illustrate a variety of"},{"Start":"00:11.160 ","End":"00:16.225","Text":"situations in which we can compute an area using a definite integral."},{"Start":"00:16.225 ","End":"00:22.760","Text":"1 example of a problem would be that you are given a couple of functions."},{"Start":"00:22.760 ","End":"00:31.190","Text":"Let\u0027s say we\u0027re given f of x and we\u0027re given another one,"},{"Start":"00:31.190 ","End":"00:36.000","Text":"call it g of x."},{"Start":"00:40.850 ","End":"00:48.130","Text":"What we want to know is the area that\u0027s between these curves. Let me shade it."},{"Start":"00:48.130 ","End":"00:53.000","Text":"I\u0027ll show you how to solve this kind of question using definite integrals."},{"Start":"00:53.000 ","End":"00:57.800","Text":"Another kind of a problem would be that you just have 1 function,"},{"Start":"00:57.800 ","End":"00:59.510","Text":"let\u0027s say f of x,"},{"Start":"00:59.510 ","End":"01:02.165","Text":"which is something like this."},{"Start":"01:02.165 ","End":"01:09.900","Text":"You\u0027re arbitrarily given a couple of points here, a and b."},{"Start":"01:09.900 ","End":"01:16.240","Text":"What we want is the area between the x-axis and the curve,"},{"Start":"01:16.240 ","End":"01:19.750","Text":"and between x equals a and x equals b."},{"Start":"01:19.750 ","End":"01:24.470","Text":"This time, it will look something like this."},{"Start":"01:24.830 ","End":"01:30.430","Text":"Yet another variation of the problem is given here,"},{"Start":"01:30.430 ","End":"01:33.579","Text":"2 functions, f of x,"},{"Start":"01:33.579 ","End":"01:37.020","Text":"say this is f of x,"},{"Start":"01:37.020 ","End":"01:40.974","Text":"and another function g of x,"},{"Start":"01:40.974 ","End":"01:45.290","Text":"say this is g of x."},{"Start":"01:46.790 ","End":"01:52.215","Text":"We are given 2 points, a and b."},{"Start":"01:52.215 ","End":"01:58.305","Text":"Now, we want to find the area bounded by this curve,"},{"Start":"01:58.305 ","End":"02:02.500","Text":"this curve, and the limits x equals a and x equals b."},{"Start":"02:02.500 ","End":"02:06.430","Text":"This time the shading is like this."},{"Start":"02:06.430 ","End":"02:15.440","Text":"Often, we call the area S. I\u0027m not sure why the letter S and not the letter A."},{"Start":"02:15.620 ","End":"02:18.075","Text":"Anyway, that\u0027s just what\u0027s used,"},{"Start":"02:18.075 ","End":"02:28.785","Text":"S. We are actually going to focus in this tutorial on this situation here."},{"Start":"02:28.785 ","End":"02:35.195","Text":"But very similar are the other 2 and they will appear in the exercises."},{"Start":"02:35.195 ","End":"02:39.410","Text":"I can\u0027t give you the formulas for each of the 3 of them,"},{"Start":"02:39.410 ","End":"02:43.105","Text":"but we\u0027ll be concentrating on this 1."},{"Start":"02:43.105 ","End":"02:45.875","Text":"Now, if we\u0027re given this situation,"},{"Start":"02:45.875 ","End":"02:49.940","Text":"there\u0027s a bit of preliminary work that we have to do and that is to"},{"Start":"02:49.940 ","End":"02:55.070","Text":"find the x values of this point here."},{"Start":"02:55.070 ","End":"02:59.465","Text":"Let\u0027s call that one a and the x of this point here,"},{"Start":"02:59.465 ","End":"03:01.310","Text":"we\u0027ll call that b."},{"Start":"03:01.310 ","End":"03:04.930","Text":"What we do is, we just solve the equation,"},{"Start":"03:04.930 ","End":"03:09.885","Text":"f of x equals g of x."},{"Start":"03:09.885 ","End":"03:15.680","Text":"It will have at least 2 solutions and we\u0027ll identify which ones are this and this,"},{"Start":"03:15.680 ","End":"03:18.510","Text":"and that will give us our a and b."},{"Start":"03:18.850 ","End":"03:25.160","Text":"In this situation, a and b are given and in this situation also,"},{"Start":"03:25.160 ","End":"03:31.120","Text":"but here we have to just do a little bit of computation, solve an equation."},{"Start":"03:31.120 ","End":"03:33.615","Text":"When we have a and b,"},{"Start":"03:33.615 ","End":"03:36.784","Text":"next thing we have to do is to say,"},{"Start":"03:36.784 ","End":"03:38.750","Text":"which is f and which is g?"},{"Start":"03:38.750 ","End":"03:43.130","Text":"Because often you\u0027ll be given a sketch and you\u0027ll be given f and g,"},{"Start":"03:43.130 ","End":"03:47.450","Text":"but they won\u0027t say which is f and which is g. Like in here,"},{"Start":"03:47.450 ","End":"03:50.585","Text":"f is the blue one and it\u0027s the top one."},{"Start":"03:50.585 ","End":"03:56.635","Text":"F is the top function and g is the bottom function."},{"Start":"03:56.635 ","End":"04:00.050","Text":"Similarly, it\u0027s also important to note which is"},{"Start":"04:00.050 ","End":"04:03.485","Text":"the left limit of integration. That\u0027s the a."},{"Start":"04:03.485 ","End":"04:08.830","Text":"This one is the left and this one is the right."},{"Start":"04:08.830 ","End":"04:11.020","Text":"It\u0027s important to note otherwise,"},{"Start":"04:11.020 ","End":"04:12.620","Text":"you may get your answer backwards."},{"Start":"04:12.620 ","End":"04:13.670","Text":"You get it reversed,"},{"Start":"04:13.670 ","End":"04:15.080","Text":"the negative instead of positive,"},{"Start":"04:15.080 ","End":"04:17.360","Text":"and so on, because the formula,"},{"Start":"04:17.360 ","End":"04:20.060","Text":"if I just write it with f, g, a and b,"},{"Start":"04:20.060 ","End":"04:26.345","Text":"is that S is equal to the definite integral from a"},{"Start":"04:26.345 ","End":"04:32.285","Text":"to b of f of x"},{"Start":"04:32.285 ","End":"04:39.050","Text":"minus g of x, dx."},{"Start":"04:39.050 ","End":"04:41.990","Text":"But you shouldn\u0027t just think of it in terms of letters."},{"Start":"04:41.990 ","End":"04:52.460","Text":"You should really think of it in terms of the integral from the left limit"},{"Start":"04:52.460 ","End":"04:58.250","Text":"to the right limit of"},{"Start":"04:58.250 ","End":"05:02.575","Text":"the top function minus"},{"Start":"05:02.575 ","End":"05:09.110","Text":"the bottom function, dx."},{"Start":"05:09.110 ","End":"05:13.825","Text":"Don\u0027t reverse them. From left to right."},{"Start":"05:13.825 ","End":"05:16.420","Text":"Left, right, this is the top one."},{"Start":"05:16.420 ","End":"05:18.560","Text":"This is the bottom one."},{"Start":"05:19.290 ","End":"05:24.280","Text":"Just for the sake of it,"},{"Start":"05:24.280 ","End":"05:27.280","Text":"I\u0027ll just tell you what happens here."},{"Start":"05:27.280 ","End":"05:28.660","Text":"If we have just 1 function,"},{"Start":"05:28.660 ","End":"05:31.675","Text":"f of x and it has to be above the axis,"},{"Start":"05:31.675 ","End":"05:40.810","Text":"then the area here is just the integral from a to b of f of x,"},{"Start":"05:40.810 ","End":"05:44.480","Text":"dx for this situation."},{"Start":"05:48.920 ","End":"05:52.390","Text":"Take a different color here."},{"Start":"05:52.780 ","End":"05:59.420","Text":"For this situation, we have the integral also from a to b,"},{"Start":"05:59.420 ","End":"06:05.120","Text":"where this is the left and this is the right of the top one minus the bottom one."},{"Start":"06:05.120 ","End":"06:07.040","Text":"Pretty much the same as this;"},{"Start":"06:07.040 ","End":"06:12.285","Text":"f of x minus g of x, dx."},{"Start":"06:12.285 ","End":"06:15.290","Text":"It\u0027s just that here, we have the extra work of finding a and"},{"Start":"06:15.290 ","End":"06:18.260","Text":"b and the area has nice borders."},{"Start":"06:18.260 ","End":"06:20.000","Text":"It\u0027s bordered by f and g only,"},{"Start":"06:20.000 ","End":"06:24.470","Text":"whereas here it\u0027s bordered artificially by chopping it here and here."},{"Start":"06:24.470 ","End":"06:31.580","Text":"We are going to concentrate on this and I\u0027m going to erase the rest of it. They\u0027re gone."},{"Start":"06:31.580 ","End":"06:36.860","Text":"I just wanted to show you that there are a variety of basic situations."},{"Start":"06:36.860 ","End":"06:40.640","Text":"They\u0027re all similar and they will be covered in the exercises,"},{"Start":"06:40.640 ","End":"06:42.485","Text":"and you\u0027ll get the hang of it."},{"Start":"06:42.485 ","End":"06:45.230","Text":"Concentrate on this model where we have"},{"Start":"06:45.230 ","End":"06:48.110","Text":"2 functions that intersect each other in 2 places."},{"Start":"06:48.110 ","End":"06:53.539","Text":"One is above the other and we want the area enclosed by these 2 functions."},{"Start":"06:53.539 ","End":"06:56.830","Text":"I think we\u0027ll go straight to an example."},{"Start":"06:56.830 ","End":"06:58.700","Text":"I\u0027ll move to the next page,"},{"Start":"06:58.700 ","End":"07:01.460","Text":"but I\u0027ll take these formulas with me."},{"Start":"07:01.460 ","End":"07:03.845","Text":"Here we are."},{"Start":"07:03.845 ","End":"07:06.650","Text":"There\u0027s the exercise, there\u0027s a sketch,"},{"Start":"07:06.650 ","End":"07:09.305","Text":"and here\u0027s the formula that I copied."},{"Start":"07:09.305 ","End":"07:11.985","Text":"Combine the 2 into 1,"},{"Start":"07:11.985 ","End":"07:14.415","Text":"the one with symbols and the words."},{"Start":"07:14.415 ","End":"07:17.250","Text":"F minus g integral from a to b,"},{"Start":"07:17.250 ","End":"07:19.080","Text":"and I felt that this is the left, this is right."},{"Start":"07:19.080 ","End":"07:20.300","Text":"This is the top function."},{"Start":"07:20.300 ","End":"07:22.320","Text":"This is the bottom function."},{"Start":"07:22.320 ","End":"07:26.230","Text":"Actually, that should be the first thing we do is"},{"Start":"07:26.230 ","End":"07:30.400","Text":"decide which of these is f and which is g. There\u0027s a blue one and the green one,"},{"Start":"07:30.400 ","End":"07:32.275","Text":"and which is which?"},{"Start":"07:32.275 ","End":"07:37.400","Text":"Well, both of these are parabolas."},{"Start":"07:37.980 ","End":"07:42.625","Text":"We can just use our knowledge of parabolas to say which is which."},{"Start":"07:42.625 ","End":"07:46.615","Text":"We don\u0027t have to draw a sketch of each of these to decide."},{"Start":"07:46.615 ","End":"07:51.730","Text":"The obvious difference is that one of them is concave up on one of them is concave down,"},{"Start":"07:51.730 ","End":"07:53.830","Text":"one faces up on faces down."},{"Start":"07:53.830 ","End":"07:57.955","Text":"If the coefficient of x squared is positive, it\u0027s facing up."},{"Start":"07:57.955 ","End":"08:00.800","Text":"This must be f of x,"},{"Start":"08:01.020 ","End":"08:07.060","Text":"which is the x squared minus 1."},{"Start":"08:07.060 ","End":"08:09.760","Text":"For the other one,"},{"Start":"08:09.760 ","End":"08:16.765","Text":"that\u0027s minus x squared negative coefficient concave down the g of x function."},{"Start":"08:16.765 ","End":"08:20.230","Text":"That\u0027s for deciding which is which."},{"Start":"08:20.230 ","End":"08:26.320","Text":"The first thing after this we want to do is find the limits of integration."},{"Start":"08:26.320 ","End":"08:31.105","Text":"In other words, I want to find out what these points are."},{"Start":"08:31.105 ","End":"08:36.740","Text":"I don\u0027t really need the points I just need the x values of the points."},{"Start":"08:36.860 ","End":"08:40.920","Text":"The thing to do is just to compare f of x with g of x."},{"Start":"08:40.920 ","End":"08:49.870","Text":"We get the equation that x squared minus 1 equals 7 minus x squared."},{"Start":"08:49.940 ","End":"08:54.160","Text":"If I bring the xs to one side and the numbers to the other,"},{"Start":"08:54.160 ","End":"08:58.540","Text":"I\u0027ll get that 2x squared is equal to 8."},{"Start":"08:58.540 ","End":"09:01.630","Text":"X squared is equal to 4."},{"Start":"09:01.630 ","End":"09:06.265","Text":"X is equal to plus or minus 2."},{"Start":"09:06.265 ","End":"09:10.390","Text":"It must be that this is 2 and this is minus 2."},{"Start":"09:10.390 ","End":"09:12.760","Text":"As I said, I don\u0027t care about the value of y,"},{"Start":"09:12.760 ","End":"09:17.650","Text":"but I could compute it by substituting for each of these in,"},{"Start":"09:17.650 ","End":"09:21.085","Text":"doesn\u0027t matter the left-hand side or the right-hand side because they are equal."},{"Start":"09:21.085 ","End":"09:22.540","Text":"Actually, if you do it in your head,"},{"Start":"09:22.540 ","End":"09:24.340","Text":"2 squared minus 1 is 3,"},{"Start":"09:24.340 ","End":"09:28.045","Text":"same for minus 2 and 7 minus 4 is also 3."},{"Start":"09:28.045 ","End":"09:31.360","Text":"Actually, well, it looks a bit skew,"},{"Start":"09:31.360 ","End":"09:34.360","Text":"but for others that we don\u0027t need it."},{"Start":"09:34.360 ","End":"09:38.695","Text":"But if you want to know that this is equal to 3."},{"Start":"09:38.695 ","End":"09:44.810","Text":"Now the area in question that\u0027s enclosed out to shade it."},{"Start":"09:45.140 ","End":"09:49.950","Text":"There it is in yellow and what do we call it?"},{"Start":"09:49.950 ","End":"09:57.730","Text":"S. If I take this formula that I provided, we can substitute,"},{"Start":"09:57.730 ","End":"10:01.675","Text":"in this case the a is minus 2,"},{"Start":"10:01.675 ","End":"10:03.415","Text":"the b is plus 2,"},{"Start":"10:03.415 ","End":"10:08.410","Text":"the top function is f of x,"},{"Start":"10:08.410 ","End":"10:09.850","Text":"which is x squared minus 1."},{"Start":"10:09.850 ","End":"10:12.010","Text":"The bottom one is g of x, which is this,"},{"Start":"10:12.010 ","End":"10:14.470","Text":"and altogether throw them in the formula."},{"Start":"10:14.470 ","End":"10:20.110","Text":"We get that the area is equal to the integral from"},{"Start":"10:20.110 ","End":"10:26.540","Text":"minus 2 to 2 of the top function."},{"Start":"10:28.410 ","End":"10:35.950","Text":"It\u0027s yes, I chose badly because it\u0027s going to come out the other way."},{"Start":"10:35.950 ","End":"10:38.245","Text":"But the most important is not the f and the g,"},{"Start":"10:38.245 ","End":"10:40.900","Text":"the top minus the bottom."},{"Start":"10:40.900 ","End":"10:42.400","Text":"That\u0027s what\u0027s important."},{"Start":"10:42.400 ","End":"10:44.755","Text":"You know what? I\u0027ll relabel them."},{"Start":"10:44.755 ","End":"10:50.455","Text":"Here we are now recording this one g and this one f,"},{"Start":"10:50.455 ","End":"10:57.250","Text":"and so it will be that f is above g. We take f of x,"},{"Start":"10:57.250 ","End":"11:04.540","Text":"which is 7 minus x squared minus g of x,"},{"Start":"11:04.540 ","End":"11:05.755","Text":"which is the lower one,"},{"Start":"11:05.755 ","End":"11:09.355","Text":"which is x squared minus 1."},{"Start":"11:09.355 ","End":"11:14.530","Text":"Let me just put these in brackets, dx."},{"Start":"11:14.530 ","End":"11:17.155","Text":"That\u0027s the expression for the area."},{"Start":"11:17.155 ","End":"11:20.725","Text":"Now, hope you remember how to do definite integrals."},{"Start":"11:20.725 ","End":"11:25.720","Text":"We\u0027ll simplify the integral from minus 2 to 2."},{"Start":"11:25.720 ","End":"11:27.700","Text":"Let\u0027s see what we have inside."},{"Start":"11:27.700 ","End":"11:32.079","Text":"We have 7 minus minus 1 is 8,"},{"Start":"11:32.079 ","End":"11:35.290","Text":"and we have minus x squared minus x squared,"},{"Start":"11:35.290 ","End":"11:40.990","Text":"which is minus 2x squared dx."},{"Start":"11:40.990 ","End":"11:44.890","Text":"Now the way to do a definite integral is"},{"Start":"11:44.890 ","End":"11:49.075","Text":"first to do the indefinite integral and an antiderivative."},{"Start":"11:49.075 ","End":"11:53.810","Text":"The integral of 8 would be 8x."},{"Start":"11:53.970 ","End":"11:57.340","Text":"The integral of minus 20 x squared,"},{"Start":"11:57.340 ","End":"11:58.900","Text":"we raise the power by 1, 2,"},{"Start":"11:58.900 ","End":"12:00.685","Text":"3 and divide by 8,"},{"Start":"12:00.685 ","End":"12:04.915","Text":"so it\u0027s minus 2/3x cubed."},{"Start":"12:04.915 ","End":"12:09.430","Text":"Now, we do not need a constant with definite integrals."},{"Start":"12:09.430 ","End":"12:11.605","Text":"Let\u0027s say we put a constant in."},{"Start":"12:11.605 ","End":"12:15.325","Text":"Let\u0027s say, I\u0027ll just call it C in general."},{"Start":"12:15.325 ","End":"12:20.560","Text":"Now, the definite integral says that what we do is compute"},{"Start":"12:20.560 ","End":"12:30.100","Text":"this expression and then evaluate it between the limits minus 2 and 2."},{"Start":"12:30.100 ","End":"12:33.160","Text":"I\u0027ll remind you what this terminology means."},{"Start":"12:33.160 ","End":"12:35.320","Text":"I just like to say that there\u0027s"},{"Start":"12:35.320 ","End":"12:40.180","Text":"no more old fashioned terminology that doesn\u0027t write it this way."},{"Start":"12:40.180 ","End":"12:46.410","Text":"It\u0027s written also as square brackets,"},{"Start":"12:46.410 ","End":"12:52.680","Text":"8x minus 2/3x cubed plus whatever it was."},{"Start":"12:52.680 ","End":"12:57.695","Text":"Close square brackets and put the numbers here and here."},{"Start":"12:57.695 ","End":"13:03.235","Text":"This is not what should be confusing you if you see it in another book this way."},{"Start":"13:03.235 ","End":"13:06.895","Text":"Same thing. The idea is take the indefinite integral,"},{"Start":"13:06.895 ","End":"13:08.725","Text":"substitute the upper limit,"},{"Start":"13:08.725 ","End":"13:11.695","Text":"substitute the lower limit and subtract."},{"Start":"13:11.695 ","End":"13:14.065","Text":"We\u0027re going to get back to do that here."},{"Start":"13:14.065 ","End":"13:21.490","Text":"Plug-in the 2, and that will give us 8 times 2"},{"Start":"13:21.490 ","End":"13:28.180","Text":"minus 2/3 of 2 cubed"},{"Start":"13:28.180 ","End":"13:33.610","Text":"plus C. But normally we\u0027re not going to write the C that takes care of this path."},{"Start":"13:33.610 ","End":"13:34.780","Text":"I will go for this part,"},{"Start":"13:34.780 ","End":"13:36.865","Text":"the minus 2, which will,"},{"Start":"13:36.865 ","End":"13:38.815","Text":"and we put a minus sign."},{"Start":"13:38.815 ","End":"13:44.935","Text":"Then we get 8 times minus 2 minus 2/3 of"},{"Start":"13:44.935 ","End":"13:54.220","Text":"minus 2 cubed plus C. This is equal 2."},{"Start":"13:54.220 ","End":"13:58.150","Text":"Now, notice that here we have a plus C and here we"},{"Start":"13:58.150 ","End":"14:02.260","Text":"have minus C. The Cs actually cancels out."},{"Start":"14:02.260 ","End":"14:04.300","Text":"This is the reason why it doesn\u0027t matter."},{"Start":"14:04.300 ","End":"14:05.860","Text":"We don\u0027t have to put a constant in."},{"Start":"14:05.860 ","End":"14:07.405","Text":"Whatever constant we put,"},{"Start":"14:07.405 ","End":"14:09.850","Text":"If we increase both of these by a constant."},{"Start":"14:09.850 ","End":"14:12.370","Text":"When we take the difference of two values,"},{"Start":"14:12.370 ","End":"14:15.025","Text":"it\u0027s not gonna make any difference."},{"Start":"14:15.025 ","End":"14:17.740","Text":"The computation, let\u0027s see,"},{"Start":"14:17.740 ","End":"14:20.485","Text":"8 times 2 is 16."},{"Start":"14:20.485 ","End":"14:25.615","Text":"I don\u0027t know if I want to waste my time on computation or just do it very quickly."},{"Start":"14:25.615 ","End":"14:31.490","Text":"This is 8 times 2 over 3 is 16 over 3 is 5 and 1/3."},{"Start":"14:31.620 ","End":"14:36.580","Text":"Here I see, I get everything the same as here."},{"Start":"14:36.580 ","End":"14:39.010","Text":"The sign minus minus makes it a plus."},{"Start":"14:39.010 ","End":"14:40.630","Text":"I get exactly the same thing here,"},{"Start":"14:40.630 ","End":"14:42.040","Text":"so another plus 16,"},{"Start":"14:42.040 ","End":"14:45.940","Text":"another minus 5 and 1/3 altogether,"},{"Start":"14:45.940 ","End":"14:49.315","Text":"32 minus 10 and 2/3,"},{"Start":"14:49.315 ","End":"14:52.540","Text":"which is 21 and 1/3."},{"Start":"14:52.540 ","End":"14:54.625","Text":"Check it if you don\u0027t believe me."},{"Start":"14:54.625 ","End":"14:57.110","Text":"Whenever you do a definite integral,"},{"Start":"14:57.110 ","End":"14:58.580","Text":"using an indefinite integral,"},{"Start":"14:58.580 ","End":"15:00.005","Text":"you do not need the constant."},{"Start":"15:00.005 ","End":"15:01.370","Text":"I could have done without this,"},{"Start":"15:01.370 ","End":"15:03.200","Text":"I could have done without this."},{"Start":"15:03.200 ","End":"15:05.840","Text":"Didn\u0027t need this, didn\u0027t need this."},{"Start":"15:05.840 ","End":"15:08.315","Text":"Just find any indefinite integral,"},{"Start":"15:08.315 ","End":"15:10.205","Text":"substitute the upper limit."},{"Start":"15:10.205 ","End":"15:14.000","Text":"The lower limit, subtract them that you definite integral."},{"Start":"15:14.000 ","End":"15:16.580","Text":"This is the answer."},{"Start":"15:16.580 ","End":"15:19.265","Text":"We are done."},{"Start":"15:19.265 ","End":"15:22.820","Text":"But there you have to do lots of examples."},{"Start":"15:22.820 ","End":"15:28.835","Text":"There are examples after the theory and all kinds of situations."},{"Start":"15:28.835 ","End":"15:33.155","Text":"Variations of this area between two curves and so on."},{"Start":"15:33.155 ","End":"15:34.565","Text":"Do the exercises."},{"Start":"15:34.565 ","End":"15:38.070","Text":"It is important. I\u0027m done now."}],"Thumbnail":null,"ID":4485},{"Watched":false,"Name":"Exercise 1","Duration":"6m 17s","ChapterTopicVideoID":4689,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"Let me go over this problem in general."},{"Start":"00:02.550 ","End":"00:05.745","Text":"I\u0027ll explain it and give our general strategy."},{"Start":"00:05.745 ","End":"00:08.070","Text":"We have 2 functions,"},{"Start":"00:08.070 ","End":"00:11.085","Text":"both parabolas as in the figure."},{"Start":"00:11.085 ","End":"00:14.160","Text":"The first thing I\u0027m going to do is find out which is f"},{"Start":"00:14.160 ","End":"00:17.325","Text":"and which is g. After I\u0027ve done that,"},{"Start":"00:17.325 ","End":"00:20.100","Text":"we have to find their point of intersection,"},{"Start":"00:20.100 ","End":"00:22.740","Text":"which means this point here,"},{"Start":"00:22.740 ","End":"00:25.920","Text":"that\u0027s where the 2 parabolas intersect."},{"Start":"00:25.920 ","End":"00:28.845","Text":"Then we have an area problem,"},{"Start":"00:28.845 ","End":"00:31.245","Text":"the shaded area in the figure,"},{"Start":"00:31.245 ","End":"00:33.210","Text":"we have to compute it."},{"Start":"00:33.210 ","End":"00:37.100","Text":"The way we\u0027re going to do that is we\u0027re going to use the the point of"},{"Start":"00:37.100 ","End":"00:41.840","Text":"intersection to break it up into 2 areas,"},{"Start":"00:41.840 ","End":"00:44.725","Text":"part 1 and part 2."},{"Start":"00:44.725 ","End":"00:47.720","Text":"This 1 will be under 1 function,"},{"Start":"00:47.720 ","End":"00:49.460","Text":"this form B under the other function,"},{"Start":"00:49.460 ","End":"00:51.965","Text":"and we\u0027ll break it up into 2 separate integrals."},{"Start":"00:51.965 ","End":"00:53.779","Text":"That\u0027s the general strategy."},{"Start":"00:53.779 ","End":"00:59.195","Text":"Now let\u0027s see which is f and which is g. There are several ways of doing this,"},{"Start":"00:59.195 ","End":"01:02.285","Text":"easiest probably is to figure out the y"},{"Start":"01:02.285 ","End":"01:07.490","Text":"intercept because I see that this point is higher than this point."},{"Start":"01:07.490 ","End":"01:12.995","Text":"The y-intercept is what happens when you set x equals 0."},{"Start":"01:12.995 ","End":"01:19.010","Text":"For f, I get 0 plus 0 plus 6 is 6 and for g I get 14."},{"Start":"01:19.010 ","End":"01:24.930","Text":"Now, 14 is higher than 6 so this has got to be the 14,"},{"Start":"01:24.930 ","End":"01:26.445","Text":"this is going to be the 6."},{"Start":"01:26.445 ","End":"01:31.610","Text":"That means that this is the function f and this is the function"},{"Start":"01:31.610 ","End":"01:39.095","Text":"g. Another way of doing it would have been to check the coordinate of the apex,"},{"Start":"01:39.095 ","End":"01:42.180","Text":"the minimum here, minus b/2a."},{"Start":"01:42.180 ","End":"01:44.640","Text":"If I do minus b/2a here,"},{"Start":"01:44.640 ","End":"01:47.840","Text":"I get minus 2,"},{"Start":"01:47.840 ","End":"01:50.310","Text":"which fits in with this."},{"Start":"01:51.170 ","End":"01:56.615","Text":"I mean if I compute minus b/2a here I get plus 4/2, which is 2."},{"Start":"01:56.615 ","End":"02:04.280","Text":"Once again, confirms that this is f and this is g. For the point of intersection,"},{"Start":"02:04.280 ","End":"02:08.690","Text":"I\u0027m going to equate f and g. For part a,"},{"Start":"02:08.690 ","End":"02:14.210","Text":"I\u0027m going to get the equation that x squared plus 4x plus 6,"},{"Start":"02:14.210 ","End":"02:16.370","Text":"which is F, is equal to g,"},{"Start":"02:16.370 ","End":"02:21.320","Text":"which is x squared minus 4x plus 14."},{"Start":"02:21.320 ","End":"02:24.464","Text":"Bring everything to the left-hand side,"},{"Start":"02:24.464 ","End":"02:28.859","Text":"and x squared minus x squared just cancels,"},{"Start":"02:28.859 ","End":"02:32.060","Text":"4x minus minus 4x is 8x,"},{"Start":"02:32.060 ","End":"02:36.980","Text":"6 less 14 is minus 8 equals 0,"},{"Start":"02:36.980 ","End":"02:40.250","Text":"which gives me immediately that x equals 1."},{"Start":"02:40.250 ","End":"02:44.615","Text":"This point here is the point where x is 1."},{"Start":"02:44.615 ","End":"02:46.850","Text":"But we don\u0027t just want the x,"},{"Start":"02:46.850 ","End":"02:52.264","Text":"we want the point of intersection so I have to also say what y equals."},{"Start":"02:52.264 ","End":"02:56.600","Text":"Well, y is what a cool f of x or g of x in this case,"},{"Start":"02:56.600 ","End":"02:57.815","Text":"they\u0027re both the same."},{"Start":"02:57.815 ","End":"03:00.530","Text":"So if I plug in 1 to either one of these,"},{"Start":"03:00.530 ","End":"03:05.230","Text":"if I put it in here, I get 1 plus 4 plus 6, which is 11."},{"Start":"03:05.230 ","End":"03:06.950","Text":"Note that if I put it here,"},{"Start":"03:06.950 ","End":"03:10.640","Text":"I get 1 minus 4 plus 14 is also 11."},{"Start":"03:10.640 ","End":"03:12.920","Text":"This point is what we\u0027re looking for,"},{"Start":"03:12.920 ","End":"03:15.350","Text":"is the point 1,"},{"Start":"03:15.350 ","End":"03:19.375","Text":"11, and that\u0027s the answer for part a."},{"Start":"03:19.375 ","End":"03:22.005","Text":"Now let\u0027s get on to part b."},{"Start":"03:22.005 ","End":"03:25.100","Text":"What we\u0027re going to do is take the integral of f from"},{"Start":"03:25.100 ","End":"03:29.210","Text":"minus 2-1 and the integral of g from 1-2."},{"Start":"03:29.210 ","End":"03:31.655","Text":"Let me write that down."},{"Start":"03:31.655 ","End":"03:41.060","Text":"What we have in b for the area is the integral from minus 2-1 of f of x dx,"},{"Start":"03:41.060 ","End":"03:45.830","Text":"but this one is f and this one is g so I\u0027ve"},{"Start":"03:45.830 ","End":"03:51.750","Text":"got x squared plus 4x plus 6 dx."},{"Start":"03:51.750 ","End":"03:54.390","Text":"That\u0027s for the bit here to"},{"Start":"03:54.390 ","End":"03:59.495","Text":"the left and the second bit which is to the right of this line,"},{"Start":"03:59.495 ","End":"04:05.670","Text":"is the integral from 1-2 of the other one,"},{"Start":"04:05.670 ","End":"04:13.290","Text":"g, x squared minus 4x plus 14 dx."},{"Start":"04:13.290 ","End":"04:16.010","Text":"Now for the integration,"},{"Start":"04:16.010 ","End":"04:20.330","Text":"x squared gives me x cubed over 3."},{"Start":"04:20.330 ","End":"04:24.495","Text":"This gives me 4x squared over 2, so 2x."},{"Start":"04:24.495 ","End":"04:32.605","Text":"Here I have 6x and this I have to take between minus 2 and 1 and the other one,"},{"Start":"04:32.605 ","End":"04:39.475","Text":"x cubed over 3 minus 4x squared over 2 minus 2x\u0027s plus 14x."},{"Start":"04:39.475 ","End":"04:44.385","Text":"This one, we take between 1 and 2."},{"Start":"04:44.385 ","End":"04:47.310","Text":"What we get is,"},{"Start":"04:47.310 ","End":"04:49.100","Text":"we\u0027re going to get a bunch of stuff,"},{"Start":"04:49.100 ","End":"04:50.840","Text":"first of all, put in the 1."},{"Start":"04:50.840 ","End":"04:55.490","Text":"So I\u0027ve got 1/3 plus 2,"},{"Start":"04:55.490 ","End":"04:59.660","Text":"and plus 6. That\u0027s for the one."},{"Start":"04:59.660 ","End":"05:03.080","Text":"Now substitute the minus 2 and I have to subtract it."},{"Start":"05:03.080 ","End":"05:07.250","Text":"So minus 2 gives me minus 8/3."},{"Start":"05:07.250 ","End":"05:11.105","Text":"Here I get plus 8,"},{"Start":"05:11.105 ","End":"05:15.275","Text":"and here I get minus 12."},{"Start":"05:15.275 ","End":"05:17.945","Text":"That\u0027s the first term."},{"Start":"05:17.945 ","End":"05:21.515","Text":"I\u0027m going to get here another plus something, minus something."},{"Start":"05:21.515 ","End":"05:25.760","Text":"Here I put in 2 and I\u0027ve got"},{"Start":"05:25.760 ","End":"05:34.785","Text":"8/3 minus 8 plus 14 times 2 is 28."},{"Start":"05:34.785 ","End":"05:38.220","Text":"Then minus, when I put in 1,"},{"Start":"05:38.220 ","End":"05:45.430","Text":"I get 1/3 minus 2 plus 14."},{"Start":"05:45.740 ","End":"05:49.160","Text":"I\u0027m thinking I\u0027ll spare you the boredom of"},{"Start":"05:49.160 ","End":"05:52.630","Text":"going through all this and just give you the answer."},{"Start":"05:52.630 ","End":"05:56.105","Text":"This bit, which is the first area,"},{"Start":"05:56.105 ","End":"05:58.955","Text":"comes out to be 15."},{"Start":"05:58.955 ","End":"06:01.535","Text":"This bit which is the second area,"},{"Start":"06:01.535 ","End":"06:06.245","Text":"the one on the right comes out to be 10 and 1/3."},{"Start":"06:06.245 ","End":"06:11.455","Text":"Finally, we get 25 and 1/3,"},{"Start":"06:11.455 ","End":"06:18.010","Text":"and that is the shaded area and we are done."}],"Thumbnail":null,"ID":4697},{"Watched":false,"Name":"Exercise 2","Duration":"9m 56s","ChapterTopicVideoID":4690,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"I have here a 3-part question and each part builds on the previous."},{"Start":"00:05.370 ","End":"00:10.230","Text":"We start off by having an equation of a parabola as here."},{"Start":"00:10.230 ","End":"00:11.865","Text":"It\u0027s face down parabola,"},{"Start":"00:11.865 ","End":"00:16.020","Text":"negative coefficient of the x^2."},{"Start":"00:16.020 ","End":"00:20.565","Text":"In part a, we have to find the coordinates of the maximum point to the function."},{"Start":"00:20.565 ","End":"00:23.100","Text":"That\u0027s the coordinates of this point,"},{"Start":"00:23.100 ","End":"00:26.820","Text":"which is also the vertex of the parabola."},{"Start":"00:26.820 ","End":"00:30.075","Text":"We have to find the equation of the tangent,"},{"Start":"00:30.075 ","End":"00:34.350","Text":"that\u0027s this line here at its maximum."},{"Start":"00:34.350 ","End":"00:40.085","Text":"It\u0027s no secret that the tangent at a maximum or minimum is always horizontal."},{"Start":"00:40.085 ","End":"00:46.700","Text":"I mean, the slope here is 0 because maximum or minimum points were critical points,"},{"Start":"00:46.700 ","End":"00:50.645","Text":"the derivative 0, so this is going to be horizontal."},{"Start":"00:50.645 ","End":"00:52.980","Text":"We can use that or not."},{"Start":"00:52.980 ","End":"00:58.055","Text":"In part C, we have to find the area that\u0027s shaded."},{"Start":"00:58.055 ","End":"01:02.250","Text":"Here\u0027s the tangent, here\u0027s the x-axis,"},{"Start":"01:02.250 ","End":"01:07.530","Text":"here\u0027s the y-axis, that\u0027s what we call the axis and the part of the graph itself."},{"Start":"01:07.530 ","End":"01:10.290","Text":"It\u0027s bounded on 4 sides."},{"Start":"01:10.290 ","End":"01:15.560","Text":"This will probably do by using integration for area."},{"Start":"01:15.560 ","End":"01:20.360","Text":"What we\u0027ll do is we\u0027ll take this rectangle here"},{"Start":"01:20.360 ","End":"01:24.875","Text":"and subtract from it the area that is here,"},{"Start":"01:24.875 ","End":"01:28.250","Text":"which is going to be our general strategy."},{"Start":"01:28.250 ","End":"01:29.990","Text":"Shading doesn\u0027t look very good,"},{"Start":"01:29.990 ","End":"01:32.150","Text":"but you get the idea."},{"Start":"01:32.150 ","End":"01:39.905","Text":"Let\u0027s get to it and start with part a and coordinates of the maximum."},{"Start":"01:39.905 ","End":"01:49.190","Text":"Now, I can use a shortcut and say that I know that the maximum is always attained minus b"},{"Start":"01:49.190 ","End":"01:53.390","Text":"over 2a when I have ax squared plus bx plus C. That just"},{"Start":"01:53.390 ","End":"01:58.230","Text":"gives me minus 6 over minus 2, which is 3."},{"Start":"01:58.230 ","End":"02:01.010","Text":"But if you\u0027re not happy with that or it\u0027s too short,"},{"Start":"02:01.010 ","End":"02:04.760","Text":"then we can derive and set to 0."},{"Start":"02:04.760 ","End":"02:07.075","Text":"Let me do it the long way then."},{"Start":"02:07.075 ","End":"02:10.880","Text":"I put a line here to say, that\u0027s 1 way."},{"Start":"02:10.880 ","End":"02:17.790","Text":"The other way is to say y prime equals minus 2x plus 6."},{"Start":"02:17.790 ","End":"02:19.714","Text":"At a maximum point,"},{"Start":"02:19.714 ","End":"02:22.550","Text":"we have that y prime equals 0,"},{"Start":"02:22.550 ","End":"02:26.690","Text":"which gives me the minus 2x plus 6 equals 0."},{"Start":"02:26.690 ","End":"02:27.920","Text":"If you solve this,"},{"Start":"02:27.920 ","End":"02:31.115","Text":"you get x equals 3."},{"Start":"02:31.115 ","End":"02:33.830","Text":"We check that it is indeed a maximum."},{"Start":"02:33.830 ","End":"02:35.420","Text":"I can see from the picture,"},{"Start":"02:35.420 ","End":"02:36.470","Text":"I know it\u0027s an inverted,"},{"Start":"02:36.470 ","End":"02:39.235","Text":"but if you really want to do it the hard way,"},{"Start":"02:39.235 ","End":"02:45.430","Text":"the long way, then we have a table for xy prime, and y."},{"Start":"02:45.430 ","End":"02:49.250","Text":"Here are critical points which are suspects for maximum,"},{"Start":"02:49.250 ","End":"02:50.540","Text":"which is x equals 3,"},{"Start":"02:50.540 ","End":"02:52.070","Text":"is the only suspect."},{"Start":"02:52.070 ","End":"02:54.305","Text":"Here, the derivative is 0."},{"Start":"02:54.305 ","End":"02:57.500","Text":"We take a point, the left value to the right,"},{"Start":"02:57.500 ","End":"03:00.965","Text":"plug it into y prime, which is here."},{"Start":"03:00.965 ","End":"03:05.860","Text":"If x is 2, I get minus 4 plus 6, it\u0027s positive."},{"Start":"03:05.860 ","End":"03:09.450","Text":"At 4, it comes out to be negative."},{"Start":"03:09.450 ","End":"03:13.910","Text":"The function itself is increasing here, decreasing here."},{"Start":"03:13.910 ","End":"03:16.010","Text":"The general shape is like this,"},{"Start":"03:16.010 ","End":"03:17.990","Text":"which is a maximum."},{"Start":"03:17.990 ","End":"03:22.345","Text":"This is part a, the short way and the long way."},{"Start":"03:22.345 ","End":"03:27.590","Text":"We\u0027re not quite done because we\u0027ve only got the x of the point in either case,"},{"Start":"03:27.590 ","End":"03:30.080","Text":"I mean, we have that x equals 3,"},{"Start":"03:30.080 ","End":"03:32.820","Text":"but we need what y equals."},{"Start":"03:32.820 ","End":"03:39.575","Text":"Y equals just what happens when you put x equals 3 into the original function."},{"Start":"03:39.575 ","End":"03:42.260","Text":"I am substituting and I\u0027ve got,"},{"Start":"03:42.260 ","End":"03:43.400","Text":"well, I\u0027ll write it out."},{"Start":"03:43.400 ","End":"03:45.800","Text":"It\u0027s minus x is 3,"},{"Start":"03:45.800 ","End":"03:53.215","Text":"so it\u0027s 3 squared plus 6 times 3 minus 5."},{"Start":"03:53.215 ","End":"03:57.260","Text":"This is equal to 3 squared is 9,"},{"Start":"03:57.260 ","End":"03:59.809","Text":"so it\u0027s minus 9 plus 18."},{"Start":"03:59.809 ","End":"04:04.100","Text":"We\u0027re already up to 9 minus 5 is 4."},{"Start":"04:04.100 ","End":"04:07.430","Text":"I guess y is 4,"},{"Start":"04:07.430 ","End":"04:14.530","Text":"so altogether between here and here we get the 0.3 , 4."},{"Start":"04:14.530 ","End":"04:22.070","Text":"I can even write it in the graph here that this is the 0.3,4,"},{"Start":"04:22.070 ","End":"04:26.000","Text":"which means that this is 3 and this is 4."},{"Start":"04:26.000 ","End":"04:33.065","Text":"Now, we want in part b to find the equation of the tangent."},{"Start":"04:33.065 ","End":"04:35.030","Text":"Now, as I say,"},{"Start":"04:35.030 ","End":"04:39.605","Text":"the tangent is horizontal at a maximum or minimum."},{"Start":"04:39.605 ","End":"04:42.005","Text":"The slope is 0."},{"Start":"04:42.005 ","End":"04:47.974","Text":"I can use the equation that y minus y1"},{"Start":"04:47.974 ","End":"04:54.725","Text":"equals f prime of x1 times x minus x1."},{"Start":"04:54.725 ","End":"05:01.875","Text":"Except that I know that f prime of x1 is 0 at a maximum point."},{"Start":"05:01.875 ","End":"05:06.380","Text":"Sure, if you take f equals the quadratic that we had,"},{"Start":"05:06.380 ","End":"05:08.705","Text":"if we call this f of x,"},{"Start":"05:08.705 ","End":"05:11.510","Text":"then sure, f prime of x is 0."},{"Start":"05:11.510 ","End":"05:13.570","Text":"I mean, that\u0027s how we got the 3."},{"Start":"05:13.570 ","End":"05:16.385","Text":"If you want me to check it again,"},{"Start":"05:16.385 ","End":"05:18.770","Text":"y prime is minus 2x plus 6."},{"Start":"05:18.770 ","End":"05:22.235","Text":"If I put in, X1 is 3,"},{"Start":"05:22.235 ","End":"05:25.005","Text":"that will be like my x1,"},{"Start":"05:25.005 ","End":"05:27.170","Text":"y1, and sure I get 0."},{"Start":"05:27.170 ","End":"05:31.610","Text":"Just get that y equals y1. That\u0027s 0."},{"Start":"05:31.610 ","End":"05:33.950","Text":"Y equals y1 and y1 is 4,"},{"Start":"05:33.950 ","End":"05:36.835","Text":"so we just got y equals 4."},{"Start":"05:36.835 ","End":"05:39.590","Text":"That\u0027s the equation. I mean,"},{"Start":"05:39.590 ","End":"05:41.029","Text":"let\u0027s look at the picture."},{"Start":"05:41.029 ","End":"05:45.820","Text":"Horizontal line passes through 4, y equals 4."},{"Start":"05:45.820 ","End":"05:49.450","Text":"That\u0027s part b. In part c,"},{"Start":"05:49.450 ","End":"05:52.235","Text":"we have to figure out this area."},{"Start":"05:52.235 ","End":"05:59.570","Text":"What I want for the shaded is the shaded is like this bit here."},{"Start":"05:59.570 ","End":"06:07.730","Text":"I\u0027m claiming that this equals the whole rectangle minus this bit here."},{"Start":"06:07.730 ","End":"06:12.590","Text":"Symbolic representation that, I mean to figure out the area here,"},{"Start":"06:12.590 ","End":"06:15.395","Text":"figure out the rectangle and subtract."},{"Start":"06:15.395 ","End":"06:19.020","Text":"For the rectangle, so I\u0027ll call this the area,"},{"Start":"06:19.020 ","End":"06:20.960","Text":"I\u0027ll call it s equals."},{"Start":"06:20.960 ","End":"06:24.040","Text":"Now this rectangle is 3 by 4."},{"Start":"06:24.040 ","End":"06:27.740","Text":"3 times 4 minus,"},{"Start":"06:27.740 ","End":"06:30.440","Text":"and this bit is the integral."},{"Start":"06:30.440 ","End":"06:32.905","Text":"I don\u0027t know this point,"},{"Start":"06:32.905 ","End":"06:36.050","Text":"so let\u0027s just think about that."},{"Start":"06:36.050 ","End":"06:38.225","Text":"If I said f equals 0,"},{"Start":"06:38.225 ","End":"06:43.370","Text":"I get x squared minus 6x plus 5 equals 0."},{"Start":"06:43.370 ","End":"06:45.395","Text":"That\u0027s where the parabola cuts."},{"Start":"06:45.395 ","End":"06:49.110","Text":"This factorizes into x minus 1, x minus 5,"},{"Start":"06:49.110 ","End":"06:50.720","Text":"or you could use the formula,"},{"Start":"06:50.720 ","End":"06:53.570","Text":"you get x equals 1 or x equals 5."},{"Start":"06:53.570 ","End":"06:55.685","Text":"This is 1 and this is 5."},{"Start":"06:55.685 ","End":"06:57.575","Text":"Then the event which is what I want."},{"Start":"06:57.575 ","End":"07:06.770","Text":"I need the integral from 1 to 3 of this function, which is f of x,"},{"Start":"07:06.770 ","End":"07:10.355","Text":"which is minus x^2 plus"},{"Start":"07:10.355 ","End":"07:17.520","Text":"6x minus 5 dx."},{"Start":"07:17.520 ","End":"07:22.640","Text":"What we get is 12 minus the"},{"Start":"07:22.640 ","End":"07:28.070","Text":"integral of this is going to be, well, you know what?"},{"Start":"07:28.070 ","End":"07:30.680","Text":"I\u0027m going to change the signs or just use"},{"Start":"07:30.680 ","End":"07:33.739","Text":"a different color here to show you what I\u0027m doing."},{"Start":"07:33.739 ","End":"07:36.635","Text":"I\u0027m going to make this a plus."},{"Start":"07:36.635 ","End":"07:40.760","Text":"This becomes a plus,"},{"Start":"07:40.760 ","End":"07:43.130","Text":"this becomes a minus,"},{"Start":"07:43.130 ","End":"07:45.320","Text":"and this becomes a plus,"},{"Start":"07:45.320 ","End":"07:48.785","Text":"just reversing all the signs."},{"Start":"07:48.785 ","End":"07:58.360","Text":"That means that what I get is 12 plus something between 1 and 3."},{"Start":"07:58.360 ","End":"08:02.855","Text":"Let\u0027s see, x^2 is x^3 over 3 in the integral."},{"Start":"08:02.855 ","End":"08:09.585","Text":"The minus 6x gives me 6x^2 over 2,"},{"Start":"08:09.585 ","End":"08:16.770","Text":"which is 3x minus and the 5 gives me 5 x."},{"Start":"08:16.770 ","End":"08:21.799","Text":"What this gives me is 12 plus now if I put in 3,"},{"Start":"08:21.799 ","End":"08:28.520","Text":"3^3 over 3 is 9 minus 3 times 3 is 9,"},{"Start":"08:28.520 ","End":"08:32.165","Text":"plus 5 times 3 is 15."},{"Start":"08:32.165 ","End":"08:35.284","Text":"Less. When I put in 1,"},{"Start":"08:35.284 ","End":"08:40.275","Text":"I get 1 third minus 3,"},{"Start":"08:40.275 ","End":"08:45.865","Text":"2 missing here, then this is minus 9, but 27."},{"Start":"08:45.865 ","End":"08:54.250","Text":"Fixed. 1 third minus 3 plus 5."},{"Start":"08:54.680 ","End":"09:00.075","Text":"We get 12 plus, let\u0027s see."},{"Start":"09:00.075 ","End":"09:02.355","Text":"9 and 15 is 24,"},{"Start":"09:02.355 ","End":"09:06.930","Text":"minus 27 is minus 3."},{"Start":"09:06.930 ","End":"09:10.615","Text":"Over here I have minus."},{"Start":"09:10.615 ","End":"09:12.415","Text":"Let\u0027s see what we have."},{"Start":"09:12.415 ","End":"09:20.935","Text":"5 minus 3 is 2 plus 1/3 is 2 and a 1/3."},{"Start":"09:20.935 ","End":"09:24.650","Text":"This will equal altogether,"},{"Start":"09:24.650 ","End":"09:30.525","Text":"I have 12 minus 5 and 1/3,"},{"Start":"09:30.525 ","End":"09:35.045","Text":"which gives me 6 and 2/3."},{"Start":"09:35.045 ","End":"09:44.840","Text":"So this is the answer for part C. We have answered all together part a,"},{"Start":"09:44.840 ","End":"09:48.215","Text":"which is here, part B,"},{"Start":"09:48.215 ","End":"09:52.700","Text":"which is here, and part c,"},{"Start":"09:52.700 ","End":"09:57.360","Text":"which is here. We are done."}],"Thumbnail":null,"ID":4698},{"Watched":false,"Name":"Exercise 3","Duration":"8m 23s","ChapterTopicVideoID":4691,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Here I\u0027m given 2 graphs, 2 functions."},{"Start":"00:03.570 ","End":"00:05.790","Text":"The first one is f of x,"},{"Start":"00:05.790 ","End":"00:09.420","Text":"which is also y equals this,"},{"Start":"00:09.420 ","End":"00:11.475","Text":"and that\u0027s got an x squared in it."},{"Start":"00:11.475 ","End":"00:12.720","Text":"That\u0027s the parabola."},{"Start":"00:12.720 ","End":"00:16.380","Text":"This one is f, the other one is a straight line y,"},{"Start":"00:16.380 ","End":"00:19.905","Text":"I could also call it some function of x, g of x."},{"Start":"00:19.905 ","End":"00:22.500","Text":"In fact, I\u0027ll do that so I can give them names,"},{"Start":"00:22.500 ","End":"00:25.410","Text":"f and g. This is the equation of a straight line,"},{"Start":"00:25.410 ","End":"00:26.580","Text":"so that\u0027s this one,"},{"Start":"00:26.580 ","End":"00:30.210","Text":"so we know which is which, parabola and line."},{"Start":"00:30.210 ","End":"00:38.075","Text":"Now, these 2 form an area that\u0027s shaded here that goes between the x-axis,"},{"Start":"00:38.075 ","End":"00:41.760","Text":"the line, and the parabola."},{"Start":"00:41.780 ","End":"00:45.120","Text":"We just have to find the area."},{"Start":"00:45.120 ","End":"00:49.310","Text":"My strategy will be to find the point of intersection"},{"Start":"00:49.310 ","End":"00:54.065","Text":"here between the line and the parabola."},{"Start":"00:54.065 ","End":"00:57.920","Text":"Although I see that the line cuts the parabola twice,"},{"Start":"00:57.920 ","End":"01:00.320","Text":"this one is not really relevant."},{"Start":"01:00.320 ","End":"01:01.460","Text":"This is the one we want,"},{"Start":"01:01.460 ","End":"01:04.760","Text":"we\u0027ll take the one that\u0027s the smaller one."},{"Start":"01:04.760 ","End":"01:08.920","Text":"Then we\u0027ll break the shaded area into 2 bits,"},{"Start":"01:08.920 ","End":"01:11.295","Text":"we\u0027ll take the x value here,"},{"Start":"01:11.295 ","End":"01:15.725","Text":"and then we\u0027ll take from here to here as 1 bit,"},{"Start":"01:15.725 ","End":"01:17.210","Text":"and from here to here another bit."},{"Start":"01:17.210 ","End":"01:19.700","Text":"In other words, the integral of the line from here to"},{"Start":"01:19.700 ","End":"01:22.685","Text":"here and the curve from here to here."},{"Start":"01:22.685 ","End":"01:26.585","Text":"But we also don\u0027t know this point or this point."},{"Start":"01:26.585 ","End":"01:31.160","Text":"This point I get by intersecting the line with the x-axis."},{"Start":"01:31.160 ","End":"01:36.515","Text":"This point is the minimum point of the parabola,"},{"Start":"01:36.515 ","End":"01:41.060","Text":"which I can do by differentiating and equating to 0,"},{"Start":"01:41.060 ","End":"01:45.440","Text":"or I can just find it by minus b over 2a,"},{"Start":"01:45.440 ","End":"01:47.840","Text":"and the y of this point,"},{"Start":"01:47.840 ","End":"01:50.210","Text":"it better be 0,"},{"Start":"01:50.210 ","End":"01:52.535","Text":"otherwise, something won\u0027t make sense."},{"Start":"01:52.535 ","End":"01:57.505","Text":"Anyway, let me just get to it and we\u0027ll see if there\u0027s any problems."},{"Start":"01:57.505 ","End":"02:01.755","Text":"Like I said, we want to find this point first."},{"Start":"02:01.755 ","End":"02:05.070","Text":"We\u0027ll intersect the 2 curves,"},{"Start":"02:05.070 ","End":"02:09.630","Text":"f and g. I\u0027ll set x minus 2 squared,"},{"Start":"02:09.630 ","End":"02:10.935","Text":"which is this one,"},{"Start":"02:10.935 ","End":"02:14.100","Text":"to equal this one, which is,"},{"Start":"02:14.100 ","End":"02:17.159","Text":"I don\u0027t like decimal here I\u0027ll put a fraction,"},{"Start":"02:17.159 ","End":"02:21.980","Text":"a 1/2x plus a 1/2, open the brackets."},{"Start":"02:21.980 ","End":"02:30.570","Text":"I\u0027ve got x squared minus 4x, sorry, plus 4."},{"Start":"02:30.570 ","End":"02:35.255","Text":"This equals 1/2x plus a 1/2."},{"Start":"02:35.255 ","End":"02:40.120","Text":"What I think I\u0027ll do is I\u0027ll multiply both sides by 2."},{"Start":"02:40.120 ","End":"02:41.410","Text":"But at the same time,"},{"Start":"02:41.410 ","End":"02:42.940","Text":"I\u0027ll bring this over to the other side."},{"Start":"02:42.940 ","End":"02:44.830","Text":"I think we can handle that."},{"Start":"02:44.830 ","End":"02:46.420","Text":"If I double this,"},{"Start":"02:46.420 ","End":"02:49.100","Text":"I\u0027ve got 2x squared,"},{"Start":"02:49.100 ","End":"02:52.170","Text":"and then here I have plus 8x,"},{"Start":"02:52.170 ","End":"02:56.490","Text":"but also minus the x from doubling this."},{"Start":"02:56.490 ","End":"03:00.230","Text":"Or I could bring it over first and say 3 1/2 times 2,"},{"Start":"03:00.230 ","End":"03:02.815","Text":"either way, you get 7x."},{"Start":"03:02.815 ","End":"03:07.980","Text":"Then plus 8 minus 1 is also plus 7,"},{"Start":"03:07.980 ","End":"03:12.870","Text":"so this equals 0."},{"Start":"03:12.870 ","End":"03:16.670","Text":"Whoops, I just noticed this should be a minus."},{"Start":"03:16.670 ","End":"03:18.785","Text":"Let me just do some repairs."},{"Start":"03:18.785 ","End":"03:20.735","Text":"That\u0027s going to be a minus."},{"Start":"03:20.735 ","End":"03:23.435","Text":"If this is minus 4x,"},{"Start":"03:23.435 ","End":"03:25.975","Text":"that means it\u0027s minus 8x,"},{"Start":"03:25.975 ","End":"03:31.220","Text":"and then minus another x going to be minus 9x."},{"Start":"03:31.220 ","End":"03:33.775","Text":"I think we\u0027re straight now."},{"Start":"03:33.775 ","End":"03:36.230","Text":"This is a quadratic equation."},{"Start":"03:36.230 ","End":"03:41.495","Text":"Let me just give you the answers because we\u0027re not here to do quadratic equations."},{"Start":"03:41.495 ","End":"03:47.575","Text":"x will be equal to 1, and 3 1/2."},{"Start":"03:47.575 ","End":"03:51.840","Text":"Clearly, this is the 1 and the 3 1/2,"},{"Start":"03:51.840 ","End":"03:54.330","Text":"not that we need it is this one,"},{"Start":"03:54.330 ","End":"03:57.025","Text":"so that\u0027s where these 2 intersect."},{"Start":"03:57.025 ","End":"04:02.240","Text":"The other thing we have to find is this point here."},{"Start":"04:02.240 ","End":"04:04.095","Text":"But I\u0027ll just say,"},{"Start":"04:04.095 ","End":"04:05.360","Text":"if the picture\u0027s right,"},{"Start":"04:05.360 ","End":"04:07.385","Text":"then y is 0."},{"Start":"04:07.385 ","End":"04:10.020","Text":"I can use the tangent, or I can use the minimum,"},{"Start":"04:10.020 ","End":"04:13.130","Text":"or I can just use the fact that y is 0 here."},{"Start":"04:13.130 ","End":"04:20.390","Text":"If y is 0, then this point becomes x minus 2 squared,"},{"Start":"04:20.390 ","End":"04:24.685","Text":"which is y is equal to 0."},{"Start":"04:24.685 ","End":"04:28.725","Text":"So x is equal to 2."},{"Start":"04:28.725 ","End":"04:34.930","Text":"This is the point where x is 2 and y is 0."},{"Start":"04:34.930 ","End":"04:38.075","Text":"As I said, we could have found the minimum point"},{"Start":"04:38.075 ","End":"04:41.870","Text":"by differentiating and setting equal to 0,"},{"Start":"04:41.870 ","End":"04:43.235","Text":"you get the same thing."},{"Start":"04:43.235 ","End":"04:45.690","Text":"This is x is equal 2."},{"Start":"04:45.690 ","End":"04:52.565","Text":"The last thing we have to find is this point here,"},{"Start":"04:52.565 ","End":"04:58.505","Text":"which is what I get when I intersect the line with the x-axis."},{"Start":"04:58.505 ","End":"05:02.600","Text":"Look, if I just set y equals 0 here,"},{"Start":"05:02.600 ","End":"05:05.560","Text":"I\u0027ve got for this bit,"},{"Start":"05:05.560 ","End":"05:13.790","Text":"y which is 0, so 0 equals 0.5x plus 0.5."},{"Start":"05:13.790 ","End":"05:21.200","Text":"So x equals minus 0.5 over 0.5,"},{"Start":"05:21.200 ","End":"05:23.255","Text":"which is minus 1."},{"Start":"05:23.255 ","End":"05:26.080","Text":"This is the point minus 1."},{"Start":"05:26.080 ","End":"05:30.030","Text":"The 3 points I really needed are here,"},{"Start":"05:30.030 ","End":"05:32.595","Text":"minus 1, 1, and 2."},{"Start":"05:32.595 ","End":"05:38.330","Text":"Then these 2 are used for this part of the integral and we use this straight line,"},{"Start":"05:38.330 ","End":"05:41.000","Text":"and from here to here, we use the parabola."},{"Start":"05:41.000 ","End":"05:50.290","Text":"Altogether, what I get is the integral from minus 1 to 1 of"},{"Start":"05:50.290 ","End":"05:54.340","Text":"the straight line bit of the"},{"Start":"05:54.340 ","End":"06:01.220","Text":"1/2x plus 1/2dx plus the remainder,"},{"Start":"06:01.220 ","End":"06:03.065","Text":"which is from 1 to 2,"},{"Start":"06:03.065 ","End":"06:05.390","Text":"taken from the parabola,"},{"Start":"06:05.390 ","End":"06:09.350","Text":"which is just off the picture,"},{"Start":"06:09.350 ","End":"06:15.150","Text":"which is x minus 2 squared dx."},{"Start":"06:15.150 ","End":"06:19.010","Text":"That should give us the shaded area."},{"Start":"06:19.010 ","End":"06:22.670","Text":"Let\u0027s see, do a bit of integration here."},{"Start":"06:22.670 ","End":"06:30.015","Text":"Integration of 1/2x is 1/4x squared here,"},{"Start":"06:30.015 ","End":"06:36.665","Text":"plus 1/2 x, and this we want to take between minus 1 and 1."},{"Start":"06:36.665 ","End":"06:41.930","Text":"The other bit will be integral of x minus 2 squared is"},{"Start":"06:41.930 ","End":"06:49.070","Text":"just 1/3 of x minus 2 cubed because instead of x I have x minus 2,"},{"Start":"06:49.070 ","End":"06:50.720","Text":"but the inner derivative is 1,"},{"Start":"06:50.720 ","End":"06:54.185","Text":"so you can just do it this way."},{"Start":"06:54.185 ","End":"06:57.900","Text":"This I need between 1 and 2."},{"Start":"06:59.210 ","End":"07:01.940","Text":"Now just some substitution."},{"Start":"07:01.940 ","End":"07:04.520","Text":"If I put in x equals 1,"},{"Start":"07:04.520 ","End":"07:12.240","Text":"I get 1/4 plus 1/2 is 3/4 less,"},{"Start":"07:12.240 ","End":"07:15.005","Text":"what happens if I put in minus 1?"},{"Start":"07:15.005 ","End":"07:18.395","Text":"Minus 1 gives me, again,"},{"Start":"07:18.395 ","End":"07:22.580","Text":"plus 1/4, but minus 1/2,"},{"Start":"07:22.580 ","End":"07:25.255","Text":"which is minus 1/4."},{"Start":"07:25.255 ","End":"07:29.210","Text":"Then for here I also get something minus something."},{"Start":"07:29.210 ","End":"07:30.935","Text":"If I put in 2,"},{"Start":"07:30.935 ","End":"07:34.985","Text":"I\u0027ve got 1/3 of,"},{"Start":"07:34.985 ","End":"07:39.330","Text":"2 minus 2 is 0 cubed,"},{"Start":"07:39.330 ","End":"07:41.490","Text":"which is 0, of course,"},{"Start":"07:41.490 ","End":"07:45.080","Text":"and minus, if I put in 1,"},{"Start":"07:45.080 ","End":"07:49.435","Text":"I\u0027ve got 1/3 of minus 1 cubed."},{"Start":"07:49.435 ","End":"07:51.615","Text":"This is what I have to compute."},{"Start":"07:51.615 ","End":"07:53.370","Text":"Let\u0027s see, 3/4 minus,"},{"Start":"07:53.370 ","End":"07:55.770","Text":"minus a quarter is 1."},{"Start":"07:55.770 ","End":"07:59.670","Text":"This bit is 0."},{"Start":"07:59.670 ","End":"08:04.410","Text":"I can write it plus 0 minus,"},{"Start":"08:04.410 ","End":"08:07.440","Text":"minus 1 cubed is minus 1, minus,"},{"Start":"08:07.440 ","End":"08:10.920","Text":"minus is plus, plus a 1/3,"},{"Start":"08:10.920 ","End":"08:16.260","Text":"so I make it 1 and a 1/3."},{"Start":"08:16.260 ","End":"08:18.380","Text":"Unless I made a mistake somewhere,"},{"Start":"08:18.380 ","End":"08:24.270","Text":"this should be the answer for the shaded area, and we\u0027re done."}],"Thumbnail":null,"ID":4699},{"Watched":false,"Name":"Exercise 4","Duration":"5m 11s","ChapterTopicVideoID":4692,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"In this exercise, we\u0027re given functions f and g in the picture."},{"Start":"00:05.220 ","End":"00:11.025","Text":"The first thing I want to do even before I read the question is decide which is which."},{"Start":"00:11.025 ","End":"00:12.825","Text":"Well, that\u0027s pretty simple."},{"Start":"00:12.825 ","End":"00:17.680","Text":"Because if we look at the coefficient of x squared and see if it\u0027s positive or negative,"},{"Start":"00:17.680 ","End":"00:20.630","Text":"that tells us whether it\u0027s face up or face down."},{"Start":"00:20.630 ","End":"00:22.680","Text":"This is a face-up parabola."},{"Start":"00:22.680 ","End":"00:24.570","Text":"The coefficient is positive."},{"Start":"00:24.570 ","End":"00:29.340","Text":"This one has to be f and the other one is face down,"},{"Start":"00:29.340 ","End":"00:34.610","Text":"this one has to be g. Now we\u0027ve got which is which and these 2 parabolas"},{"Start":"00:34.610 ","End":"00:41.165","Text":"intersect at A and B. I also want to note that the vertex of the parabola,"},{"Start":"00:41.165 ","End":"00:42.440","Text":"f of x equals x squared,"},{"Start":"00:42.440 ","End":"00:44.720","Text":"it\u0027s a well known function,"},{"Start":"00:44.720 ","End":"00:49.770","Text":"this is the vertex at 0,0."},{"Start":"00:49.770 ","End":"00:53.150","Text":"The first thing we have to do is to find the coordinates of A and"},{"Start":"00:53.150 ","End":"00:57.665","Text":"B. I\u0027m going to do this by equating the 2 functions,"},{"Start":"00:57.665 ","End":"01:03.380","Text":"f and g. The second thing we\u0027ll have to do would be to compute the area that\u0027s shaded"},{"Start":"01:03.380 ","End":"01:11.480","Text":"here that\u0027s bounded by the functions and the x-axis on the line x equals 4."},{"Start":"01:11.480 ","End":"01:13.970","Text":"Notice that this point here,"},{"Start":"01:13.970 ","End":"01:18.405","Text":"this point A, if I draw a straight line down here,"},{"Start":"01:18.405 ","End":"01:21.290","Text":"we\u0027ll have to compute the area separately to"},{"Start":"01:21.290 ","End":"01:24.170","Text":"the left of the line and to the right of the line because in"},{"Start":"01:24.170 ","End":"01:27.230","Text":"this part I\u0027m working off the function f and"},{"Start":"01:27.230 ","End":"01:30.380","Text":"in this part I\u0027m working off the function g. Let me write it again."},{"Start":"01:30.380 ","End":"01:37.320","Text":"This is f and this is g. We are just going to split it."},{"Start":"01:37.320 ","End":"01:39.284","Text":"Let\u0027s get started."},{"Start":"01:39.284 ","End":"01:43.190","Text":"On A, points of intersection,"},{"Start":"01:43.190 ","End":"01:46.760","Text":"we just equate f of x equals g of x of A and B,"},{"Start":"01:46.760 ","End":"01:53.275","Text":"so x squared is equal to minus x squared plus 18,"},{"Start":"01:53.275 ","End":"01:56.355","Text":"2x squared equals 18,"},{"Start":"01:56.355 ","End":"01:58.635","Text":"x squared equals 9,"},{"Start":"01:58.635 ","End":"02:02.385","Text":"x is plus or minus 3."},{"Start":"02:02.385 ","End":"02:07.849","Text":"Now we know that the coordinate of B"},{"Start":"02:07.849 ","End":"02:13.620","Text":"is minus 3 and the x coordinate of A is 3."},{"Start":"02:13.620 ","End":"02:15.570","Text":"We\u0027re done with part a."},{"Start":"02:15.570 ","End":"02:18.085","Text":"Let me just highlight that."},{"Start":"02:18.085 ","End":"02:21.330","Text":"We have that x is plus or minus 3."},{"Start":"02:21.330 ","End":"02:26.145","Text":"Now, let\u0027s go to part b."},{"Start":"02:26.145 ","End":"02:29.445","Text":"Here we have to compute the area."},{"Start":"02:29.445 ","End":"02:31.125","Text":"We\u0027ll break it up into 2 bits."},{"Start":"02:31.125 ","End":"02:35.630","Text":"The left bit is the integral and this of course is 0,"},{"Start":"02:35.630 ","End":"02:41.745","Text":"from 0 to 3 of f of x dx,"},{"Start":"02:41.745 ","End":"02:47.250","Text":"and then from 3-4, we take it from g of x."},{"Start":"02:47.250 ","End":"02:51.110","Text":"Let\u0027s just spell out what f and g are. Take a quick look again."},{"Start":"02:51.110 ","End":"02:54.935","Text":"F of x is x squared g of x is minus x squared plus 18."},{"Start":"02:54.935 ","End":"03:02.745","Text":"Here we have the integral of 0-3 of x squared dx."},{"Start":"03:02.745 ","End":"03:12.415","Text":"Here we have the integral from 3-4 of minus x squared plus 18 dx."},{"Start":"03:12.415 ","End":"03:18.100","Text":"The integral of x squared is x cubed over 3,"},{"Start":"03:18.100 ","End":"03:22.195","Text":"which we have to take between 0 and 3 and here we have"},{"Start":"03:22.195 ","End":"03:31.515","Text":"minus x cubed over 3 plus 18x between 3 and 4."},{"Start":"03:31.515 ","End":"03:34.120","Text":"Let\u0027s evaluate. We need to plug-in the top"},{"Start":"03:34.120 ","End":"03:38.515","Text":"one and subtract what we get when we substitute the bottom one."},{"Start":"03:38.515 ","End":"03:41.260","Text":"Let\u0027s see here, it\u0027s 3 cubed over 3,"},{"Start":"03:41.260 ","End":"03:48.240","Text":"which is 9 minus 0 cubed over 3 which is 0 plus, let\u0027s see."},{"Start":"03:48.240 ","End":"03:52.135","Text":"The other integral is, if I put in 4,"},{"Start":"03:52.135 ","End":"03:59.265","Text":"I get minus 4 cubed over 3 is minus 64/3."},{"Start":"03:59.265 ","End":"04:03.680","Text":"18 times 4 is 72."},{"Start":"04:03.680 ","End":"04:06.920","Text":"Now I subtract what happens when I put in 3,"},{"Start":"04:06.920 ","End":"04:12.440","Text":"so this is minus 3 cubed over 3 which"},{"Start":"04:12.440 ","End":"04:19.290","Text":"is minus 9 plus 18 times 3 is 54."},{"Start":"04:19.290 ","End":"04:22.055","Text":"Let\u0027s see, what do we have here?"},{"Start":"04:22.055 ","End":"04:30.545","Text":"We have 9 plus 64/3 is 21 and 1/3,"},{"Start":"04:30.545 ","End":"04:38.340","Text":"72 minus 21 and 1/3 is got to be 50 and 2/3."},{"Start":"04:38.420 ","End":"04:43.575","Text":"Here I have 54 minus 9 is 45,"},{"Start":"04:43.575 ","End":"04:48.710","Text":"so it\u0027s minus 45 equals 9."},{"Start":"04:48.710 ","End":"04:51.230","Text":"That\u0027s the first integral on the left."},{"Start":"04:51.230 ","End":"04:54.830","Text":"The other one is just 5 and 2/3,"},{"Start":"04:54.830 ","End":"04:56.390","Text":"the 2 shaded bits."},{"Start":"04:56.390 ","End":"05:03.810","Text":"Finally, it comes out to be 14 and 2/3."},{"Start":"05:03.810 ","End":"05:05.765","Text":"I\u0027m going to highlight this."},{"Start":"05:05.765 ","End":"05:08.400","Text":"This is our answer to part b,"},{"Start":"05:08.400 ","End":"05:12.360","Text":"this is the answer to part a and we are done."}],"Thumbnail":null,"ID":4700},{"Watched":false,"Name":"Exercise 5","Duration":"6m 47s","ChapterTopicVideoID":4693,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"Here we\u0027re given 2 functions and they\u0027re sketched in this figure here,"},{"Start":"00:04.770 ","End":"00:09.105","Text":"the 2 graphs, I like to know which is which right away."},{"Start":"00:09.105 ","End":"00:13.320","Text":"Notice that the top one is a quadratic,"},{"Start":"00:13.320 ","End":"00:15.675","Text":"it\u0027s called x squared and no higher."},{"Start":"00:15.675 ","End":"00:17.640","Text":"This is going to be a parabola."},{"Start":"00:17.640 ","End":"00:19.530","Text":"There is only 1 parabola,"},{"Start":"00:19.530 ","End":"00:21.195","Text":"and that is this one."},{"Start":"00:21.195 ","End":"00:31.875","Text":"This one is the function y equals minus x squared plus 3x plus 2,"},{"Start":"00:31.875 ","End":"00:35.910","Text":"and the other one, which is not a parabola, it\u0027s a cubic,"},{"Start":"00:35.910 ","End":"00:43.710","Text":"is y equals x cubed minus 3x plus 2."},{"Start":"00:43.710 ","End":"00:45.600","Text":"Now we know which is which."},{"Start":"00:45.600 ","End":"00:48.650","Text":"In part a, we\u0027re going to have to find the x coordinates of"},{"Start":"00:48.650 ","End":"00:53.150","Text":"the intersection points and notice that there were actually 3 intersection points,"},{"Start":"00:53.150 ","End":"00:58.610","Text":"at least according to the sketch: 1, 2, and 3."},{"Start":"00:58.610 ","End":"01:04.940","Text":"The second thing we\u0027ll have to do is compute the shaded area just by the description,"},{"Start":"01:04.940 ","End":"01:06.905","Text":"the area bounded by the graphs."},{"Start":"01:06.905 ","End":"01:10.475","Text":"We might think that you\u0027d have this area here too,"},{"Start":"01:10.475 ","End":"01:15.620","Text":"but the sketch leaves no doubt as the figure as to which they mean,"},{"Start":"01:15.620 ","End":"01:17.755","Text":"they mean from here to here."},{"Start":"01:17.755 ","End":"01:25.955","Text":"What we\u0027re going to do is once we found the x coordinate of this point from part a,"},{"Start":"01:25.955 ","End":"01:29.285","Text":"and then this is probably going to come out 0."},{"Start":"01:29.285 ","End":"01:34.535","Text":"Then we\u0027ll take the integral from here to here of the top function,"},{"Start":"01:34.535 ","End":"01:38.315","Text":"which is this one,"},{"Start":"01:38.315 ","End":"01:40.820","Text":"minus the bottom function,"},{"Start":"01:40.820 ","End":"01:42.290","Text":"which is this one."},{"Start":"01:42.290 ","End":"01:44.750","Text":"Like, well, and I call them names,"},{"Start":"01:44.750 ","End":"01:46.535","Text":"let\u0027s called this one f,"},{"Start":"01:46.535 ","End":"01:49.560","Text":"let\u0027s call this one g,"},{"Start":"01:49.560 ","End":"01:51.840","Text":"and then this will be,"},{"Start":"01:51.840 ","End":"01:53.520","Text":"so I\u0027ll know which is which of this,"},{"Start":"01:53.520 ","End":"01:55.620","Text":"f is the x squared one,"},{"Start":"01:55.620 ","End":"02:01.095","Text":"so that\u0027s this, and g is the x cubed one."},{"Start":"02:01.095 ","End":"02:03.915","Text":"No, this is f,"},{"Start":"02:03.915 ","End":"02:06.480","Text":"and let me erase that."},{"Start":"02:06.480 ","End":"02:13.170","Text":"This is g. Let\u0027s get to it now."},{"Start":"02:13.170 ","End":"02:18.320","Text":"Part a, usual when we have to intersect 2 functions,"},{"Start":"02:18.320 ","End":"02:21.740","Text":"we just equate them at these 3 points, f equals g,"},{"Start":"02:21.740 ","End":"02:27.875","Text":"so we have minus x squared plus 3x plus 2 is equal to the other one,"},{"Start":"02:27.875 ","End":"02:32.150","Text":"x cubed minus 3x plus 2."},{"Start":"02:32.150 ","End":"02:35.690","Text":"Let\u0027s bring everything to the left-hand side,"},{"Start":"02:35.690 ","End":"02:37.499","Text":"and then I\u0027ll switch sides."},{"Start":"02:37.499 ","End":"02:40.035","Text":"I get x cubed,"},{"Start":"02:40.035 ","End":"02:42.260","Text":"bring this to the right,"},{"Start":"02:42.260 ","End":"02:45.005","Text":"will be plus x squared,"},{"Start":"02:45.005 ","End":"02:49.925","Text":"and then I\u0027ll have minus 3x minus 3x is minus 6x,"},{"Start":"02:49.925 ","End":"02:53.980","Text":"but the 2 cancels, equals 0,"},{"Start":"02:53.980 ","End":"02:58.005","Text":"can take x outside the brackets, so it\u0027s x,"},{"Start":"02:58.005 ","End":"03:04.125","Text":"x squared plus x minus 6 equals 0."},{"Start":"03:04.125 ","End":"03:06.735","Text":"We have 3 possible solutions,"},{"Start":"03:06.735 ","End":"03:13.810","Text":"either x is 0 or x is a solution to this quadratic equation,"},{"Start":"03:13.810 ","End":"03:18.550","Text":"and I\u0027m not going to waste time solving the quadratic equation."},{"Start":"03:18.550 ","End":"03:25.459","Text":"If you check it, you\u0027ll see that the solutions are minus 3 and 2,"},{"Start":"03:25.459 ","End":"03:30.430","Text":"x equals 0, x equals minus 3, or x equals 2."},{"Start":"03:30.430 ","End":"03:35.585","Text":"In other words, this is the point minus 3,"},{"Start":"03:35.585 ","End":"03:39.405","Text":"this is the point 0, what it looks like in the picture,"},{"Start":"03:39.405 ","End":"03:44.255","Text":"and this is the point where x is equal to 2."},{"Start":"03:44.255 ","End":"03:48.685","Text":"But what interests as part b is the minus 3 and the 0."},{"Start":"03:48.685 ","End":"03:53.590","Text":"Let me just highlight that, for part a."},{"Start":"03:53.590 ","End":"03:56.180","Text":"These are the intersection points,"},{"Start":"03:56.180 ","End":"03:58.475","Text":"the x coordinates just 3 of them."},{"Start":"03:58.475 ","End":"04:06.520","Text":"On to b, and in b I\u0027m going to take the integral from minus 3-0 of g minus f,"},{"Start":"04:06.520 ","End":"04:13.820","Text":"so what I need is the integral from minus 3-0,"},{"Start":"04:13.820 ","End":"04:15.500","Text":"the top one g,"},{"Start":"04:15.500 ","End":"04:24.770","Text":"which is x cubed minus 3x plus 2 minus f,"},{"Start":"04:24.770 ","End":"04:26.195","Text":"which is the lower one,"},{"Start":"04:26.195 ","End":"04:29.480","Text":"which is minus x squared plus"},{"Start":"04:29.480 ","End":"04:39.640","Text":"3x plus 2 minus x squared plus 3x plus 2, all this dx."},{"Start":"04:39.640 ","End":"04:45.370","Text":"Now, I noticed that we\u0027ve already done the subtraction,"},{"Start":"04:45.370 ","End":"04:47.070","Text":"and what we get,"},{"Start":"04:47.070 ","End":"04:49.354","Text":"I mean, at least we look at this line."},{"Start":"04:49.354 ","End":"04:52.450","Text":"This is a subtraction of this minus this,"},{"Start":"04:52.450 ","End":"04:58.165","Text":"so we get that this is equal to the integral from minus 3-0"},{"Start":"04:58.165 ","End":"05:07.390","Text":"of x cubed plus x squared minus 6x dx."},{"Start":"05:07.390 ","End":"05:09.425","Text":"Let\u0027s do the integration,"},{"Start":"05:09.425 ","End":"05:15.090","Text":"the integration here we get x to the 4th over 4, here,"},{"Start":"05:15.090 ","End":"05:17.655","Text":"x cubed over 3,"},{"Start":"05:17.655 ","End":"05:21.450","Text":"and here 6x squared over 2,"},{"Start":"05:21.450 ","End":"05:25.215","Text":"which is 3x squared with a minus."},{"Start":"05:25.215 ","End":"05:29.150","Text":"This we have to evaluate between minus 3 and 0,"},{"Start":"05:29.150 ","End":"05:31.159","Text":"which means substitute the 0,"},{"Start":"05:31.159 ","End":"05:35.670","Text":"substitute the minus 3 and subtract this one from this one."},{"Start":"05:35.670 ","End":"05:39.885","Text":"Let\u0027s see, if we put in minus 3,"},{"Start":"05:39.885 ","End":"05:43.545","Text":"we\u0027ll get 81 over 4,"},{"Start":"05:43.545 ","End":"05:48.700","Text":"81 over 4 is 20 and a 1/4."},{"Start":"05:48.700 ","End":"05:50.690","Text":"If I put in here,"},{"Start":"05:50.690 ","End":"05:53.510","Text":"I\u0027ve got minus 27 over 3,"},{"Start":"05:53.510 ","End":"05:56.330","Text":"which is minus 9."},{"Start":"05:56.330 ","End":"05:59.285","Text":"Here if I put in minus 3,"},{"Start":"05:59.285 ","End":"06:08.880","Text":"I have got minus 3 times plus 9 is minus 27."},{"Start":"06:08.880 ","End":"06:14.265","Text":"If I put in the 0, I got 0 plus 0 minus 0,"},{"Start":"06:14.265 ","End":"06:17.685","Text":"all these terms come out to be 0."},{"Start":"06:17.685 ","End":"06:21.290","Text":"I end up with just this,"},{"Start":"06:21.290 ","End":"06:24.679","Text":"which is, let\u0027s see,"},{"Start":"06:24.679 ","End":"06:32.115","Text":"20 and a 1/4 minus 9 and 27 is 36,"},{"Start":"06:32.115 ","End":"06:40.320","Text":"36 minus 20 and a 1/4 is 15 and 3/4."},{"Start":"06:40.320 ","End":"06:48.850","Text":"This would be the answer for part b, and we\u0027re done."}],"Thumbnail":null,"ID":4701},{"Watched":false,"Name":"Exercise 6","Duration":"6m ","ChapterTopicVideoID":4694,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"Here we are given a function which is actually"},{"Start":"00:02.850 ","End":"00:06.465","Text":"a parabola and that the sketch of it is this."},{"Start":"00:06.465 ","End":"00:11.475","Text":"We know it passes through the point 2, 8, that\u0027s given."},{"Start":"00:11.475 ","End":"00:15.150","Text":"The point A is the point 2, 8,"},{"Start":"00:15.150 ","End":"00:18.480","Text":"which for 1 thing means that if we drop a perpendicular,"},{"Start":"00:18.480 ","End":"00:21.125","Text":"then this is the point 2,"},{"Start":"00:21.125 ","End":"00:23.295","Text":"known as the x coordinate."},{"Start":"00:23.295 ","End":"00:27.245","Text":"Now, the first thing we have to do is find the parameter a."},{"Start":"00:27.245 ","End":"00:29.495","Text":"After that in part B,"},{"Start":"00:29.495 ","End":"00:32.030","Text":"we have to find the point B."},{"Start":"00:32.030 ","End":"00:34.370","Text":"We see the parabola goes through 0, 0."},{"Start":"00:34.370 ","End":"00:37.460","Text":"That\u0027s clear because when x is 0, whatever a is,"},{"Start":"00:37.460 ","End":"00:40.730","Text":"f of x is 0, so this point we know,"},{"Start":"00:40.730 ","End":"00:42.500","Text":"but it also cuts at another point."},{"Start":"00:42.500 ","End":"00:43.970","Text":"Once we have a,"},{"Start":"00:43.970 ","End":"00:47.540","Text":"we\u0027ll be able to find point B. Lastly,"},{"Start":"00:47.540 ","End":"00:50.480","Text":"we\u0027ll have to compute the shaded area."},{"Start":"00:50.480 ","End":"00:52.550","Text":"When we get to it, I\u0027ll show you."},{"Start":"00:52.550 ","End":"00:56.820","Text":"Meanwhile, let\u0027s start with question A,"},{"Start":"00:56.820 ","End":"00:58.660","Text":"to find the parameter a."},{"Start":"00:58.660 ","End":"01:00.365","Text":"Well, the most obvious thing is,"},{"Start":"01:00.365 ","End":"01:03.110","Text":"what does it mean that the function passes through this point?"},{"Start":"01:03.110 ","End":"01:05.090","Text":"It means that if I substitute 2,"},{"Start":"01:05.090 ","End":"01:09.140","Text":"8, then the equation will be satisfied."},{"Start":"01:09.140 ","End":"01:16.520","Text":"You know what, let\u0027s also call f of x as y. I substitute x equals 2, y equals 8."},{"Start":"01:16.520 ","End":"01:20.215","Text":"That means that 8 is equal to,"},{"Start":"01:20.215 ","End":"01:26.280","Text":"minus 2 squared plus a times 2."},{"Start":"01:26.280 ","End":"01:29.700","Text":"That\u0027s what happens when x equals 2, y equals 8."},{"Start":"01:29.700 ","End":"01:39.345","Text":"What I get is that 8 equals minus 4 plus 2a,"},{"Start":"01:39.345 ","End":"01:43.455","Text":"2a is 12, a is equal to 6."},{"Start":"01:43.455 ","End":"01:48.210","Text":"That\u0027s part A and I\u0027ll highlight this."},{"Start":"01:48.210 ","End":"01:51.225","Text":"Now onto part B."},{"Start":"01:51.225 ","End":"01:58.295","Text":"In part B, we have to find where the function is equal to 0."},{"Start":"01:58.295 ","End":"02:01.025","Text":"At least we know 1 of the solutions,"},{"Start":"02:01.025 ","End":"02:03.365","Text":"but we\u0027ll need the other 1 too."},{"Start":"02:03.365 ","End":"02:05.600","Text":"What I have to do is say,"},{"Start":"02:05.600 ","End":"02:12.210","Text":"minus x squared plus ax equals 0."},{"Start":"02:12.210 ","End":"02:13.880","Text":"But we know that a is 6,"},{"Start":"02:13.880 ","End":"02:17.350","Text":"so let me erase that and put a 6 instead."},{"Start":"02:17.350 ","End":"02:19.830","Text":"Now we get that,"},{"Start":"02:19.830 ","End":"02:21.830","Text":"if I take minus x out,"},{"Start":"02:21.830 ","End":"02:30.315","Text":"I\u0027ve got minus x times x minus 6 is equal to 0,"},{"Start":"02:30.315 ","End":"02:36.795","Text":"so x is either equal to 0 or 6."},{"Start":"02:36.795 ","End":"02:40.050","Text":"That x equals 0 or 6."},{"Start":"02:40.050 ","End":"02:42.015","Text":"Well, the 0 we know about it already,"},{"Start":"02:42.015 ","End":"02:44.479","Text":"because the x function cuts at the origin,"},{"Start":"02:44.479 ","End":"02:47.030","Text":"it\u0027s the other 1 that we\u0027re interested in."},{"Start":"02:47.030 ","End":"02:51.510","Text":"The point B, is going to be the point 6,"},{"Start":"02:51.510 ","End":"02:55.845","Text":"0, just like the origin is 0, 0."},{"Start":"02:55.845 ","End":"03:02.180","Text":"That completes part B. I\u0027m just going to highlight the solution."},{"Start":"03:02.180 ","End":"03:05.680","Text":"That is B is 6, 0."},{"Start":"03:05.680 ","End":"03:11.010","Text":"Now onto part C. I\u0027m going to need the diagram really."},{"Start":"03:11.010 ","End":"03:14.650","Text":"I think I\u0027ll do part C over here."},{"Start":"03:14.650 ","End":"03:17.450","Text":"What I need is this shaded area,"},{"Start":"03:17.450 ","End":"03:24.050","Text":"which I\u0027ll call S. But I\u0027m going to break it up into 2 pieces according to this line."},{"Start":"03:24.050 ","End":"03:29.125","Text":"This piece will be S_1 and this piece will be S_2."},{"Start":"03:29.125 ","End":"03:35.370","Text":"Part C, let\u0027s first of all compute S_1."},{"Start":"03:35.370 ","End":"03:43.700","Text":"This is the integral from 0-2 of the parabola,"},{"Start":"03:43.700 ","End":"03:52.980","Text":"which is minus x squared plus 6x dx."},{"Start":"03:52.980 ","End":"03:54.580","Text":"This is equal to,"},{"Start":"03:54.580 ","End":"03:59.224","Text":"the integral of minus x squared is minus x cubed over 3,"},{"Start":"03:59.224 ","End":"04:02.530","Text":"plus 6x squared over 2,"},{"Start":"04:02.530 ","End":"04:05.325","Text":"which is 3x squared."},{"Start":"04:05.325 ","End":"04:08.620","Text":"All this between 0 and 2."},{"Start":"04:08.620 ","End":"04:12.050","Text":"Now when I plug in the 0, everything is 0,"},{"Start":"04:12.050 ","End":"04:14.080","Text":"so I just need to put in 2,"},{"Start":"04:14.080 ","End":"04:19.275","Text":"so this is equal to minus 8 over 3."},{"Start":"04:19.275 ","End":"04:21.050","Text":"Here, if I put in 2,"},{"Start":"04:21.050 ","End":"04:23.315","Text":"I get 3 times 4 is 12."},{"Start":"04:23.315 ","End":"04:29.015","Text":"It\u0027s 12 minus 2 and 2/3,"},{"Start":"04:29.015 ","End":"04:32.725","Text":"which is 9 and 1/3."},{"Start":"04:32.725 ","End":"04:35.285","Text":"That\u0027s just this part here."},{"Start":"04:35.285 ","End":"04:38.415","Text":"Now, what about S_2?"},{"Start":"04:38.415 ","End":"04:45.115","Text":"This, I don\u0027t need to use integration for because I\u0027m using the area of a triangle,"},{"Start":"04:45.115 ","End":"04:49.915","Text":"and the area of a triangle is 1/2 base times height."},{"Start":"04:49.915 ","End":"04:54.920","Text":"Let\u0027s see. The base is from here to here,"},{"Start":"04:54.920 ","End":"04:58.535","Text":"which is 6 minus 2, which is 4."},{"Start":"04:58.535 ","End":"05:02.240","Text":"I\u0027ll just write that the base equals 4."},{"Start":"05:02.240 ","End":"05:04.385","Text":"That\u0027s this part here,"},{"Start":"05:04.385 ","End":"05:06.185","Text":"and I\u0027ll highlight it."},{"Start":"05:06.185 ","End":"05:10.280","Text":"This is the base of the triangle because this is 90 degrees."},{"Start":"05:10.280 ","End":"05:12.650","Text":"This is the height of the triangle."},{"Start":"05:12.650 ","End":"05:18.350","Text":"The height is the y component of this, which is 8."},{"Start":"05:18.350 ","End":"05:22.175","Text":"S_2 is 1/2 base times height,"},{"Start":"05:22.175 ","End":"05:26.865","Text":"1/2 times 4 times 8,"},{"Start":"05:26.865 ","End":"05:29.745","Text":"the 4 was 6 minus 2,"},{"Start":"05:29.745 ","End":"05:34.110","Text":"so this is equal to 16."},{"Start":"05:34.110 ","End":"05:37.905","Text":"Altogether, what I have is that S,"},{"Start":"05:37.905 ","End":"05:40.875","Text":"which is S_1 plus S_2,"},{"Start":"05:40.875 ","End":"05:46.260","Text":"total area is 16 plus 9 and 1/3,"},{"Start":"05:46.260 ","End":"05:51.510","Text":"and I make that 25 and 1/3."},{"Start":"05:51.510 ","End":"05:54.600","Text":"Let\u0027s just highlight that answer,"},{"Start":"05:54.600 ","End":"05:57.840","Text":"25 and 1/3 is the area."},{"Start":"05:57.840 ","End":"06:01.750","Text":"The answer for part C. We are done."}],"Thumbnail":null,"ID":4702},{"Watched":false,"Name":"Exercise 7","Duration":"7m 18s","ChapterTopicVideoID":4695,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.535","Text":"Here, I\u0027m given 2 functions."},{"Start":"00:02.535 ","End":"00:04.050","Text":"I don\u0027t know yet which is which,"},{"Start":"00:04.050 ","End":"00:05.400","Text":"but soon I\u0027ll know."},{"Start":"00:05.400 ","End":"00:09.090","Text":"In part a, I have to find the intersection points of"},{"Start":"00:09.090 ","End":"00:13.890","Text":"the function with the y-axis and I\u0027ve already marked them here in blue."},{"Start":"00:13.890 ","End":"00:19.740","Text":"In part b, I have to find the intersection point of the 2 functions,"},{"Start":"00:19.740 ","End":"00:23.940","Text":"which is already been highlighted here but not in color."},{"Start":"00:23.940 ","End":"00:30.119","Text":"The final thing will be to find the areas S_1 and S_2,"},{"Start":"00:30.119 ","End":"00:34.100","Text":"or more specifically, we\u0027re asked for S_1/S_2."},{"Start":"00:34.100 ","End":"00:37.230","Text":"Let\u0027s get started with part a."},{"Start":"00:37.230 ","End":"00:39.810","Text":"Now, the intersection with the y-axis,"},{"Start":"00:39.810 ","End":"00:42.830","Text":"which means x equals 0. Let\u0027s see."},{"Start":"00:42.830 ","End":"00:46.940","Text":"We need to know what f of 0 equals and what g of 0 equals."},{"Start":"00:46.940 ","End":"00:49.100","Text":"Let\u0027s see which is the bigger one."},{"Start":"00:49.100 ","End":"00:54.455","Text":"F of 0 is e to the power of minus 0 plus 2,"},{"Start":"00:54.455 ","End":"00:56.510","Text":"which is just e squared."},{"Start":"00:56.510 ","End":"01:00.305","Text":"G of 0 is e^0,"},{"Start":"01:00.305 ","End":"01:02.345","Text":"which is equal to 1."},{"Start":"01:02.345 ","End":"01:08.320","Text":"Now, e squared is certainly bigger than 1 because e is bigger than 1,"},{"Start":"01:08.320 ","End":"01:09.650","Text":"and so it\u0027s squared,"},{"Start":"01:09.650 ","End":"01:11.255","Text":"it has to be bigger than 1."},{"Start":"01:11.255 ","End":"01:15.395","Text":"On the calculator, this is roughly 8 or something like that."},{"Start":"01:15.395 ","End":"01:19.850","Text":"Anyway, this will be e squared and this will be 1,"},{"Start":"01:19.850 ","End":"01:25.070","Text":"which also shows us that this one is the function f,"},{"Start":"01:25.070 ","End":"01:28.790","Text":"because that\u0027s the one that gave us the e squared,"},{"Start":"01:28.790 ","End":"01:31.220","Text":"and this one has to be"},{"Start":"01:31.220 ","End":"01:37.985","Text":"therefore g because that\u0027s the one that gave us 1 when we substituted x equals 0."},{"Start":"01:37.985 ","End":"01:39.935","Text":"Now we know which is which."},{"Start":"01:39.935 ","End":"01:44.900","Text":"To be pedantic, I have to say that for part a,"},{"Start":"01:44.900 ","End":"01:53.600","Text":"I should really say that the intersection points are 0,1 for g,"},{"Start":"01:53.600 ","End":"02:03.099","Text":"and 0,e squared for f. That would be a full answer."},{"Start":"02:03.470 ","End":"02:13.710","Text":"Onto part b, we have to just find the intersection of the 2 functions."},{"Start":"02:13.710 ","End":"02:19.740","Text":"We just equate e to the minus x plus 2 equals e^x."},{"Start":"02:19.740 ","End":"02:24.545","Text":"Here we are, e to the minus x plus 2 equals e^x."},{"Start":"02:24.545 ","End":"02:27.334","Text":"Now if the exponents are equal,"},{"Start":"02:27.334 ","End":"02:29.470","Text":"the numbers are equal too."},{"Start":"02:29.470 ","End":"02:32.790","Text":"Minus x plus 2 equals x,"},{"Start":"02:32.790 ","End":"02:36.960","Text":"and so x equals 1 if you figured it out,"},{"Start":"02:36.960 ","End":"02:38.925","Text":"because 2x equals 2."},{"Start":"02:38.925 ","End":"02:40.995","Text":"X equals 1."},{"Start":"02:40.995 ","End":"02:43.130","Text":"Here, x is 1,"},{"Start":"02:43.130 ","End":"02:49.009","Text":"but we were actually asked for the intersection point so when x is 1,"},{"Start":"02:49.009 ","End":"02:51.515","Text":"y is equal to,"},{"Start":"02:51.515 ","End":"02:58.030","Text":"you can substitute in either you get e^1 is e. If we substitute here,"},{"Start":"02:58.030 ","End":"03:01.385","Text":"you also get e. Actually, for part b,"},{"Start":"03:01.385 ","End":"03:04.760","Text":"the intersection point is 1,e,"},{"Start":"03:04.760 ","End":"03:08.815","Text":"and that\u0027s what I\u0027m going to highlight."},{"Start":"03:08.815 ","End":"03:15.560","Text":"Now on to part c. I have to say that I copied this question,"},{"Start":"03:15.560 ","End":"03:18.070","Text":"and it seems a bit ambiguous to me."},{"Start":"03:18.070 ","End":"03:23.525","Text":"S_1, does it include S_2 because the lines certainly continue?"},{"Start":"03:23.525 ","End":"03:25.130","Text":"I\u0027m not sure."},{"Start":"03:25.130 ","End":"03:28.565","Text":"I just have to make a judgment call here."},{"Start":"03:28.565 ","End":"03:31.780","Text":"Let\u0027s say that S_1 is just the top bit."},{"Start":"03:31.780 ","End":"03:38.620","Text":"I\u0027ll highlight the border and we\u0027ll let S_2 be a different color."},{"Start":"03:39.890 ","End":"03:46.100","Text":"What we need now is to do some integration."},{"Start":"03:46.100 ","End":"03:50.300","Text":"Let\u0027s find what S_1 is and let\u0027s find what S_2 is."},{"Start":"03:50.300 ","End":"03:52.865","Text":"I\u0027ll start with the easier one."},{"Start":"03:52.865 ","End":"03:57.080","Text":"Seems to me that S_2 is easier because it\u0027s the area below a curve,"},{"Start":"03:57.080 ","End":"03:59.390","Text":"so I don\u0027t need any subtractions."},{"Start":"03:59.390 ","End":"04:02.495","Text":"For part c, I just need the integral."},{"Start":"04:02.495 ","End":"04:10.925","Text":"This is obviously 0, from 0 to 1 of g. Let me write that, c is,"},{"Start":"04:10.925 ","End":"04:17.990","Text":"I need the integral from 0 to 1 of g of x dx,"},{"Start":"04:17.990 ","End":"04:22.660","Text":"and g is e^x,"},{"Start":"04:22.660 ","End":"04:31.350","Text":"so it\u0027s just the integral from 0 to 1 of e^x dx."},{"Start":"04:31.400 ","End":"04:38.535","Text":"This equals, the integral of e^x is just e^x itself between 0 and 1,"},{"Start":"04:38.535 ","End":"04:41.520","Text":"and this is equal to, when x is 1,"},{"Start":"04:41.520 ","End":"04:48.135","Text":"it\u0027s just e^1 is e. E^0 is 1, e minus 1."},{"Start":"04:48.135 ","End":"04:51.285","Text":"We have here S_2."},{"Start":"04:51.285 ","End":"04:53.985","Text":"Let me just separate here."},{"Start":"04:53.985 ","End":"04:58.065","Text":"I think I\u0027ll do S_1 on the same page."},{"Start":"04:58.065 ","End":"05:08.000","Text":"S_1 is going to be the integral from 0 to 1 of the difference."},{"Start":"05:08.000 ","End":"05:10.595","Text":"I\u0027ll just show you. Take f of x,"},{"Start":"05:10.595 ","End":"05:13.430","Text":"subtract g of x. I can read it off here."},{"Start":"05:13.430 ","End":"05:20.375","Text":"This is f and this is g. We get the integral of f of x,"},{"Start":"05:20.375 ","End":"05:28.255","Text":"that\u0027s e to the minus x plus 2 minus g, which is e^x."},{"Start":"05:28.255 ","End":"05:29.835","Text":"This is the integral I want,"},{"Start":"05:29.835 ","End":"05:32.260","Text":"from 0 to 1."},{"Start":"05:34.540 ","End":"05:40.280","Text":"I should have said this is equal to f minus g. Just to be consistent,"},{"Start":"05:40.280 ","End":"05:43.100","Text":"f of x minus g of x,"},{"Start":"05:43.100 ","End":"05:44.990","Text":"which is equal to,"},{"Start":"05:44.990 ","End":"05:50.195","Text":"the integral of this is e to the minus x plus 2,"},{"Start":"05:50.195 ","End":"05:53.825","Text":"divided by the inner derivative."},{"Start":"05:53.825 ","End":"05:55.615","Text":"Because it\u0027s a linear function,"},{"Start":"05:55.615 ","End":"05:59.055","Text":"it can do that, divided by minus 1."},{"Start":"05:59.055 ","End":"06:02.270","Text":"Less e^x doesn\u0027t need to be divided by anything."},{"Start":"06:02.270 ","End":"06:04.700","Text":"The integral of e^x is just e^x."},{"Start":"06:04.700 ","End":"06:08.075","Text":"Take this between 0 and 1."},{"Start":"06:08.075 ","End":"06:12.080","Text":"I get this equal altogether."},{"Start":"06:12.080 ","End":"06:14.435","Text":"If I just do the computation,"},{"Start":"06:14.435 ","End":"06:22.835","Text":"it is plus e squared minus 2e plus 1,"},{"Start":"06:22.835 ","End":"06:27.945","Text":"and this is S_1."},{"Start":"06:27.945 ","End":"06:31.755","Text":"Now I have S_1 and I have S_2."},{"Start":"06:31.755 ","End":"06:37.814","Text":"S2 is this, and S1 is this."},{"Start":"06:37.814 ","End":"06:45.915","Text":"What I need is S_1/S_2, the ratio."},{"Start":"06:45.915 ","End":"06:48.730","Text":"Let\u0027s just compute that."},{"Start":"06:50.060 ","End":"06:58.205","Text":"S_1/S_2 is e squared minus 2e plus 1 over e minus 1."},{"Start":"06:58.205 ","End":"07:03.485","Text":"Notice that this is just e minus 1 squared."},{"Start":"07:03.485 ","End":"07:05.735","Text":"Multiply this out and you\u0027ll get this,"},{"Start":"07:05.735 ","End":"07:07.760","Text":"over e minus 1."},{"Start":"07:07.760 ","End":"07:11.780","Text":"I can divide top and bottom by e minus 1,"},{"Start":"07:11.780 ","End":"07:15.320","Text":"and this is just equal to e minus 1."},{"Start":"07:15.320 ","End":"07:19.740","Text":"We\u0027ve answered all 3 parts. That\u0027s it."}],"Thumbnail":null,"ID":4703},{"Watched":false,"Name":"Exercise 8","Duration":"14m 30s","ChapterTopicVideoID":4696,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.825","Text":"Here, we\u0027re given a function which is f of x or y equals e to the minus 2x."},{"Start":"00:06.825 ","End":"00:09.075","Text":"I\u0027ll just label it here."},{"Start":"00:09.075 ","End":"00:14.145","Text":"Y equals e to the minus 2x."},{"Start":"00:14.145 ","End":"00:19.650","Text":"We have a tangent at the point x equals minus 1, somewhere here."},{"Start":"00:19.650 ","End":"00:21.165","Text":"It\u0027s hard to see exactly."},{"Start":"00:21.165 ","End":"00:26.205","Text":"This is where x is equal to minus 1."},{"Start":"00:26.205 ","End":"00:28.470","Text":"There\u0027s the tangent."},{"Start":"00:28.470 ","End":"00:31.155","Text":"Label it tangent."},{"Start":"00:31.155 ","End":"00:33.360","Text":"But we don\u0027t know its equation."},{"Start":"00:33.360 ","End":"00:34.880","Text":"That brings us to part A,"},{"Start":"00:34.880 ","End":"00:38.000","Text":"which is to find the equation of the tangent."},{"Start":"00:38.000 ","End":"00:39.665","Text":"Then when we found the tangent,"},{"Start":"00:39.665 ","End":"00:41.570","Text":"then that will help us in part B,"},{"Start":"00:41.570 ","End":"00:43.670","Text":"to compute the shaded area."},{"Start":"00:43.670 ","End":"00:45.130","Text":"Like it says here,"},{"Start":"00:45.130 ","End":"00:48.770","Text":"it\u0027s between the curve e to the minus 2x,"},{"Start":"00:48.770 ","End":"00:53.390","Text":"the x-axis, the y-axis, and the tangent."},{"Start":"00:53.390 ","End":"00:57.130","Text":"Let\u0027s get started with part A."},{"Start":"00:57.130 ","End":"01:03.170","Text":"In part A, there\u0027s a standard formula for the equation of a tangent."},{"Start":"01:03.170 ","End":"01:06.890","Text":"In general, let me just write the general formula."},{"Start":"01:06.890 ","End":"01:17.675","Text":"It\u0027s y minus y_1 is equal to f prime of x_1 times x minus x_1,"},{"Start":"01:17.675 ","End":"01:25.010","Text":"where x_1y_1 is the point on the curve where we draw the tangent and f prime,"},{"Start":"01:25.010 ","End":"01:26.900","Text":"of course, is the derivative at that point."},{"Start":"01:26.900 ","End":"01:33.460","Text":"In our case, we have that x_1 is equal to minus 1."},{"Start":"01:33.460 ","End":"01:42.090","Text":"We\u0027re still missing the y_1 and the f prime of x_1. Let\u0027s see."},{"Start":"01:42.090 ","End":"01:44.250","Text":"What we need is,"},{"Start":"01:44.250 ","End":"01:45.750","Text":"first of all, y_1,"},{"Start":"01:45.750 ","End":"01:48.585","Text":"which is f of x_1."},{"Start":"01:48.585 ","End":"01:58.700","Text":"F of minus 1 is equal to e to the minus 2 times minus 1,"},{"Start":"01:58.700 ","End":"02:02.010","Text":"which is just e squared."},{"Start":"02:02.010 ","End":"02:08.220","Text":"Back to there, I have that y_1 is e squared."},{"Start":"02:08.220 ","End":"02:11.730","Text":"For this, I need in general the derivative."},{"Start":"02:11.730 ","End":"02:16.790","Text":"The general derivative for f prime of x is the derivative of this,"},{"Start":"02:16.790 ","End":"02:22.290","Text":"which is going to be minus 2e to the minus 2x."},{"Start":"02:22.310 ","End":"02:26.850","Text":"F prime of minus 1,"},{"Start":"02:26.850 ","End":"02:28.335","Text":"which is our x_1,"},{"Start":"02:28.335 ","End":"02:30.675","Text":"will be minus 2."},{"Start":"02:30.675 ","End":"02:32.865","Text":"This bit will come out the same,"},{"Start":"02:32.865 ","End":"02:36.360","Text":"e to the minus 2 times minus 1 is plus 2."},{"Start":"02:36.360 ","End":"02:39.890","Text":"I also have this bit,"},{"Start":"02:39.890 ","End":"02:44.360","Text":"which is minus 2e squared."},{"Start":"02:44.360 ","End":"02:46.220","Text":"Now that I have these 3 things,"},{"Start":"02:46.220 ","End":"02:47.900","Text":"I can plug them in here,"},{"Start":"02:47.900 ","End":"02:54.660","Text":"and I can say that the tangent is and I\u0027m looking at this formula here,"},{"Start":"02:54.660 ","End":"03:00.180","Text":"y minus y_1 is e squared,"},{"Start":"03:00.180 ","End":"03:04.590","Text":"equals f prime of x_1, which is this."},{"Start":"03:04.590 ","End":"03:06.345","Text":"This is my x_1,"},{"Start":"03:06.345 ","End":"03:11.355","Text":"is minus 2e squared. That\u0027s this bit."},{"Start":"03:11.355 ","End":"03:17.055","Text":"Then x minus x_1 is minus 1."},{"Start":"03:17.055 ","End":"03:20.280","Text":"Let me just simplify that a bit."},{"Start":"03:20.280 ","End":"03:27.605","Text":"Let\u0027s see, we get that y equals and I\u0027m going to bring this over to the other side,"},{"Start":"03:27.605 ","End":"03:28.730","Text":"but first of all, let\u0027s see."},{"Start":"03:28.730 ","End":"03:34.070","Text":"The coefficient of x is going to be minus 2e squared x."},{"Start":"03:34.070 ","End":"03:37.655","Text":"That it\u0027s negative is good because we can see it\u0027s decreasing."},{"Start":"03:37.655 ","End":"03:42.560","Text":"Then the 3 coefficient here is going to be, this is minus,"},{"Start":"03:42.560 ","End":"03:44.390","Text":"minus 1 is plus 1,"},{"Start":"03:44.390 ","End":"03:47.440","Text":"so it\u0027s minus 2e squared,"},{"Start":"03:47.440 ","End":"03:49.650","Text":"but plus the e squared."},{"Start":"03:49.650 ","End":"03:53.070","Text":"It\u0027s just minus e squared."},{"Start":"03:53.070 ","End":"04:00.390","Text":"This is the equation of the tangent and I\u0027ll just highlight that."},{"Start":"04:00.390 ","End":"04:07.590","Text":"That\u0027s the answer to part A. I\u0027d like to just write that in the diagram as well."},{"Start":"04:07.590 ","End":"04:09.499","Text":"We\u0027re talking about the tangent."},{"Start":"04:09.499 ","End":"04:11.270","Text":"I don\u0027t need this bit here."},{"Start":"04:11.270 ","End":"04:12.680","Text":"I\u0027m about to write the tangent."},{"Start":"04:12.680 ","End":"04:14.450","Text":"You know what? I\u0027m going to clean up a bit."},{"Start":"04:14.450 ","End":"04:17.880","Text":"I don\u0027t need all of this stuff either anymore."},{"Start":"04:17.960 ","End":"04:24.125","Text":"Let me just write that the tangent, which is this,"},{"Start":"04:24.125 ","End":"04:31.560","Text":"is y equals minus 2e squared,"},{"Start":"04:31.560 ","End":"04:35.835","Text":"x minus e squared."},{"Start":"04:35.835 ","End":"04:39.795","Text":"Before I properly start part B,"},{"Start":"04:39.795 ","End":"04:43.735","Text":"I\u0027d like to compute this point here."},{"Start":"04:43.735 ","End":"04:49.710","Text":"This point is where the tangent hits the x-axis."},{"Start":"04:49.710 ","End":"04:55.130","Text":"All I have to do is let y equals 0 here."},{"Start":"04:55.130 ","End":"04:56.720","Text":"Let\u0027s solve that."},{"Start":"04:56.720 ","End":"04:58.010","Text":"To find this point,"},{"Start":"04:58.010 ","End":"05:00.740","Text":"we could let y equals 0,"},{"Start":"05:00.740 ","End":"05:07.010","Text":"so 0 equals minus 2e squared x minus e squared."},{"Start":"05:07.010 ","End":"05:13.205","Text":"If I bring this to the other side and then divide by 2e squared,"},{"Start":"05:13.205 ","End":"05:20.945","Text":"I\u0027ll get that x equals minus e squared over 2e squared,"},{"Start":"05:20.945 ","End":"05:24.140","Text":"which is just minus a 1/2."},{"Start":"05:24.140 ","End":"05:28.550","Text":"I can label this point now as x."},{"Start":"05:28.550 ","End":"05:34.000","Text":"This is where x equals minus 1/2 and this of course is 0."},{"Start":"05:34.000 ","End":"05:36.905","Text":"Now, let\u0027s get onto part B."},{"Start":"05:36.905 ","End":"05:38.795","Text":"Before I jump into it,"},{"Start":"05:38.795 ","End":"05:41.810","Text":"I just want to share my thoughts with you that there are"},{"Start":"05:41.810 ","End":"05:46.190","Text":"actually several ways we could compute this shaded area."},{"Start":"05:46.190 ","End":"05:51.615","Text":"I\u0027m going to just outline it from here up to here."},{"Start":"05:51.615 ","End":"05:57.220","Text":"Then down here, here, and here."},{"Start":"05:57.220 ","End":"06:00.200","Text":"There are many ways that I could break this up."},{"Start":"06:00.200 ","End":"06:03.140","Text":"I can\u0027t do this in one single integral,"},{"Start":"06:03.140 ","End":"06:05.555","Text":"but here are some ideas."},{"Start":"06:05.555 ","End":"06:13.820","Text":"1 possibility is to draw a vertical line here and then to break it up into 2 bits,"},{"Start":"06:13.820 ","End":"06:15.170","Text":"this bit and this bit."},{"Start":"06:15.170 ","End":"06:19.545","Text":"Here, I have the integral of just the curve."},{"Start":"06:19.545 ","End":"06:22.870","Text":"From here to here, the integral of the curve minus the line."},{"Start":"06:22.870 ","End":"06:25.370","Text":"That\u0027s certainly feasible."},{"Start":"06:25.370 ","End":"06:33.020","Text":"The other thing we could do is to take the area of this whole thing,"},{"Start":"06:33.020 ","End":"06:38.510","Text":"including this, and then subtract the area of this triangle."},{"Start":"06:38.510 ","End":"06:45.695","Text":"There\u0027s even a third possibility and that possibility would be to continue this line."},{"Start":"06:45.695 ","End":"06:49.969","Text":"I just extended this tangent till it hits the y-axis."},{"Start":"06:49.969 ","End":"06:51.965","Text":"I extended the y-axis."},{"Start":"06:51.965 ","End":"06:55.430","Text":"The third approach, which is not the most"},{"Start":"06:55.430 ","End":"06:59.599","Text":"obvious but it\u0027s a challenge and I think we should go with it,"},{"Start":"06:59.599 ","End":"07:03.545","Text":"is to compute the whole area from here,"},{"Start":"07:03.545 ","End":"07:06.020","Text":"down to here, and to here."},{"Start":"07:06.020 ","End":"07:09.230","Text":"In other words, including this triangle."},{"Start":"07:09.230 ","End":"07:12.950","Text":"Then to subtract the triangle, once again,"},{"Start":"07:12.950 ","End":"07:16.730","Text":"I\u0027m taking the whole outer contour,"},{"Start":"07:16.730 ","End":"07:19.249","Text":"which will be the upper function,"},{"Start":"07:19.249 ","End":"07:23.900","Text":"which is our curve minus the tangent line from minus 1 to 0,"},{"Start":"07:23.900 ","End":"07:26.540","Text":"and then I\u0027ll compute the triangle separately,"},{"Start":"07:26.540 ","End":"07:27.650","Text":"not even with an integral,"},{"Start":"07:27.650 ","End":"07:30.725","Text":"just with geometry, and we\u0027ll go with that approach."},{"Start":"07:30.725 ","End":"07:34.820","Text":"But I showed you that there are at least 3 things,"},{"Start":"07:34.820 ","End":"07:37.475","Text":"3 ways that come to mind of doing it,"},{"Start":"07:37.475 ","End":"07:41.855","Text":"and there is usually more than 1 way and you can be creative."},{"Start":"07:41.855 ","End":"07:45.245","Text":"This is the approach we are going with."},{"Start":"07:45.245 ","End":"07:52.235","Text":"First of all, let\u0027s compute the area of the whole thing, including this."},{"Start":"07:52.235 ","End":"07:55.640","Text":"Just to emphasize, I\u0027m going to actually do"},{"Start":"07:55.640 ","End":"07:59.150","Text":"some shading to show you that it\u0027s this minus this."},{"Start":"07:59.150 ","End":"08:04.595","Text":"It\u0027s as if we are taking a vertical shading like this,"},{"Start":"08:04.595 ","End":"08:08.645","Text":"of this whole thing up to here."},{"Start":"08:08.645 ","End":"08:11.420","Text":"You can see this way that we\u0027re taking"},{"Start":"08:11.420 ","End":"08:15.665","Text":"the difference of the upper minus the lower in vertical strips."},{"Start":"08:15.665 ","End":"08:23.705","Text":"What we get is the integral from minus 1 to 0,"},{"Start":"08:23.705 ","End":"08:27.919","Text":"of I need the upper minus the lower."},{"Start":"08:27.919 ","End":"08:34.100","Text":"The upper is just our function e to the minus 2x."},{"Start":"08:34.100 ","End":"08:37.700","Text":"Then we have to subtract this 1 which is the tangent."},{"Start":"08:37.700 ","End":"08:45.185","Text":"E to the minus 2x minus tangent line,"},{"Start":"08:45.185 ","End":"08:55.385","Text":"which is minus 2e squared x minus e squared of the brackets here, dx."},{"Start":"08:55.385 ","End":"08:57.290","Text":"That will give us the outer thing."},{"Start":"08:57.290 ","End":"09:00.900","Text":"Then we have to subtract the triangle at the end."},{"Start":"09:01.990 ","End":"09:04.760","Text":"Now we don\u0027t need the diagram anymore,"},{"Start":"09:04.760 ","End":"09:06.799","Text":"it\u0027s just an integration problem."},{"Start":"09:06.799 ","End":"09:12.830","Text":"We have this is equal to the integral from minus 1 to 0."},{"Start":"09:12.830 ","End":"09:16.070","Text":"I will simplify before I do the integration."},{"Start":"09:16.070 ","End":"09:20.150","Text":"I copied something wrong that\u0027s supposed to be an x."},{"Start":"09:20.150 ","End":"09:23.060","Text":"Let\u0027s take the x term first."},{"Start":"09:23.060 ","End":"09:29.195","Text":"I have minus minus is plus I have 2e squared x,"},{"Start":"09:29.195 ","End":"09:35.630","Text":"and then I have plus e squared and I have,"},{"Start":"09:35.630 ","End":"09:36.890","Text":"I think this is plus, yeah,"},{"Start":"09:36.890 ","End":"09:42.900","Text":"plus e to the minus 2x dx."},{"Start":"09:45.760 ","End":"09:53.170","Text":"This is just a constant times x. I raise the power by 1,"},{"Start":"09:53.170 ","End":"09:55.195","Text":"it\u0027s x squared over 2."},{"Start":"09:55.195 ","End":"09:59.745","Text":"You can see that I end up with just e squared x squared."},{"Start":"09:59.745 ","End":"10:02.300","Text":"Should be twice this and over 2,"},{"Start":"10:02.300 ","End":"10:03.800","Text":"but the 2s cancel."},{"Start":"10:03.800 ","End":"10:06.050","Text":"The next 1 gives me,"},{"Start":"10:06.050 ","End":"10:09.605","Text":"it\u0027s a constant, so it\u0027s e squared x."},{"Start":"10:09.605 ","End":"10:15.530","Text":"This 1, the integral of e to the minus 2x,"},{"Start":"10:15.530 ","End":"10:18.530","Text":"if it was just e to the x, e to the something,"},{"Start":"10:18.530 ","End":"10:21.740","Text":"you\u0027d start off with just e to that same something."},{"Start":"10:21.740 ","End":"10:23.870","Text":"Because it\u0027s a linear function of x."},{"Start":"10:23.870 ","End":"10:26.255","Text":"We divide by the coefficient of x."},{"Start":"10:26.255 ","End":"10:29.615","Text":"So this would be over minus 2."},{"Start":"10:29.615 ","End":"10:34.400","Text":"Now the 0 have to put in the limits of integration,"},{"Start":"10:34.400 ","End":"10:37.130","Text":"so here the minus 1 and 0,"},{"Start":"10:37.130 ","End":"10:39.140","Text":"which means that we have to substitute 0,"},{"Start":"10:39.140 ","End":"10:41.630","Text":"substitute minus 1 and subtract."},{"Start":"10:41.630 ","End":"10:45.035","Text":"Let\u0027s see. If we put in 0,"},{"Start":"10:45.035 ","End":"10:48.365","Text":"we get this is 0 and this is 0."},{"Start":"10:48.365 ","End":"10:54.065","Text":"We\u0027re just up to each of the 0 over minus 2,"},{"Start":"10:54.065 ","End":"10:58.445","Text":"e to the 0 is 1, so it\u0027s just minus a 1/2."},{"Start":"10:58.445 ","End":"11:01.190","Text":"That takes care of the 0 part."},{"Start":"11:01.190 ","End":"11:05.705","Text":"Now after subtract the part with the minus 1."},{"Start":"11:05.705 ","End":"11:07.940","Text":"The minus 1 squared is 1,"},{"Start":"11:07.940 ","End":"11:09.935","Text":"so that\u0027s e squared."},{"Start":"11:09.935 ","End":"11:12.260","Text":"Minus 1 is minus 1,"},{"Start":"11:12.260 ","End":"11:14.509","Text":"so that\u0027s minus e squared."},{"Start":"11:14.509 ","End":"11:17.884","Text":"Then minus 2x is plus 2,"},{"Start":"11:17.884 ","End":"11:22.685","Text":"so this is e squared over minus 2,"},{"Start":"11:22.685 ","End":"11:27.215","Text":"which is minus 1/2 e squared."},{"Start":"11:27.215 ","End":"11:29.810","Text":"What does this leave us with?"},{"Start":"11:29.810 ","End":"11:35.270","Text":"It leaves us, that this cancels with this,"},{"Start":"11:35.270 ","End":"11:39.350","Text":"and the minus and the minus will make it come out a plus."},{"Start":"11:39.350 ","End":"11:42.755","Text":"This minus with this minus will make it plus."},{"Start":"11:42.755 ","End":"11:51.965","Text":"What we get is 1/2 e squared minus 1/2."},{"Start":"11:51.965 ","End":"11:55.460","Text":"That\u0027s an intermediate result."},{"Start":"11:55.460 ","End":"11:57.830","Text":"Now I need the other bit,"},{"Start":"11:57.830 ","End":"12:01.700","Text":"which is the triangle bit. Let\u0027s see."},{"Start":"12:01.700 ","End":"12:04.770","Text":"Yes, the green triangle."},{"Start":"12:05.950 ","End":"12:10.685","Text":"This is the base,"},{"Start":"12:10.685 ","End":"12:14.105","Text":"this would be the height,"},{"Start":"12:14.105 ","End":"12:16.250","Text":"and I\u0027m going to need 1/2 base times height."},{"Start":"12:16.250 ","End":"12:20.330","Text":"Now, the base from here is equal"},{"Start":"12:20.330 ","End":"12:27.200","Text":"to 1/2 because the distance from minus 1/2 to 0 is 1/2. The height."},{"Start":"12:27.200 ","End":"12:29.540","Text":"That\u0027s a good question."},{"Start":"12:29.540 ","End":"12:31.610","Text":"What is the height?"},{"Start":"12:31.610 ","End":"12:34.670","Text":"That\u0027s another side exercise."},{"Start":"12:34.670 ","End":"12:37.340","Text":"I\u0027m going to continue working over here,"},{"Start":"12:37.340 ","End":"12:38.990","Text":"so it\u0027ll be close to the picture."},{"Start":"12:38.990 ","End":"12:43.340","Text":"I\u0027ll just put a dividing line here. The height."},{"Start":"12:43.340 ","End":"12:48.440","Text":"What I have to do is find the intersection of the tangent with the y-axis."},{"Start":"12:48.440 ","End":"12:51.604","Text":"The y-axis is where x equals 0."},{"Start":"12:51.604 ","End":"12:55.190","Text":"We can do it in our heads really,"},{"Start":"12:55.190 ","End":"12:58.895","Text":"because if I take this tangent and put x equals 0,"},{"Start":"12:58.895 ","End":"13:03.300","Text":"I get that y equals minus e squared."},{"Start":"13:07.120 ","End":"13:13.250","Text":"Here this height is plus e squared,"},{"Start":"13:13.250 ","End":"13:16.295","Text":"I\u0027m sorry, lengths are always positive."},{"Start":"13:16.295 ","End":"13:23.210","Text":"The area is equal to 1/2 times base times height,"},{"Start":"13:23.210 ","End":"13:28.475","Text":"which is times this 1/2 times e squared."},{"Start":"13:28.475 ","End":"13:36.785","Text":"Altogether, this is equal to 1/4 e squared."},{"Start":"13:36.785 ","End":"13:42.380","Text":"Let\u0027s see what we had for the whole thing, including the triangle."},{"Start":"13:42.380 ","End":"13:45.575","Text":"What I have to do is take this and subtract this."},{"Start":"13:45.575 ","End":"13:53.480","Text":"The answer will be this thing less this 1/2 e"},{"Start":"13:53.480 ","End":"14:02.675","Text":"squared minus 1/2 minus 1/4 e squared."},{"Start":"14:02.675 ","End":"14:08.330","Text":"It\u0027s 1/4 e squared minus 1/2."},{"Start":"14:08.330 ","End":"14:11.089","Text":"For those who like decimals,"},{"Start":"14:11.089 ","End":"14:15.755","Text":"this is approximately equal to 1.347."},{"Start":"14:15.755 ","End":"14:19.550","Text":"But I\u0027m going to leave the answer like this."},{"Start":"14:19.550 ","End":"14:27.560","Text":"I\u0027ve now got the answer to part b here and to part a over here,"},{"Start":"14:27.560 ","End":"14:31.050","Text":"and so we are done."}],"Thumbnail":null,"ID":4704},{"Watched":false,"Name":"Exercise 9","Duration":"6m 45s","ChapterTopicVideoID":4697,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.429","Text":"In this exercise we\u0027re given a function,"},{"Start":"00:02.429 ","End":"00:06.450","Text":"y equals the cosine of 2x."},{"Start":"00:06.450 ","End":"00:08.760","Text":"That\u0027s the curve here."},{"Start":"00:08.760 ","End":"00:11.625","Text":"Let me just write it that here,"},{"Start":"00:11.625 ","End":"00:16.365","Text":"y equals cosine 2x."},{"Start":"00:16.365 ","End":"00:23.240","Text":"The domain is from 0 to pi over 4."},{"Start":"00:23.240 ","End":"00:29.060","Text":"We need to check later that really at pi over 4 does give 0."},{"Start":"00:29.060 ","End":"00:30.800","Text":"We can do it in our heads."},{"Start":"00:30.800 ","End":"00:32.585","Text":"If x is pi over 4,"},{"Start":"00:32.585 ","End":"00:35.270","Text":"twice pi over 4 is pi over 2."},{"Start":"00:35.270 ","End":"00:40.310","Text":"Cosine pi over 2 is cosine 90 degrees, which is 0."},{"Start":"00:40.310 ","End":"00:46.760","Text":"We\u0027re okay that this is really the point pi over 4 comma 0. Just checking."},{"Start":"00:46.760 ","End":"00:51.449","Text":"A tangent is drawn at this point,"},{"Start":"00:51.449 ","End":"00:53.700","Text":"so this is the point pi over 4, 0."},{"Start":"00:53.700 ","End":"00:57.350","Text":"We got a tangent here and not"},{"Start":"00:57.350 ","End":"00:59.510","Text":"clear whether the continuation is part"},{"Start":"00:59.510 ","End":"01:02.015","Text":"of the tangent of the curve, and it doesn\u0027t matter."},{"Start":"01:02.015 ","End":"01:06.560","Text":"We have a tangent and we have to find its equation."},{"Start":"01:06.560 ","End":"01:12.680","Text":"Let me just for the moment call it tangent and that\u0027s what we have to find in part a,"},{"Start":"01:12.680 ","End":"01:20.045","Text":"and then in part b we have to compute the shaded area which is between the y-axis,"},{"Start":"01:20.045 ","End":"01:23.489","Text":"the curve, and the tangent."},{"Start":"01:23.890 ","End":"01:28.025","Text":"Let\u0027s get to part a first."},{"Start":"01:28.025 ","End":"01:34.020","Text":"There is a standard formula for the tangent to the curve at a point,"},{"Start":"01:34.020 ","End":"01:39.465","Text":"and the formula is that y minus y1"},{"Start":"01:39.465 ","End":"01:45.390","Text":"is f prime of x1 times x minus x1."},{"Start":"01:45.390 ","End":"01:50.820","Text":"Now, x1,y1 are the point at which we draw the tangent,"},{"Start":"01:50.820 ","End":"01:58.250","Text":"so in our case x1,y1 is the point pi over 4 comma 0."},{"Start":"01:58.250 ","End":"02:01.290","Text":"We already checked that y is 0."},{"Start":"02:01.660 ","End":"02:07.200","Text":"Well, this is my f of x, the cosine 2x."},{"Start":"02:07.200 ","End":"02:08.975","Text":"Let me just write it here."},{"Start":"02:08.975 ","End":"02:16.025","Text":"F of x is cosine 2x and so in general,"},{"Start":"02:16.025 ","End":"02:19.830","Text":"f prime of x will be equal to,"},{"Start":"02:19.830 ","End":"02:24.100","Text":"the derivative of cosine is minus sine,"},{"Start":"02:24.100 ","End":"02:28.290","Text":"but we also have an inner derivative which is 2."},{"Start":"02:28.290 ","End":"02:34.535","Text":"Now f prime of x1 is f prime of pi over 4,"},{"Start":"02:34.535 ","End":"02:40.725","Text":"and the sine of pi over 2 is 1."},{"Start":"02:40.725 ","End":"02:44.325","Text":"This is equal to minus 2."},{"Start":"02:44.325 ","End":"02:46.640","Text":"Now I have everything I need."},{"Start":"02:46.640 ","End":"02:57.170","Text":"I have x1,y1 and I have the f prime of x1 and so I can use this formula."},{"Start":"02:57.170 ","End":"02:59.885","Text":"I\u0027ll go back over here."},{"Start":"02:59.885 ","End":"03:08.615","Text":"I have that the tangent is given by y minus 0"},{"Start":"03:08.615 ","End":"03:18.610","Text":"equals minus 2 times x minus pi over 4."},{"Start":"03:18.610 ","End":"03:26.310","Text":"This is really just the equation of a line given a point in a slope."},{"Start":"03:26.310 ","End":"03:28.635","Text":"Let\u0027s see what we get,"},{"Start":"03:28.635 ","End":"03:31.350","Text":"y, if I bring everything, well,"},{"Start":"03:31.350 ","End":"03:40.185","Text":"the 0 is nothing and so y equals minus 2x plus pi over 2."},{"Start":"03:40.185 ","End":"03:50.665","Text":"This would be the answer to part a and I\u0027ll highlight it and now on to part b."},{"Start":"03:50.665 ","End":"03:52.830","Text":"Now as for the area,"},{"Start":"03:52.830 ","End":"03:54.740","Text":"which I\u0027m going to do as an integral,"},{"Start":"03:54.740 ","End":"03:57.770","Text":"I\u0027m going to take the integral from 0 to pi over"},{"Start":"03:57.770 ","End":"04:01.220","Text":"4 of the upper function minus the lower function."},{"Start":"04:01.220 ","End":"04:05.240","Text":"In other words, the tangent minus the given function."},{"Start":"04:05.240 ","End":"04:08.230","Text":"It\u0027s like vertical stripes."},{"Start":"04:08.230 ","End":"04:11.310","Text":"Upper minus the lower and so on,"},{"Start":"04:11.310 ","End":"04:14.660","Text":"here it\u0027s meaningless put so close together."},{"Start":"04:14.660 ","End":"04:22.890","Text":"In any event, we get the integral from 0 to pi over 4."},{"Start":"04:22.890 ","End":"04:29.755","Text":"The top one which is minus 2x plus pi over 2,"},{"Start":"04:29.755 ","End":"04:39.330","Text":"minus the bottom one which is cosine 2x and dx."},{"Start":"04:40.480 ","End":"04:43.204","Text":"Let\u0027s just do the integral."},{"Start":"04:43.204 ","End":"04:50.745","Text":"Minus 2x gives us x squared with a minus of course."},{"Start":"04:50.745 ","End":"04:52.670","Text":"Pi over 2 is a constant,"},{"Start":"04:52.670 ","End":"04:55.615","Text":"so it\u0027s pi over 2x,"},{"Start":"04:55.615 ","End":"05:04.125","Text":"and the integral of cosine 2x is almost sine 2x,"},{"Start":"05:04.125 ","End":"05:08.250","Text":"but we need to divide by 2 the inner"},{"Start":"05:08.250 ","End":"05:12.950","Text":"derivative when it\u0027s a linear function just divide by the coefficient of x,"},{"Start":"05:12.950 ","End":"05:14.510","Text":"so that\u0027s the 2 there."},{"Start":"05:14.510 ","End":"05:21.560","Text":"All this has to be taken between 0 and pi over 4."},{"Start":"05:21.560 ","End":"05:24.550","Text":"What we get is first of all,"},{"Start":"05:24.550 ","End":"05:28.030","Text":"let\u0027s put in the pi over 4,"},{"Start":"05:28.030 ","End":"05:37.710","Text":"so we get minus pi squared over 16 plus pi over"},{"Start":"05:37.710 ","End":"05:47.595","Text":"2 times pi over 4 is pi squared over 8 and minus a 1/2."},{"Start":"05:47.595 ","End":"05:53.380","Text":"Let\u0027s see, sine of 2x is sine of pi over 2, which is 1."},{"Start":"05:53.380 ","End":"05:55.670","Text":"It\u0027s just minus a 1/2."},{"Start":"05:55.670 ","End":"05:57.390","Text":"That\u0027s for the pi over 4."},{"Start":"05:57.390 ","End":"06:01.380","Text":"Now for the 0 we get, if x is 0,"},{"Start":"06:01.380 ","End":"06:06.020","Text":"we\u0027ve got minus 0 plus pi over 2 times"},{"Start":"06:06.020 ","End":"06:12.305","Text":"0 is 0 and minus sine of twice 0,"},{"Start":"06:12.305 ","End":"06:18.215","Text":"that\u0027s also a 0, so all we get is what\u0027s from here."},{"Start":"06:18.215 ","End":"06:21.680","Text":"Now, an 8th minus a 16th is a 16th,"},{"Start":"06:21.680 ","End":"06:29.800","Text":"so this is pi squared over 16 minus 1/2."},{"Start":"06:29.800 ","End":"06:35.845","Text":"This is roughly 0.117 for those who like numerical answers."},{"Start":"06:35.845 ","End":"06:39.920","Text":"I am going to just highlight this bit as the answer"},{"Start":"06:39.920 ","End":"06:44.015","Text":"to part b as the area of the shaded bit."},{"Start":"06:44.015 ","End":"06:46.800","Text":"We are done."}],"Thumbnail":null,"ID":4705},{"Watched":false,"Name":"Exercise 10","Duration":"3m 49s","ChapterTopicVideoID":4698,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.995","Text":"Here we have another area problem to be done with integration."},{"Start":"00:04.995 ","End":"00:16.085","Text":"This time we have the function y equals 1 over 2x minus 1, which is this."},{"Start":"00:16.085 ","End":"00:21.585","Text":"Then we have the horizontal line y equals 1."},{"Start":"00:21.585 ","End":"00:26.295","Text":"We have the vertical line x equals 3."},{"Start":"00:26.295 ","End":"00:28.890","Text":"Obviously the vertical 1 is x equals"},{"Start":"00:28.890 ","End":"00:32.550","Text":"something and the horizontal 1 is y equals something."},{"Start":"00:32.550 ","End":"00:37.310","Text":"We have to figure out the shaded area in the figure,"},{"Start":"00:37.310 ","End":"00:41.270","Text":"which means that we\u0027ll need to find this point."},{"Start":"00:41.270 ","End":"00:43.955","Text":"Everything else is known."},{"Start":"00:43.955 ","End":"00:48.905","Text":"We just need this point so that we can do an integral from here to here,"},{"Start":"00:48.905 ","End":"00:51.985","Text":"of the upper, minus the lower."},{"Start":"00:51.985 ","End":"00:57.920","Text":"This point here is just the intersection of y equals 1 with y equals this,"},{"Start":"00:57.920 ","End":"00:59.720","Text":"1 over 2x minus 1."},{"Start":"00:59.720 ","End":"01:04.910","Text":"If we want to find this point, I\u0027ll write it this way."},{"Start":"01:04.910 ","End":"01:09.920","Text":"We have a set of equations y equals 1 over 2x minus 1,"},{"Start":"01:09.920 ","End":"01:12.185","Text":"and y equals 1."},{"Start":"01:12.185 ","End":"01:14.915","Text":"Let\u0027s see if we can solve for x."},{"Start":"01:14.915 ","End":"01:17.945","Text":"The right-hand sides are equal."},{"Start":"01:17.945 ","End":"01:23.420","Text":"We get 1 over 2x minus 1,"},{"Start":"01:23.420 ","End":"01:26.915","Text":"is equal to 1."},{"Start":"01:26.915 ","End":"01:32.795","Text":"2x minus 1 is equal to 1."},{"Start":"01:32.795 ","End":"01:36.380","Text":"Just multiplying, and then 2x equals 2,"},{"Start":"01:36.380 ","End":"01:39.020","Text":"so x equals 1."},{"Start":"01:39.020 ","End":"01:45.310","Text":"Now I know that this is the point where x equals 1."},{"Start":"01:45.310 ","End":"01:54.975","Text":"The integral becomes the integral from 1 to 3 of the upper,"},{"Start":"01:54.975 ","End":"01:58.400","Text":"which is 1 minus the lower,"},{"Start":"01:58.400 ","End":"02:03.105","Text":"which is 1 over 2x minus 1dx."},{"Start":"02:03.105 ","End":"02:11.940","Text":"We get the integral of 1 is just x."},{"Start":"02:11.940 ","End":"02:16.494","Text":"The integral of 2x minus 1,"},{"Start":"02:16.494 ","End":"02:25.035","Text":"is going to be the natural log of absolute value of 2x minus 1."},{"Start":"02:25.035 ","End":"02:28.325","Text":"But this is a linear function of x."},{"Start":"02:28.325 ","End":"02:31.400","Text":"We have to divide by the inner derivative,"},{"Start":"02:31.400 ","End":"02:34.820","Text":"which is 2, so is a half here."},{"Start":"02:34.820 ","End":"02:40.365","Text":"All this between 1 and 3."},{"Start":"02:40.365 ","End":"02:44.110","Text":"Let\u0027s see, if we put in 3,"},{"Start":"02:44.110 ","End":"02:52.410","Text":"we\u0027ll get twice 3 minus 1 is 5."},{"Start":"02:52.410 ","End":"02:58.204","Text":"Minus a half natural log of 5."},{"Start":"02:58.204 ","End":"03:01.055","Text":"That\u0027s the top limit."},{"Start":"03:01.055 ","End":"03:02.975","Text":"The bottom limit is 1."},{"Start":"03:02.975 ","End":"03:05.945","Text":"We\u0027ve got 1 minus."},{"Start":"03:05.945 ","End":"03:11.265","Text":"Now 2x minus 1 is 1."},{"Start":"03:11.265 ","End":"03:14.175","Text":"Natural log of 1 is 0,"},{"Start":"03:14.175 ","End":"03:17.865","Text":"so this part is 0."},{"Start":"03:17.865 ","End":"03:22.800","Text":"What we get is 3 minus 1 is 2,"},{"Start":"03:22.800 ","End":"03:29.130","Text":"minus a half natural log of 5."},{"Start":"03:29.130 ","End":"03:31.430","Text":"For those who like numerical,"},{"Start":"03:31.430 ","End":"03:39.050","Text":"this is approximately equal to 1.1953."},{"Start":"03:39.050 ","End":"03:46.790","Text":"But I\u0027m going to just leave this as the answer because that\u0027s more exact."},{"Start":"03:46.790 ","End":"03:50.160","Text":"We\u0027re done."}],"Thumbnail":null,"ID":4706},{"Watched":false,"Name":"Exercise 11","Duration":"8m 9s","ChapterTopicVideoID":4699,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"Here we have another 1 of these word problems,"},{"Start":"00:02.370 ","End":"00:04.860","Text":"with areas and definite integrals."},{"Start":"00:04.860 ","End":"00:06.645","Text":"I\u0027ll explain what\u0027s going on."},{"Start":"00:06.645 ","End":"00:08.925","Text":"But given the graph of a function,"},{"Start":"00:08.925 ","End":"00:11.355","Text":"and that\u0027s this function here,"},{"Start":"00:11.355 ","End":"00:13.215","Text":"let\u0027s just write what it is."},{"Start":"00:13.215 ","End":"00:19.725","Text":"I\u0027ll use y instead of f of x. y equals e^2x minus e^x,"},{"Start":"00:19.725 ","End":"00:21.735","Text":"so that\u0027s that graph."},{"Start":"00:21.735 ","End":"00:25.535","Text":"Now, it has a minimum point."},{"Start":"00:25.535 ","End":"00:27.150","Text":"We\u0027re told that it does,"},{"Start":"00:27.150 ","End":"00:29.030","Text":"and the picture shows that it does,"},{"Start":"00:29.030 ","End":"00:30.410","Text":"and that\u0027s what we\u0027re talking about."},{"Start":"00:30.410 ","End":"00:32.870","Text":"This is the minimum point."},{"Start":"00:32.870 ","End":"00:37.070","Text":"In part A, we have to find the x-coordinate of this minimum point."},{"Start":"00:37.070 ","End":"00:39.550","Text":"So this is what we\u0027re going to look for."},{"Start":"00:39.550 ","End":"00:43.890","Text":"In part B, they\u0027re talking about dropping the perpendicular,"},{"Start":"00:43.890 ","End":"00:46.380","Text":"just means connecting this line here,"},{"Start":"00:46.380 ","End":"00:48.095","Text":"and then we get an area,"},{"Start":"00:48.095 ","End":"00:52.250","Text":"because here we take the line where x equals a."},{"Start":"00:52.250 ","End":"00:56.895","Text":"So we have now bounded on 4 sides, the x-axis,"},{"Start":"00:56.895 ","End":"00:59.700","Text":"this perpendicular, the function,"},{"Start":"00:59.700 ","End":"01:02.355","Text":"and the line x equals a."},{"Start":"01:02.355 ","End":"01:05.435","Text":"This line here is x equals a."},{"Start":"01:05.435 ","End":"01:13.960","Text":"We\u0027re given this area that it\u0027s equal to 3e^2a,"},{"Start":"01:13.960 ","End":"01:17.900","Text":"basically what it says here, minus e^a."},{"Start":"01:17.900 ","End":"01:19.610","Text":"I see it\u0027s hard to read."},{"Start":"01:19.610 ","End":"01:24.290","Text":"The final thing we\u0027re given also is an inequality,"},{"Start":"01:24.290 ","End":"01:27.315","Text":"a is less than natural logarithm of 1/2."},{"Start":"01:27.315 ","End":"01:30.040","Text":"I guess that\u0027s going to be used somewhere."},{"Start":"01:30.140 ","End":"01:33.325","Text":"Let\u0027s get started with part A."},{"Start":"01:33.325 ","End":"01:35.615","Text":"We have to find the minimum point."},{"Start":"01:35.615 ","End":"01:37.295","Text":"We know how to do minimum."},{"Start":"01:37.295 ","End":"01:42.720","Text":"We take f prime of x first of all,"},{"Start":"01:42.720 ","End":"01:48.495","Text":"and that\u0027s equal to 2e^2x minus e^x."},{"Start":"01:48.495 ","End":"01:51.150","Text":"For minimum, I need a critical point,"},{"Start":"01:51.150 ","End":"01:54.150","Text":"so f prime of x equals 0."},{"Start":"01:54.150 ","End":"01:57.200","Text":"That will give me suspects for the critical point."},{"Start":"01:57.200 ","End":"02:04.875","Text":"I get that 2e^2x minus e^x equals 0."},{"Start":"02:04.875 ","End":"02:08.630","Text":"Let\u0027s take e^x outside the brackets."},{"Start":"02:08.630 ","End":"02:13.245","Text":"You\u0027ve got e^x times 2."},{"Start":"02:13.245 ","End":"02:18.400","Text":"Now, this e^2x is just e^x times e^x."},{"Start":"02:18.400 ","End":"02:20.120","Text":"We\u0027ve done this kind of thing before,"},{"Start":"02:20.120 ","End":"02:21.545","Text":"I\u0027m not going to elaborate it."},{"Start":"02:21.545 ","End":"02:30.260","Text":"We\u0027re left with 2e^x minus 1 is equal to 0. e^x is never 0,"},{"Start":"02:30.260 ","End":"02:31.550","Text":"it\u0027s a positive function,"},{"Start":"02:31.550 ","End":"02:33.275","Text":"I can cross that off."},{"Start":"02:33.275 ","End":"02:35.090","Text":"From this thing equals 0,"},{"Start":"02:35.090 ","End":"02:39.025","Text":"I\u0027ll get e^x equals 1/2."},{"Start":"02:39.025 ","End":"02:44.710","Text":"I\u0027ll call that 0.5 because I see we\u0027re using decimals."},{"Start":"02:44.710 ","End":"02:49.490","Text":"If I take the natural logarithm of both sides,"},{"Start":"02:49.490 ","End":"02:57.460","Text":"then I\u0027ll get that x equals natural log of 0.5."},{"Start":"02:57.770 ","End":"03:00.720","Text":"That\u0027s the x for this point,"},{"Start":"03:00.720 ","End":"03:08.565","Text":"and now I can replace it with natural log of 0.5."},{"Start":"03:08.565 ","End":"03:10.190","Text":"Now I see what this means."},{"Start":"03:10.190 ","End":"03:11.900","Text":"If a is less than that,"},{"Start":"03:11.900 ","End":"03:14.980","Text":"that means that a is to the left of it."},{"Start":"03:14.980 ","End":"03:17.595","Text":"It makes sense like in the picture."},{"Start":"03:17.595 ","End":"03:21.840","Text":"Now we\u0027ll continue with part B."},{"Start":"03:21.840 ","End":"03:26.700","Text":"In part B, what we have to do is compute this integral,"},{"Start":"03:26.700 ","End":"03:28.910","Text":"and of course, it will be in terms of a,"},{"Start":"03:28.910 ","End":"03:31.270","Text":"the parameter which we don\u0027t know yet."},{"Start":"03:31.270 ","End":"03:33.945","Text":"The areas below the x-axis."},{"Start":"03:33.945 ","End":"03:37.760","Text":"The area\u0027s going to be minus the integral."},{"Start":"03:37.760 ","End":"03:42.410","Text":"I can say that if this is the area,"},{"Start":"03:42.410 ","End":"03:45.150","Text":"I\u0027ll write that down here,"},{"Start":"03:46.570 ","End":"03:52.960","Text":"3e^2a minus e^a is equal to minus the"},{"Start":"03:52.960 ","End":"04:00.455","Text":"integral from a to natural log of 0.5 of the function,"},{"Start":"04:00.455 ","End":"04:05.215","Text":"which is e^2x minus e^x."},{"Start":"04:05.215 ","End":"04:10.285","Text":"e^2x minus e^x, dx."},{"Start":"04:10.285 ","End":"04:12.265","Text":"Here\u0027s an equation."},{"Start":"04:12.265 ","End":"04:19.245","Text":"Let\u0027s solve the integral and then the equation for a. Let\u0027s see."},{"Start":"04:19.245 ","End":"04:25.990","Text":"This is equal to minus and I can do the integral."},{"Start":"04:25.990 ","End":"04:29.260","Text":"The integral of this is 1/2e^2x,"},{"Start":"04:29.490 ","End":"04:33.200","Text":"integral of this is e^x."},{"Start":"04:33.260 ","End":"04:39.050","Text":"All this is taken between the limits of this and this."},{"Start":"04:39.050 ","End":"04:43.490","Text":"But I\u0027m going to use the old trick that if you have a minus,"},{"Start":"04:43.490 ","End":"04:47.840","Text":"you can get rid of the minus and write the limits in the opposite order."},{"Start":"04:47.840 ","End":"04:52.205","Text":"I\u0027m going to take my eraser, erase the minus,"},{"Start":"04:52.205 ","End":"04:55.269","Text":"and just write the limits in reverse,"},{"Start":"04:55.269 ","End":"05:00.425","Text":"a at the top, and natural log of 0.5 at the bottom."},{"Start":"05:00.425 ","End":"05:02.840","Text":"Because when you do a subtraction in the reverse order,"},{"Start":"05:02.840 ","End":"05:05.210","Text":"It\u0027s like having a minus."},{"Start":"05:05.590 ","End":"05:11.495","Text":"Let\u0027s substitute the top 1 which is a."},{"Start":"05:11.495 ","End":"05:19.520","Text":"We have 1/2e^2a minus e^a."},{"Start":"05:19.520 ","End":"05:23.044","Text":"Now, I\u0027m going to substitute this natural log."},{"Start":"05:23.044 ","End":"05:30.380","Text":"Well, we know that e to the power of natural log of 0.5,"},{"Start":"05:30.380 ","End":"05:34.145","Text":"reading it from here is just 0.5."},{"Start":"05:34.145 ","End":"05:37.810","Text":"This bit is 0.5."},{"Start":"05:37.810 ","End":"05:41.265","Text":"Now this is e^x,"},{"Start":"05:41.265 ","End":"05:43.680","Text":"this piece is e^x squared,"},{"Start":"05:43.680 ","End":"05:50.595","Text":"so I can write 1/2 of 0.5 squared."},{"Start":"05:50.595 ","End":"05:53.440","Text":"What we have is the equation,"},{"Start":"05:53.440 ","End":"05:54.740","Text":"this is just a number."},{"Start":"05:54.740 ","End":"05:56.345","Text":"Let\u0027s see what it is."},{"Start":"05:56.345 ","End":"05:58.160","Text":"This is a half cubed,"},{"Start":"05:58.160 ","End":"06:05.580","Text":"it\u0027s 1/8.1/8 minus 1/2 is minus 3/8."},{"Start":"06:05.580 ","End":"06:07.230","Text":"So we\u0027re going to get minus, minus,"},{"Start":"06:07.230 ","End":"06:15.705","Text":"so we\u0027re going to get plus 0.375 for this bit here."},{"Start":"06:15.705 ","End":"06:17.910","Text":"Let\u0027s see now."},{"Start":"06:17.910 ","End":"06:23.880","Text":"I\u0027m going to bring everything to the left-hand side and collect like terms."},{"Start":"06:24.970 ","End":"06:31.850","Text":"3e^2a minus 1/2e^2a, is"},{"Start":"06:31.850 ","End":"06:42.390","Text":"2.5e^2a minus e^a plus e^a, so that cancels."},{"Start":"06:43.460 ","End":"06:51.720","Text":"I\u0027ll leave it on this side, equals 0.375."},{"Start":"06:51.720 ","End":"06:59.054","Text":"That gives me that e^2a is equal to this over this,"},{"Start":"06:59.054 ","End":"07:05.580","Text":"0.15, I make it."},{"Start":"07:05.580 ","End":"07:10.339","Text":"From here, I just have to take the natural logarithm of both sides."},{"Start":"07:10.339 ","End":"07:12.440","Text":"Natural logarithm of an exponent,"},{"Start":"07:12.440 ","End":"07:14.750","Text":"e to the power of, is just the thing itself."},{"Start":"07:14.750 ","End":"07:21.785","Text":"We have 2a is the natural log of 0.15."},{"Start":"07:21.785 ","End":"07:24.440","Text":"By the way, if you\u0027re doing it with fractions,"},{"Start":"07:24.440 ","End":"07:27.260","Text":"you would get 3/20."},{"Start":"07:27.260 ","End":"07:29.495","Text":"I just went with decimal."},{"Start":"07:29.495 ","End":"07:35.145","Text":"Then from here we get that a is 1/2."},{"Start":"07:35.145 ","End":"07:40.505","Text":"At this point, we either leave the answer like this or we take a calculator."},{"Start":"07:40.505 ","End":"07:45.450","Text":"I make it minus 0.95,"},{"Start":"07:46.760 ","End":"07:50.045","Text":"and this is the answer."},{"Start":"07:50.045 ","End":"07:52.160","Text":"So a is equal to this,"},{"Start":"07:52.160 ","End":"07:55.085","Text":"and that was the answer to part B."},{"Start":"07:55.085 ","End":"07:58.565","Text":"For part A, I didn\u0027t highlight it,"},{"Start":"07:58.565 ","End":"08:05.925","Text":"but the answer was natural log of 0.5."},{"Start":"08:05.925 ","End":"08:08.610","Text":"There, we answered both A and B,"},{"Start":"08:08.610 ","End":"08:10.630","Text":"and we are done."}],"Thumbnail":null,"ID":4707},{"Watched":false,"Name":"Exercise 12","Duration":"10m 25s","ChapterTopicVideoID":4700,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"Here we have another area problem with definite integrals."},{"Start":"00:04.530 ","End":"00:07.140","Text":"Let me explain what\u0027s going on here."},{"Start":"00:07.140 ","End":"00:12.405","Text":"We\u0027re given a function let\u0027s just label it f of x, which is y,"},{"Start":"00:12.405 ","End":"00:19.495","Text":"is equal to e to the of power x plus 1 over 2."},{"Start":"00:19.495 ","End":"00:22.040","Text":"There\u0027s a point a on it."},{"Start":"00:22.040 ","End":"00:28.814","Text":"It\u0027s not labeled so let\u0027s choose some point here and call that point a."},{"Start":"00:28.814 ","End":"00:33.795","Text":"Then we\u0027re talking about the tangent to the graph at point a."},{"Start":"00:33.795 ","End":"00:35.805","Text":"Let me sketch that."},{"Start":"00:35.805 ","End":"00:38.220","Text":"Here\u0027s the tangent."},{"Start":"00:38.220 ","End":"00:44.990","Text":"What we\u0027re told is that the slope of this tangent is e squared over 2."},{"Start":"00:44.990 ","End":"00:50.755","Text":"I\u0027ll write that slope equals e squared over 2."},{"Start":"00:50.755 ","End":"00:54.105","Text":"This is the tangent."},{"Start":"00:54.105 ","End":"00:58.010","Text":"Part A we have to find the coordinates of a,"},{"Start":"00:58.010 ","End":"01:01.310","Text":"and then we have to find the equation of the tangent."},{"Start":"01:01.310 ","End":"01:08.900","Text":"Finally, we have to compute the area which is bounded by the graph of the function,"},{"Start":"01:08.900 ","End":"01:11.840","Text":"the tangent and the y-axis."},{"Start":"01:11.840 ","End":"01:15.170","Text":"I\u0027ll shade that. There it is,"},{"Start":"01:15.170 ","End":"01:21.095","Text":"and I\u0027ll label it s. Let\u0027s begin with a, of course."},{"Start":"01:21.095 ","End":"01:27.905","Text":"Let\u0027s say that the coordinates of a, x_1,"},{"Start":"01:27.905 ","End":"01:37.390","Text":"y_1, then I\u0027ll just write that here A is the point x_1, y_1."},{"Start":"01:37.390 ","End":"01:40.490","Text":"We know that because it lies on f,"},{"Start":"01:40.490 ","End":"01:42.905","Text":"that the slope of the tangent,"},{"Start":"01:42.905 ","End":"01:48.845","Text":"what we\u0027re looking for is f prime of x_1."},{"Start":"01:48.845 ","End":"01:51.275","Text":"This happens to equal,"},{"Start":"01:51.275 ","End":"01:55.045","Text":"we\u0027re given is e squared over 2."},{"Start":"01:55.045 ","End":"01:57.705","Text":"We have to find f prime of x_1."},{"Start":"01:57.705 ","End":"02:04.800","Text":"Now, if f of x is e^x plus 1 over 2,"},{"Start":"02:04.800 ","End":"02:06.810","Text":"I\u0027m doing this as a side exercise,"},{"Start":"02:06.810 ","End":"02:08.600","Text":"and f prime of x,"},{"Start":"02:08.600 ","End":"02:10.445","Text":"the derivative of this,"},{"Start":"02:10.445 ","End":"02:18.335","Text":"is also starts out to be e^x plus 1 over 2 times the inner derivative of this,"},{"Start":"02:18.335 ","End":"02:20.920","Text":"which is just 1.5."},{"Start":"02:20.920 ","End":"02:27.900","Text":"I\u0027ve now got an equation that 1.5 of"},{"Start":"02:27.900 ","End":"02:37.400","Text":"e^x _1 plus 1 over 2 is equal to e squared over 2."},{"Start":"02:37.400 ","End":"02:43.475","Text":"Now, what happens here is that I can cancel the 1.5 here,"},{"Start":"02:43.475 ","End":"02:47.190","Text":"goes with the over 2 here."},{"Start":"02:47.480 ","End":"02:52.910","Text":"That means that we have e to the something equals e to the something."},{"Start":"02:52.910 ","End":"02:55.160","Text":"Those 2 some things are equal."},{"Start":"02:55.160 ","End":"03:01.285","Text":"I can get that x_1 plus 1 over 2 is equal to 2."},{"Start":"03:01.285 ","End":"03:05.420","Text":"Therefore, if I multiply by 2 and subtract 1,"},{"Start":"03:05.420 ","End":"03:09.175","Text":"I get that x_1 is equal to,"},{"Start":"03:09.175 ","End":"03:12.715","Text":"2 times 2 minus 1 is 3."},{"Start":"03:12.715 ","End":"03:15.020","Text":"Now, I don\u0027t just want the x,"},{"Start":"03:15.020 ","End":"03:19.265","Text":"I also want the y of the point."},{"Start":"03:19.265 ","End":"03:24.035","Text":"Y_1, which is just f of x_1,"},{"Start":"03:24.035 ","End":"03:30.850","Text":"is e^x plus 1 over 2 but with x replaced by 3."},{"Start":"03:31.970 ","End":"03:43.230","Text":"We get that this is equal to e^3 plus 1 over 2,"},{"Start":"03:43.230 ","End":"03:48.050","Text":"which is equal to 3 plus 1 is 4 over 2 is 2,"},{"Start":"03:48.050 ","End":"03:50.630","Text":"which is e squared."},{"Start":"03:50.630 ","End":"03:54.560","Text":"I can write that the point a,"},{"Start":"03:54.560 ","End":"04:04.490","Text":"which is x_1, y_1 is the 0.3, e squared."},{"Start":"04:04.490 ","End":"04:09.660","Text":"This answers part A of the question."},{"Start":"04:09.660 ","End":"04:17.240","Text":"Now let\u0027s move on to part B. I\u0027m going to write down a formula in part B."},{"Start":"04:17.240 ","End":"04:27.195","Text":"In part B, there is a general formula for the equation of a tangent and that is that y"},{"Start":"04:27.195 ","End":"04:37.340","Text":"minus y_1 is equal to f prime of x _1 times x minus x_1."},{"Start":"04:37.340 ","End":"04:43.580","Text":"This is the slope of the tangent to the graph f at the point x_1, y_1."},{"Start":"04:43.580 ","End":"04:45.010","Text":"Now, in our case,"},{"Start":"04:45.010 ","End":"04:46.885","Text":"we have all the unknowns."},{"Start":"04:46.885 ","End":"04:51.075","Text":"We have that x_1 is 3,"},{"Start":"04:51.075 ","End":"04:54.825","Text":"we have y_1 is e squared."},{"Start":"04:54.825 ","End":"05:04.610","Text":"We even have f prime of x _1 is equal to e squared over 2."},{"Start":"05:04.610 ","End":"05:06.810","Text":"We have everything we need."},{"Start":"05:06.810 ","End":"05:13.535","Text":"In our case, we will get y minus e squared"},{"Start":"05:13.535 ","End":"05:21.695","Text":"equals e squared over 2 times x minus 3."},{"Start":"05:21.695 ","End":"05:24.610","Text":"I just want to simplify this a bit."},{"Start":"05:24.610 ","End":"05:30.090","Text":"I get that y is equal to."},{"Start":"05:30.090 ","End":"05:36.935","Text":"Now, the coefficient to the x is e squared over 2 so it\u0027s e squared over 2x."},{"Start":"05:36.935 ","End":"05:46.455","Text":"Here I have minus 3 over 2 e squared, plus e squared."},{"Start":"05:46.455 ","End":"05:52.260","Text":"Minus 3 over 2 plus 1 is minus 1.5 plus 1 is minus a 0.5."},{"Start":"05:52.260 ","End":"05:55.715","Text":"I\u0027m just going to write it. You can look at it later if you\u0027re not convinced."},{"Start":"05:55.715 ","End":"05:59.215","Text":"Minus e squared over 2."},{"Start":"05:59.215 ","End":"06:05.165","Text":"This is the equation of the tangent line."},{"Start":"06:05.165 ","End":"06:09.130","Text":"This is the answer to part B."},{"Start":"06:09.130 ","End":"06:11.270","Text":"The remains part C,"},{"Start":"06:11.270 ","End":"06:15.230","Text":"which is the area and integration part."},{"Start":"06:15.230 ","End":"06:19.175","Text":"I\u0027m going to scroll back up so we can see the picture."},{"Start":"06:19.175 ","End":"06:21.920","Text":"This is the point where x is 0."},{"Start":"06:21.920 ","End":"06:23.930","Text":"I mean this is the origin."},{"Start":"06:23.930 ","End":"06:28.130","Text":"Over here, if I drop a perpendicular,"},{"Start":"06:28.130 ","End":"06:33.070","Text":"this is the point where x equals 3."},{"Start":"06:33.070 ","End":"06:40.249","Text":"What I need is the integral from 0 to 3 of the curve minus the tangent."},{"Start":"06:40.249 ","End":"06:43.300","Text":"It\u0027s like vertical strips."},{"Start":"06:43.300 ","End":"06:51.000","Text":"I get the integral from 0 to 3 of f of x,"},{"Start":"06:51.000 ","End":"06:55.140","Text":"which is e^x plus"},{"Start":"06:55.140 ","End":"07:01.755","Text":"1 over 2 minus the tangent,"},{"Start":"07:01.755 ","End":"07:11.785","Text":"that was e squared over 2x over 2 times x minus e squared over 2."},{"Start":"07:11.785 ","End":"07:17.930","Text":"We better put that in square brackets and add a dx here."},{"Start":"07:17.930 ","End":"07:23.165","Text":"We get that the integral of e to the x plus 1 over"},{"Start":"07:23.165 ","End":"07:30.230","Text":"2 will just be like e to the x plus 1 over 2."},{"Start":"07:30.230 ","End":"07:32.810","Text":"This is the linear function of x."},{"Start":"07:32.810 ","End":"07:35.240","Text":"We can just divide by the inner derivative."},{"Start":"07:35.240 ","End":"07:39.290","Text":"Dividing by a half is like multiplying by 2."},{"Start":"07:39.290 ","End":"07:41.465","Text":"Then we come to this bit."},{"Start":"07:41.465 ","End":"07:47.225","Text":"Now, the integral of x is x squared over 2."},{"Start":"07:47.225 ","End":"07:53.074","Text":"In this case we get minus e squared over 2 over 2,"},{"Start":"07:53.074 ","End":"07:56.884","Text":"so that makes it over 4x squared."},{"Start":"07:56.884 ","End":"08:00.920","Text":"Now, this minus with this minus gives me a plus."},{"Start":"08:00.920 ","End":"08:05.090","Text":"This is a constant, e squared over 2,"},{"Start":"08:05.090 ","End":"08:14.795","Text":"so it\u0027s just multiplied by x. I have to evaluate this between the limits 0 and 3,"},{"Start":"08:14.795 ","End":"08:16.610","Text":"meaning I substitute 3,"},{"Start":"08:16.610 ","End":"08:19.055","Text":"then substitute 0 and subtract."},{"Start":"08:19.055 ","End":"08:21.760","Text":"First of all the 3,"},{"Start":"08:21.760 ","End":"08:28.914","Text":"we get twice e^3 plus 1 over 2 is 2,"},{"Start":"08:28.914 ","End":"08:33.990","Text":"minus e squared over 4 times 3 squared is 9."},{"Start":"08:33.990 ","End":"08:35.690","Text":"I put the 9 in front,"},{"Start":"08:35.690 ","End":"08:40.790","Text":"e squared over 4 plus x is 3,"},{"Start":"08:40.790 ","End":"08:46.585","Text":"so it\u0027s 3, e squared over 2."},{"Start":"08:46.585 ","End":"08:51.965","Text":"Then I have to subtract what happens when I put 0 in."},{"Start":"08:51.965 ","End":"08:55.385","Text":"If I put 0 in, this is nothing."},{"Start":"08:55.385 ","End":"08:58.010","Text":"Here I get something."},{"Start":"08:58.010 ","End":"09:05.160","Text":"I just get 2,e to the power of 0 plus 1 over 2 is a 0.5."},{"Start":"09:05.160 ","End":"09:10.559","Text":"Maybe I\u0027ll write the nothings in minus 0 plus 0 to show haven\u0027t forgotten."},{"Start":"09:10.630 ","End":"09:14.930","Text":"This will be the answer for the area."},{"Start":"09:14.930 ","End":"09:18.260","Text":"I just have to simplify it a bit."},{"Start":"09:18.260 ","End":"09:21.085","Text":"This is our s, which equals."},{"Start":"09:21.085 ","End":"09:23.030","Text":"In the first bracket,"},{"Start":"09:23.030 ","End":"09:24.680","Text":"everything has e squared in it."},{"Start":"09:24.680 ","End":"09:32.405","Text":"Let\u0027s see how many e squared that we have altogether to minus 9 over 4 plus 3 over 2."},{"Start":"09:32.405 ","End":"09:35.465","Text":"Let me just do that at the side here."},{"Start":"09:35.465 ","End":"09:44.465","Text":"2 minus 9 over 4 plus 3 over 2 is 1.25,"},{"Start":"09:44.465 ","End":"09:52.715","Text":"or 5 over 4 e squared."},{"Start":"09:52.715 ","End":"09:58.520","Text":"Here just minus 2e^0.5."},{"Start":"09:58.520 ","End":"10:00.080","Text":"I could leave this as the answer,"},{"Start":"10:00.080 ","End":"10:02.960","Text":"but if you\u0027re going to do approximation on the calculator,"},{"Start":"10:02.960 ","End":"10:09.200","Text":"I would write it as 1.25 e squared minus 2 and each of the half a"},{"Start":"10:09.200 ","End":"10:19.115","Text":"square root of e. This is the answer but as an approximation it comes out to be 5.939."},{"Start":"10:19.115 ","End":"10:25.960","Text":"That is the answer for part C. We\u0027re done."}],"Thumbnail":null,"ID":4708},{"Watched":false,"Name":"Exercise 13","Duration":"9m 1s","ChapterTopicVideoID":4701,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"Here we have another area problem using the definite integrals."},{"Start":"00:04.020 ","End":"00:09.750","Text":"Let\u0027s see. We\u0027re given a function f of x is 8 over x minus 2."},{"Start":"00:09.750 ","End":"00:10.800","Text":"This is the function,"},{"Start":"00:10.800 ","End":"00:12.045","Text":"I\u0027ll just write it."},{"Start":"00:12.045 ","End":"00:19.615","Text":"Instead of f of x, I\u0027ll write y equals 8 over x minus 2,"},{"Start":"00:19.615 ","End":"00:24.330","Text":"and some of the domain where x is bigger than 0."},{"Start":"00:24.330 ","End":"00:32.040","Text":"We have a tangent at the point A and the point A is given as the point 2, 2."},{"Start":"00:32.040 ","End":"00:35.580","Text":"What we have to do is, first of all,"},{"Start":"00:35.580 ","End":"00:38.010","Text":"find the equation of the tangent,"},{"Start":"00:38.010 ","End":"00:42.710","Text":"and then we\u0027re going to compute the area that\u0027s shaded here that\u0027s between the tangent,"},{"Start":"00:42.710 ","End":"00:45.085","Text":"the x-axis on the graph."},{"Start":"00:45.085 ","End":"00:47.780","Text":"I\u0027d like to remind you of the formula for"},{"Start":"00:47.780 ","End":"00:52.250","Text":"the equation of a tangent to a curve at a point."},{"Start":"00:52.250 ","End":"00:57.390","Text":"The general formula is that y minus y_1 equals"},{"Start":"00:57.390 ","End":"01:02.805","Text":"f prime of x_1 times x minus x_1,"},{"Start":"01:02.805 ","End":"01:05.870","Text":"where f is the function and x_1 and y_1 is the point."},{"Start":"01:05.870 ","End":"01:10.415","Text":"In our case, we have that x_1 is 2,"},{"Start":"01:10.415 ","End":"01:13.250","Text":"we also have that y_1 is 2,"},{"Start":"01:13.250 ","End":"01:17.390","Text":"what we don\u0027t have is f prime of x_1."},{"Start":"01:17.390 ","End":"01:20.780","Text":"Well, we know it\u0027s f prime of 2 but still,"},{"Start":"01:20.780 ","End":"01:22.795","Text":"we don\u0027t know what it is."},{"Start":"01:22.795 ","End":"01:27.315","Text":"Then here we\u0027ll start computing part a,"},{"Start":"01:27.315 ","End":"01:31.845","Text":"f of x is 8 over x minus 2."},{"Start":"01:31.845 ","End":"01:36.050","Text":"In general, f prime of x is a derivative of this,"},{"Start":"01:36.050 ","End":"01:39.665","Text":"which is minus 8 over x squared."},{"Start":"01:39.665 ","End":"01:46.040","Text":"So f prime of 2 is what we get when we substitute x equals 2,"},{"Start":"01:46.040 ","End":"01:50.300","Text":"which is minus 8 over 2 squared,"},{"Start":"01:50.300 ","End":"01:51.530","Text":"which is minus 8 over 4,"},{"Start":"01:51.530 ","End":"01:53.575","Text":"which is minus 2."},{"Start":"01:53.575 ","End":"01:55.835","Text":"Now we do have everything."},{"Start":"01:55.835 ","End":"02:00.485","Text":"We have that f prime of 2 is minus 2."},{"Start":"02:00.485 ","End":"02:02.915","Text":"I have all the quantities here."},{"Start":"02:02.915 ","End":"02:10.130","Text":"I can compute from here that the equation would be y minus,"},{"Start":"02:10.130 ","End":"02:15.350","Text":"now y_1 is 2 equals f prime of x_1,"},{"Start":"02:15.350 ","End":"02:20.220","Text":"is f prime of 2 is minus 2 times x minus."},{"Start":"02:20.220 ","End":"02:23.175","Text":"Then we have that x_1 is also 2."},{"Start":"02:23.175 ","End":"02:25.160","Text":"This is the equation of a tangent,"},{"Start":"02:25.160 ","End":"02:26.930","Text":"but let\u0027s just tidy it up a little bit."},{"Start":"02:26.930 ","End":"02:30.620","Text":"We get that y equals minus 2x,"},{"Start":"02:30.620 ","End":"02:33.544","Text":"and then we get plus 4,"},{"Start":"02:33.544 ","End":"02:38.275","Text":"but also plus 2, so plus 6."},{"Start":"02:38.275 ","End":"02:41.660","Text":"Before we get into part b properly,"},{"Start":"02:41.660 ","End":"02:44.555","Text":"we\u0027ll have to find some points."},{"Start":"02:44.555 ","End":"02:49.130","Text":"What I do know is that if I drop the perpendicular here,"},{"Start":"02:49.130 ","End":"02:52.235","Text":"that this is the point where x equals 2,"},{"Start":"02:52.235 ","End":"02:58.570","Text":"what I need to know is what x is here and what x is here."},{"Start":"02:58.570 ","End":"03:03.350","Text":"When I have that, it will be easier for me to do the integration or"},{"Start":"03:03.350 ","End":"03:08.075","Text":"to find this area which we\u0027ll call S. Now,"},{"Start":"03:08.075 ","End":"03:10.265","Text":"to find this point,"},{"Start":"03:10.265 ","End":"03:15.500","Text":"I\u0027m going to have to intersect the tangent line with the x-axis."},{"Start":"03:15.500 ","End":"03:21.235","Text":"In other words, I\u0027m going to let y equal 0 in the case of the tangent."},{"Start":"03:21.235 ","End":"03:23.660","Text":"In the case of this point,"},{"Start":"03:23.660 ","End":"03:25.450","Text":"I let the curve equals 0."},{"Start":"03:25.450 ","End":"03:30.755","Text":"In each case I\u0027m assigning y equals 0 either to the tangent or to the curve."},{"Start":"03:30.755 ","End":"03:33.500","Text":"I\u0027m going to continue here."},{"Start":"03:33.500 ","End":"03:36.035","Text":"It says part b."},{"Start":"03:36.035 ","End":"03:39.470","Text":"Let\u0027s see. Let\u0027s take this one first."},{"Start":"03:39.470 ","End":"03:41.555","Text":"For the tangent to be 0,"},{"Start":"03:41.555 ","End":"03:45.690","Text":"we can let y equals 0 here,"},{"Start":"03:45.690 ","End":"03:51.435","Text":"so we get 0 equals minus 2x plus 6."},{"Start":"03:51.435 ","End":"03:55.410","Text":"This gives us that 2x equals 6,"},{"Start":"03:55.410 ","End":"03:58.395","Text":"so x equals 3."},{"Start":"03:58.395 ","End":"04:00.230","Text":"For the other one,"},{"Start":"04:00.230 ","End":"04:02.570","Text":"we get that the original curve,"},{"Start":"04:02.570 ","End":"04:03.590","Text":"let\u0027s see if I can see it."},{"Start":"04:03.590 ","End":"04:06.755","Text":"Yeah, here it is. 8 over x minus 2."},{"Start":"04:06.755 ","End":"04:11.680","Text":"If I set f of x or y equals 0 on this one,"},{"Start":"04:11.680 ","End":"04:18.000","Text":"so I get 0 equals 8 over x minus 2."},{"Start":"04:18.000 ","End":"04:21.045","Text":"Again, we can do it in our heads."},{"Start":"04:21.045 ","End":"04:23.340","Text":"8 over x equals 2,"},{"Start":"04:23.340 ","End":"04:26.715","Text":"and we could just ask that 8 over what gives me 2,"},{"Start":"04:26.715 ","End":"04:30.180","Text":"and that would be 4, you could solve it."},{"Start":"04:30.180 ","End":"04:38.450","Text":"We have that this is the point where x equals 3 and this is the point where x equals 4."},{"Start":"04:38.450 ","End":"04:42.425","Text":"There\u0027s more than one way to tackle this."},{"Start":"04:42.425 ","End":"04:46.730","Text":"One thing you might do is draw a vertical line through"},{"Start":"04:46.730 ","End":"04:51.170","Text":"the point 3 and compute this bit separately and this bit separately."},{"Start":"04:51.170 ","End":"04:53.930","Text":"But actually, there\u0027s a shorter thing you can do,"},{"Start":"04:53.930 ","End":"04:58.490","Text":"and that is to compute the area of,"},{"Start":"04:58.490 ","End":"05:03.365","Text":"I\u0027ll highlight it, from here along here,"},{"Start":"05:03.365 ","End":"05:06.460","Text":"and then along the curve."},{"Start":"05:06.460 ","End":"05:11.600","Text":"This will be the area under this curve between 2 and 4."},{"Start":"05:11.600 ","End":"05:16.610","Text":"Then after that, we could subtract from this bit,"},{"Start":"05:16.610 ","End":"05:22.090","Text":"then we could subtract this bit and then get left with S. That\u0027s"},{"Start":"05:22.090 ","End":"05:24.770","Text":"how I\u0027m going to do it because the area of the triangle we\u0027ll be able to do"},{"Start":"05:24.770 ","End":"05:28.295","Text":"without integration just using regular geometry."},{"Start":"05:28.295 ","End":"05:29.990","Text":"What I\u0027m saying is,"},{"Start":"05:29.990 ","End":"05:33.875","Text":"and I think I\u0027ll do it over here so I can keep the picture."},{"Start":"05:33.875 ","End":"05:35.760","Text":"What I want to do is, first of all,"},{"Start":"05:35.760 ","End":"05:43.870","Text":"take the integral from 2-4 under this curve,"},{"Start":"05:43.870 ","End":"05:46.285","Text":"which is the curve f,"},{"Start":"05:46.285 ","End":"05:51.645","Text":"which is 8 over x minus 2 dx."},{"Start":"05:51.645 ","End":"05:55.720","Text":"Then, later on, we\u0027ll take away the area of the triangle."},{"Start":"05:55.720 ","End":"06:00.805","Text":"I\u0027m getting the whole bit that\u0027s highlighted in yellow first."},{"Start":"06:00.805 ","End":"06:04.920","Text":"This is equal to, let\u0027s see,"},{"Start":"06:04.920 ","End":"06:10.660","Text":"the integral of 8 over x is 8 times natural log of x."},{"Start":"06:10.660 ","End":"06:18.825","Text":"We don\u0027t need the absolute value because we\u0027re in the x is bigger than 0 domain."},{"Start":"06:18.825 ","End":"06:23.170","Text":"The integral of a constant is just that"},{"Start":"06:23.170 ","End":"06:28.330","Text":"constant times x. I have all this between 2 and 4."},{"Start":"06:28.330 ","End":"06:38.010","Text":"This is equal to 8 log of 4 minus twice 4 is 8,"},{"Start":"06:38.010 ","End":"06:47.265","Text":"takeaway, now put into 8 natural log of 2 minus twice 2 is 4."},{"Start":"06:47.265 ","End":"06:57.640","Text":"This is equal to 8 times natural log of 4 minus natural log of 2."},{"Start":"06:57.640 ","End":"07:04.495","Text":"Then minus 8 plus 4 is minus 4."},{"Start":"07:04.495 ","End":"07:09.380","Text":"Now, this bit here can be simplified because using the rules of logarithms,"},{"Start":"07:09.380 ","End":"07:13.569","Text":"this is the natural log of 4 over 2,"},{"Start":"07:13.569 ","End":"07:16.170","Text":"which is natural log of 2."},{"Start":"07:16.170 ","End":"07:24.805","Text":"What we get is 8 natural log of 2 minus 4."},{"Start":"07:24.805 ","End":"07:29.150","Text":"So far we\u0027ve gotten the bit that\u0027s highlighted in yellow."},{"Start":"07:29.150 ","End":"07:31.460","Text":"Now we need the triangle."},{"Start":"07:31.460 ","End":"07:35.270","Text":"For the triangle, let\u0027s see,"},{"Start":"07:35.270 ","End":"07:39.680","Text":"this is the base and this is the height."},{"Start":"07:39.680 ","End":"07:43.205","Text":"So 1/2 base times height,"},{"Start":"07:43.205 ","End":"07:51.230","Text":"which means that we get 1/2 times the base is 1,"},{"Start":"07:51.230 ","End":"07:54.870","Text":"3 minus 2 is 1, times 1,"},{"Start":"07:54.870 ","End":"07:59.540","Text":"times the height is just the y-coordinate of A, which is 2."},{"Start":"07:59.540 ","End":"08:02.540","Text":"This is altogether equal to 1."},{"Start":"08:02.540 ","End":"08:08.560","Text":"Now what I have to do is take this and subtract this."},{"Start":"08:08.560 ","End":"08:17.220","Text":"If I take that difference I\u0027ll get what is my S. My S is equal to this minus this,"},{"Start":"08:17.220 ","End":"08:21.945","Text":"so it\u0027s 8 natural log of 2 minus 4 minus 1,"},{"Start":"08:21.945 ","End":"08:24.460","Text":"which is minus 5."},{"Start":"08:24.460 ","End":"08:28.325","Text":"For those who like approximations and decimals,"},{"Start":"08:28.325 ","End":"08:33.420","Text":"this is approximately equal to 0.545."},{"Start":"08:34.630 ","End":"08:38.215","Text":"We\u0027re actually done."},{"Start":"08:38.215 ","End":"08:42.330","Text":"We have the answer to part A."},{"Start":"08:42.330 ","End":"08:46.095","Text":"Here, the equation of the tangent."},{"Start":"08:46.095 ","End":"08:48.850","Text":"Here we have the answer to part b,"},{"Start":"08:48.850 ","End":"08:53.400","Text":"which is the area which is this precisely,"},{"Start":"08:53.400 ","End":"08:56.000","Text":"but if you like numerical approximation,"},{"Start":"08:56.000 ","End":"08:58.440","Text":"then here we have that."},{"Start":"08:58.750 ","End":"09:01.920","Text":"We are done."}],"Thumbnail":null,"ID":4709},{"Watched":false,"Name":"Exercise 14","Duration":"13m 25s","ChapterTopicVideoID":4702,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.045","Text":"Here we have a word problem to find areas using integrals,"},{"Start":"00:06.045 ","End":"00:09.390","Text":"but before that we have to draw a rough sketch."},{"Start":"00:09.390 ","End":"00:11.200","Text":"There\u0027s no picture here."},{"Start":"00:11.200 ","End":"00:13.230","Text":"We\u0027re given 2 functions,"},{"Start":"00:13.230 ","End":"00:15.480","Text":"sine x and cosine of 2x,"},{"Start":"00:15.480 ","End":"00:19.710","Text":"and we have to sketch those between 0 and pi."},{"Start":"00:19.710 ","End":"00:22.620","Text":"Let me just start by reminding you in general about"},{"Start":"00:22.620 ","End":"00:25.470","Text":"the sine and the cosine, what they look like."},{"Start":"00:25.470 ","End":"00:27.690","Text":"We\u0027ll just draw them for 1 period,"},{"Start":"00:27.690 ","End":"00:30.135","Text":"which is 0-2 pi."},{"Start":"00:30.135 ","End":"00:33.000","Text":"We\u0027ll do just a rough sketch."},{"Start":"00:33.000 ","End":"00:36.465","Text":"Here\u0027s one I\u0027m going to use for the sine,"},{"Start":"00:36.465 ","End":"00:40.605","Text":"and here\u0027s another one I\u0027m going to use for the cosine."},{"Start":"00:40.605 ","End":"00:43.745","Text":"In each case, this is y, this is x of course."},{"Start":"00:43.745 ","End":"00:48.200","Text":"Now, sine x looks something like this,"},{"Start":"00:48.200 ","End":"00:51.095","Text":"where this is 0,"},{"Start":"00:51.095 ","End":"00:55.180","Text":"this is pi over 2, this is pi,"},{"Start":"00:55.180 ","End":"01:01.054","Text":"3 pi over 2 and 2 pi that\u0027s 1 period of the sine."},{"Start":"01:01.054 ","End":"01:03.815","Text":"This is y equals sine x."},{"Start":"01:03.815 ","End":"01:07.145","Text":"A reminder of how the cosine looks."},{"Start":"01:07.145 ","End":"01:12.470","Text":"The cosine starts of at 1 and"},{"Start":"01:12.470 ","End":"01:20.620","Text":"then it goes down to 0 at 90 degrees and to minus 1 and back up and here."},{"Start":"01:20.620 ","End":"01:23.700","Text":"Again, this is just one period,"},{"Start":"01:23.700 ","End":"01:28.815","Text":"0 pi over 2 pi,"},{"Start":"01:28.815 ","End":"01:33.600","Text":"3 pi over 2, and 2 pi."},{"Start":"01:33.600 ","End":"01:35.550","Text":"That\u0027s the cosine and the cosine."},{"Start":"01:35.550 ","End":"01:38.150","Text":"Now, I want to draw these 2."},{"Start":"01:38.150 ","End":"01:39.445","Text":"It\u0027s not exactly cosine,"},{"Start":"01:39.445 ","End":"01:41.530","Text":"it\u0027s a variation of cosine,"},{"Start":"01:41.530 ","End":"01:44.255","Text":"and I want to draw them on the same set of axes."},{"Start":"01:44.255 ","End":"01:46.800","Text":"Let\u0027s do this over here."},{"Start":"01:48.320 ","End":"01:50.450","Text":"You just need a rough sketch."},{"Start":"01:50.450 ","End":"01:52.925","Text":"It really does not have to be precise."},{"Start":"01:52.925 ","End":"01:55.250","Text":"We have our y and we have our x."},{"Start":"01:55.250 ","End":"01:59.215","Text":"This time we\u0027re only going to take it between 0 and pi."},{"Start":"01:59.215 ","End":"02:04.605","Text":"Here\u0027s 0 and let\u0027s say here is pi."},{"Start":"02:04.605 ","End":"02:07.595","Text":"We also have here pi over 2,"},{"Start":"02:07.595 ","End":"02:10.100","Text":"and everything goes between 0 and 1."},{"Start":"02:10.100 ","End":"02:12.500","Text":"I should\u0027ve made a note of that here,"},{"Start":"02:12.500 ","End":"02:15.440","Text":"that sine and the cosine,"},{"Start":"02:15.440 ","End":"02:18.635","Text":"they only go from minus 1 to 1."},{"Start":"02:18.635 ","End":"02:20.165","Text":"The same here."},{"Start":"02:20.165 ","End":"02:26.220","Text":"Everything is between minus 1 and 1."},{"Start":"02:26.220 ","End":"02:28.940","Text":"Likewise, in this picture,"},{"Start":"02:28.940 ","End":"02:35.475","Text":"I\u0027m going to bound them by 1 here and minus 1 here."},{"Start":"02:35.475 ","End":"02:38.210","Text":"Now, the sine, no problem."},{"Start":"02:38.210 ","End":"02:42.505","Text":"I just take the bit from here between 0 and pi."},{"Start":"02:42.505 ","End":"02:45.810","Text":"At pi over 2, it\u0027s equal to 1."},{"Start":"02:45.810 ","End":"02:49.335","Text":"Something like this, and down here again."},{"Start":"02:49.335 ","End":"02:51.510","Text":"It doesn\u0027t have to be precise."},{"Start":"02:51.510 ","End":"02:54.000","Text":"Rough sketch."},{"Start":"02:54.000 ","End":"03:00.670","Text":"Now, the cosine, let\u0027s take the cosine of 2x."},{"Start":"03:00.670 ","End":"03:04.700","Text":"Now, you can either draw table of values and see what it is."},{"Start":"03:04.700 ","End":"03:06.200","Text":"That\u0027s one method."},{"Start":"03:06.200 ","End":"03:13.270","Text":"The other method is to remember that when we change x to ax,"},{"Start":"03:13.270 ","End":"03:17.615","Text":"we squash it inward by a factor of a."},{"Start":"03:17.615 ","End":"03:25.970","Text":"If I take the cosine of x and compress it horizontally by a factor of 2,"},{"Start":"03:25.970 ","End":"03:27.860","Text":"I\u0027ll get this whole thing,"},{"Start":"03:27.860 ","End":"03:30.815","Text":"but instead of between 0 and 2 pi,"},{"Start":"03:30.815 ","End":"03:33.335","Text":"it\u0027ll be between 0 and pi."},{"Start":"03:33.335 ","End":"03:40.005","Text":"What I\u0027ll get is something like this."},{"Start":"03:40.005 ","End":"03:42.300","Text":"Not very good. It should go to minus 1."},{"Start":"03:42.300 ","End":"03:44.175","Text":"Let me do another attempt."},{"Start":"03:44.175 ","End":"03:49.580","Text":"This is a bit better and this one is the y equals cosine 2x."},{"Start":"03:49.580 ","End":"03:54.620","Text":"But if you\u0027re not sure about transformations of functions and so on,"},{"Start":"03:54.620 ","End":"03:59.495","Text":"and what to do if you replace x by 2x and so on,"},{"Start":"03:59.495 ","End":"04:01.240","Text":"then you can just always draw a table."},{"Start":"04:01.240 ","End":"04:04.340","Text":"I\u0027ll give you an example with the cosine 2x with the sine x."},{"Start":"04:04.340 ","End":"04:06.410","Text":"It\u0027s easier. Here we have x,"},{"Start":"04:06.410 ","End":"04:08.390","Text":"here we have cosine 2x."},{"Start":"04:08.390 ","End":"04:14.805","Text":"We would take some values 0 pi over 2 and pi,"},{"Start":"04:14.805 ","End":"04:16.290","Text":"and look at the cosine."},{"Start":"04:16.290 ","End":"04:19.710","Text":"Cosine of twice 0 is cosine 0 is 1."},{"Start":"04:19.710 ","End":"04:21.180","Text":"It\u0027s pi over 2."},{"Start":"04:21.180 ","End":"04:24.045","Text":"Then we get cosine of pi,"},{"Start":"04:24.045 ","End":"04:26.400","Text":"which is minus 1,"},{"Start":"04:26.400 ","End":"04:32.120","Text":"and cosine of 2 pi is the same as cosine of 0 is also 1."},{"Start":"04:32.120 ","End":"04:34.280","Text":"But if you wanted to be more precise,"},{"Start":"04:34.280 ","End":"04:39.770","Text":"you could put also pi over 4 here and 3 pi over 4 here."},{"Start":"04:39.770 ","End":"04:42.785","Text":"Then we\u0027d get twice pi over 4 is pi over 2,"},{"Start":"04:42.785 ","End":"04:44.720","Text":"cosine of that is 0."},{"Start":"04:44.720 ","End":"04:48.230","Text":"Also this would come out 0 and then you would get this point,"},{"Start":"04:48.230 ","End":"04:49.310","Text":"this point, this point,"},{"Start":"04:49.310 ","End":"04:51.710","Text":"this point and this point and sketch it."},{"Start":"04:51.710 ","End":"04:54.200","Text":"Either way, we don\u0027t want to invest"},{"Start":"04:54.200 ","End":"04:57.860","Text":"too much time in graph sketching because that\u0027s another topic."},{"Start":"04:57.860 ","End":"05:01.010","Text":"The point is that we\u0027ve now done a"},{"Start":"05:01.010 ","End":"05:03.860","Text":"and we\u0027re now ready to tackle b and even understand it."},{"Start":"05:03.860 ","End":"05:08.515","Text":"What we have to do is to shade or highlight the area bounded by these."},{"Start":"05:08.515 ","End":"05:09.990","Text":"That\u0027s a sketching thing,"},{"Start":"05:09.990 ","End":"05:12.045","Text":"and then compute its size."},{"Start":"05:12.045 ","End":"05:15.385","Text":"As for shading or highlighting,"},{"Start":"05:15.385 ","End":"05:19.610","Text":"it looks awful, but you get the idea."},{"Start":"05:19.610 ","End":"05:23.010","Text":"This is the area we want."},{"Start":"05:25.180 ","End":"05:30.840","Text":"That\u0027s the shading part or highlighting actually,"},{"Start":"05:30.840 ","End":"05:34.480","Text":"and now we have to compute its size."},{"Start":"05:34.700 ","End":"05:40.610","Text":"The obvious thing is to take the integral from."},{"Start":"05:40.610 ","End":"05:42.710","Text":"Here we come to the thing,"},{"Start":"05:42.710 ","End":"05:47.390","Text":"is what is this point and what is this point?"},{"Start":"05:47.390 ","End":"05:51.080","Text":"Because we have to take the integral from here,"},{"Start":"05:51.080 ","End":"05:52.580","Text":"which we don\u0027t know."},{"Start":"05:52.580 ","End":"05:54.570","Text":"I mean, to here from here,"},{"Start":"05:54.570 ","End":"05:56.190","Text":"which we also don\u0027t know."},{"Start":"05:56.190 ","End":"05:58.275","Text":"Before we write an integral,"},{"Start":"05:58.275 ","End":"06:00.560","Text":"let\u0027s compute these points."},{"Start":"06:00.560 ","End":"06:02.900","Text":"I really just need dx of these points."},{"Start":"06:02.900 ","End":"06:05.900","Text":"The way I do this is just by comparing the 2 functions."},{"Start":"06:05.900 ","End":"06:09.275","Text":"I get an equation that sine x is cosine 2x."},{"Start":"06:09.275 ","End":"06:11.125","Text":"Let\u0027s start with that,"},{"Start":"06:11.125 ","End":"06:13.160","Text":"and then we\u0027ll get back."},{"Start":"06:13.160 ","End":"06:16.040","Text":"When we do get back from finding these 2 points,"},{"Start":"06:16.040 ","End":"06:19.145","Text":"I could write already what we will get."},{"Start":"06:19.145 ","End":"06:21.770","Text":"We will get the integral from"},{"Start":"06:21.770 ","End":"06:28.025","Text":"the first question mark to the second question mark of the top one,"},{"Start":"06:28.025 ","End":"06:31.145","Text":"which is sine x,"},{"Start":"06:31.145 ","End":"06:32.960","Text":"minus the lower one,"},{"Start":"06:32.960 ","End":"06:38.535","Text":"which is cosine 2x dx."},{"Start":"06:38.535 ","End":"06:42.020","Text":"Our intermediate problem now is to find these 2 points,"},{"Start":"06:42.020 ","End":"06:43.475","Text":"these 2 question marks."},{"Start":"06:43.475 ","End":"06:45.935","Text":"Let me do this at the side."},{"Start":"06:45.935 ","End":"06:49.025","Text":"What I have to do is compare this to this,"},{"Start":"06:49.025 ","End":"06:56.060","Text":"in which case I will get the equation sine x equals cosine 2x."},{"Start":"06:56.060 ","End":"07:02.720","Text":"Now, I remember my trigonometric formulas that cosine 2x can be written in terms of"},{"Start":"07:02.720 ","End":"07:11.330","Text":"sine and it\u0027s equal to 1 minus 2 sine squared x."},{"Start":"07:11.330 ","End":"07:15.020","Text":"Now, I\u0027ll write this again, sine x."},{"Start":"07:15.020 ","End":"07:24.375","Text":"Let\u0027s make a substitution and if we let s equals sine x to say s for sine,"},{"Start":"07:24.375 ","End":"07:29.665","Text":"what I get is s equals 1 minus 2s squared,"},{"Start":"07:29.665 ","End":"07:31.630","Text":"and this is a quadratic equation."},{"Start":"07:31.630 ","End":"07:34.105","Text":"If I just put everything on the left-hand side,"},{"Start":"07:34.105 ","End":"07:43.005","Text":"then I get 2s squared plus s minus 1 equals 0."},{"Start":"07:43.005 ","End":"07:47.775","Text":"This is an equation which has 2 solutions."},{"Start":"07:47.775 ","End":"07:52.915","Text":"I\u0027ll tell you what they are because I don\u0027t want to waste time with quadratic equations."},{"Start":"07:52.915 ","End":"08:01.755","Text":"In fact, the 2 solutions are s equals 1/2 or minus 1."},{"Start":"08:01.755 ","End":"08:06.615","Text":"That means that we have that sine x,"},{"Start":"08:06.615 ","End":"08:09.480","Text":"because s is sine x,"},{"Start":"08:09.480 ","End":"08:11.965","Text":"has got to equal 1/2,"},{"Start":"08:11.965 ","End":"08:17.405","Text":"or sine x equals minus 1."},{"Start":"08:17.405 ","End":"08:20.015","Text":"Now, if we look at this,"},{"Start":"08:20.015 ","End":"08:22.440","Text":"and I mean here sine x,"},{"Start":"08:22.440 ","End":"08:27.625","Text":"and in the range between 0 and pi sine x is not negative."},{"Start":"08:27.625 ","End":"08:30.440","Text":"Because it\u0027s non-negative in this range,"},{"Start":"08:30.440 ","End":"08:34.210","Text":"I have to rule out this possibility."},{"Start":"08:34.210 ","End":"08:37.575","Text":"I get that sine x equals 1/2."},{"Start":"08:37.575 ","End":"08:43.105","Text":"But in the range from 0 to pi,"},{"Start":"08:43.105 ","End":"08:48.000","Text":"there are only 2 solutions to sine x is 1/2."},{"Start":"08:48.000 ","End":"08:54.500","Text":"In degrees, it comes out to be 30 degrees and 150 degrees."},{"Start":"08:54.500 ","End":"08:58.220","Text":"You could do it on the calculator with arc sine,"},{"Start":"08:58.220 ","End":"09:00.995","Text":"but that would probably just give you the 30 degrees."},{"Start":"09:00.995 ","End":"09:06.425","Text":"But if you take the supplement of the angle which is 180 minus,"},{"Start":"09:06.425 ","End":"09:08.375","Text":"then you get this."},{"Start":"09:08.375 ","End":"09:10.310","Text":"In terms of radians,"},{"Start":"09:10.310 ","End":"09:13.040","Text":"we should write it as pi over 6,"},{"Start":"09:13.040 ","End":"09:19.535","Text":"and this as 5 pi over 6."},{"Start":"09:19.535 ","End":"09:23.630","Text":"We have the 2 question marks here."},{"Start":"09:23.630 ","End":"09:27.160","Text":"Now, erase these question marks."},{"Start":"09:27.160 ","End":"09:30.310","Text":"Now, I can write in and I\u0027ll just put"},{"Start":"09:30.310 ","End":"09:33.910","Text":"it in a different color to see that it\u0027s computed from here."},{"Start":"09:33.910 ","End":"09:35.490","Text":"I\u0027ll do it ink red."},{"Start":"09:35.490 ","End":"09:41.740","Text":"We have from pi over 6 to 5 pi over 6,"},{"Start":"09:41.740 ","End":"09:45.950","Text":"and that\u0027s all that we were waiting for to complete this exercise."},{"Start":"09:45.950 ","End":"09:52.210","Text":"Continuing, we get the integral of sine x is"},{"Start":"09:52.210 ","End":"09:58.830","Text":"minus cosine of x and the integral of cosine 2x."},{"Start":"09:58.830 ","End":"10:01.740","Text":"It would be sine 2x,"},{"Start":"10:01.740 ","End":"10:03.350","Text":"but because of the 2x,"},{"Start":"10:03.350 ","End":"10:08.100","Text":"we have to compensate by putting 1/2 here."},{"Start":"10:08.100 ","End":"10:15.845","Text":"Then we have to evaluate this between pi over 6 and 5 pi over 6."},{"Start":"10:15.845 ","End":"10:17.365","Text":"Oops, I wrote a 3."},{"Start":"10:17.365 ","End":"10:20.820","Text":"But that\u0027s also just for convenience."},{"Start":"10:20.820 ","End":"10:25.070","Text":"I\u0027m going to write that this is the 30 degrees and this is"},{"Start":"10:25.070 ","End":"10:29.925","Text":"the 150 degrees because we\u0027re so used to evaluating in degrees."},{"Start":"10:29.925 ","End":"10:34.060","Text":"Let\u0027s try the 150 degrees first."},{"Start":"10:34.060 ","End":"10:39.750","Text":"We get minus cosine of 150 is the"},{"Start":"10:39.750 ","End":"10:46.125","Text":"same as cosine of 30 degrees."},{"Start":"10:46.125 ","End":"10:50.760","Text":"That is square root of 3 over 2."},{"Start":"10:50.760 ","End":"10:58.705","Text":"But it\u0027s a minus minus because I said that the cosine of 150 is minus cosine 30."},{"Start":"10:58.705 ","End":"11:01.645","Text":"This will be plus cosine 30."},{"Start":"11:01.645 ","End":"11:07.275","Text":"Then 2x would be 300 degrees,"},{"Start":"11:07.275 ","End":"11:16.340","Text":"and sine of 300 degrees is the same as minus sine of 60 degrees."},{"Start":"11:17.120 ","End":"11:23.895","Text":"Sine of 60 degrees is also root 3 over 2."},{"Start":"11:23.895 ","End":"11:33.205","Text":"We have minus 1/2 times minus root 3 over 2 is plus root 3 over 4."},{"Start":"11:33.205 ","End":"11:36.990","Text":"That\u0027s all for the 150 degree part."},{"Start":"11:36.990 ","End":"11:39.770","Text":"Now, for the 30 degree part,"},{"Start":"11:39.770 ","End":"11:47.010","Text":"minus cosine 30 is minus root 3 over 2."},{"Start":"11:48.290 ","End":"11:53.235","Text":"Here we have minus 1/2 sine 60."},{"Start":"11:53.235 ","End":"11:56.400","Text":"Sine 60 is root 3 over 2."},{"Start":"11:56.400 ","End":"12:04.060","Text":"Minus 1/2 of it is minus root 3 over 4."},{"Start":"12:05.390 ","End":"12:11.455","Text":"What we get is we get this thing twice."},{"Start":"12:11.455 ","End":"12:16.550","Text":"Well, let\u0027s take the square root of 3 outside and let\u0027s see what we get."},{"Start":"12:16.550 ","End":"12:24.870","Text":"We get 1/2 plus 1/4 minus minus 1/2 minus minus 1/4,"},{"Start":"12:24.870 ","End":"12:27.285","Text":"so it\u0027s all pluses."},{"Start":"12:27.285 ","End":"12:33.705","Text":"This plus this, plus this, plus this is 1/2 and another 1/2."},{"Start":"12:33.705 ","End":"12:35.880","Text":"Anyway, it\u0027s 3 over 2."},{"Start":"12:35.880 ","End":"12:44.175","Text":"What we get is 3 square root of 3 over 2."},{"Start":"12:44.175 ","End":"12:48.980","Text":"For those who like decimal approximations,"},{"Start":"12:48.980 ","End":"12:53.620","Text":"I make it 2.598."},{"Start":"12:56.180 ","End":"12:59.175","Text":"Let\u0027s see where we are now."},{"Start":"12:59.175 ","End":"13:02.000","Text":"The original question asked us for the rough sketch."},{"Start":"13:02.000 ","End":"13:03.935","Text":"I can\u0027t highlight that,"},{"Start":"13:03.935 ","End":"13:05.750","Text":"but this is my rough sketch."},{"Start":"13:05.750 ","End":"13:09.845","Text":"But I can highlight the answer for part b,"},{"Start":"13:09.845 ","End":"13:16.080","Text":"which the area is this precisely,"},{"Start":"13:16.080 ","End":"13:21.080","Text":"but this is what it is approximately in decimal."},{"Start":"13:21.080 ","End":"13:25.740","Text":"This is a lovely sketch. We\u0027re done."}],"Thumbnail":null,"ID":4710},{"Watched":false,"Name":"Exercise 15","Duration":"14m 45s","ChapterTopicVideoID":4703,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"Here, we have another word problem involving areas and integrals."},{"Start":"00:05.280 ","End":"00:07.485","Text":"Let\u0027s see what\u0027s going on here."},{"Start":"00:07.485 ","End":"00:09.390","Text":"Well, we\u0027re given a function,"},{"Start":"00:09.390 ","End":"00:11.550","Text":"y equals tangent squared x,"},{"Start":"00:11.550 ","End":"00:14.475","Text":"f of x of y, it\u0027s interchangeable."},{"Start":"00:14.475 ","End":"00:16.410","Text":"We don\u0027t have a sketch."},{"Start":"00:16.410 ","End":"00:18.555","Text":"We don\u0027t need 1 just yet."},{"Start":"00:18.555 ","End":"00:20.610","Text":"We\u0027re given the domain,"},{"Start":"00:20.610 ","End":"00:25.950","Text":"which is this and in degrees that\u0027s between minus 90 degrees and 0."},{"Start":"00:25.950 ","End":"00:27.630","Text":"The first thing we have to do,"},{"Start":"00:27.630 ","End":"00:29.730","Text":"is find the equation of the tangent to"},{"Start":"00:29.730 ","End":"00:34.605","Text":"this curve at the point where x is minus Pi over 4,"},{"Start":"00:34.605 ","End":"00:37.350","Text":"which is somewhere in this interval."},{"Start":"00:37.350 ","End":"00:39.650","Text":"Then when we\u0027ve done that,"},{"Start":"00:39.650 ","End":"00:41.300","Text":"then we\u0027re ready to go to part b,"},{"Start":"00:41.300 ","End":"00:45.400","Text":"then we\u0027ll need a sketch and I\u0027ll explain part b when we get to it."},{"Start":"00:45.400 ","End":"00:47.700","Text":"Let\u0027s focus on a."},{"Start":"00:47.700 ","End":"00:52.685","Text":"Let me write the general equation for the tangent to the curve at a point."},{"Start":"00:52.685 ","End":"00:55.430","Text":"The general equation goes like this,"},{"Start":"00:55.430 ","End":"01:04.140","Text":"y minus y_1 equals f prime of x 1 times x minus x_1,"},{"Start":"01:04.140 ","End":"01:07.550","Text":"where x_1, y_1 is the point."},{"Start":"01:07.550 ","End":"01:13.980","Text":"Now, here we\u0027re just given x_1 and y_1 is f of x_1."},{"Start":"01:13.980 ","End":"01:19.370","Text":"What we\u0027re going to need is f and f prime of x_1,"},{"Start":"01:19.370 ","End":"01:21.560","Text":"which is minus Pi over 4."},{"Start":"01:21.560 ","End":"01:23.885","Text":"When I write it, it\u0027ll be clearer."},{"Start":"01:23.885 ","End":"01:31.350","Text":"We have that f of x is tangent squared of x."},{"Start":"01:31.350 ","End":"01:33.420","Text":"What we\u0027re going to need,"},{"Start":"01:33.420 ","End":"01:39.515","Text":"is f of minus Pi over 4 and I\u0027ll evaluate it in a moment."},{"Start":"01:39.515 ","End":"01:46.130","Text":"Then we\u0027ll need f prime of x and f prime of x will be"},{"Start":"01:46.130 ","End":"01:53.540","Text":"2 times tangent x times the other derivative of the squared,"},{"Start":"01:53.540 ","End":"02:03.480","Text":"which is the derivative of tangent x and that\u0027s 1 over cosine squared of x."},{"Start":"02:03.480 ","End":"02:05.705","Text":"When we\u0027ve simplified this,"},{"Start":"02:05.705 ","End":"02:14.430","Text":"then we\u0027ll have to say what is f prime of minus Pi over 4 equal to?"},{"Start":"02:14.560 ","End":"02:19.170","Text":"That will be our y_1 and then this"},{"Start":"02:19.170 ","End":"02:23.465","Text":"will be our f prime of x_1 and then we\u0027ll be able to figure out the equation."},{"Start":"02:23.465 ","End":"02:24.980","Text":"Let\u0027s go back here,"},{"Start":"02:24.980 ","End":"02:26.479","Text":"we getting ahead of ourselves."},{"Start":"02:26.479 ","End":"02:32.315","Text":"So f of minus Pi over 4 is tangent squared,"},{"Start":"02:32.315 ","End":"02:34.085","Text":"I\u0027m just reading of here,"},{"Start":"02:34.085 ","End":"02:37.310","Text":"of minus Pi over 4."},{"Start":"02:37.310 ","End":"02:42.410","Text":"Now, minus Pi over 4 helps me to think in degrees,"},{"Start":"02:42.410 ","End":"02:45.740","Text":"it\u0027s minus 45 degrees."},{"Start":"02:45.740 ","End":"02:50.750","Text":"The tangent of minus 45 degrees,"},{"Start":"02:50.750 ","End":"02:52.880","Text":"which is minus 1."},{"Start":"02:52.880 ","End":"02:57.325","Text":"It\u0027s minus 1 squared, which is 1."},{"Start":"02:57.325 ","End":"03:04.695","Text":"What we have so far is that x_1 is minus Pi over 4."},{"Start":"03:04.695 ","End":"03:07.920","Text":"We\u0027ve just figured out what y_1 is,"},{"Start":"03:07.920 ","End":"03:09.750","Text":"which is equal to 1,"},{"Start":"03:09.750 ","End":"03:12.840","Text":"and finally, we need f prime of x_1,"},{"Start":"03:12.840 ","End":"03:17.835","Text":"which is f prime of minus Pi over 4,"},{"Start":"03:17.835 ","End":"03:20.400","Text":"so that\u0027s this bit here,"},{"Start":"03:20.400 ","End":"03:24.780","Text":"and this is what we\u0027re figuring out now."},{"Start":"03:24.850 ","End":"03:27.650","Text":"What it is is,"},{"Start":"03:27.650 ","End":"03:34.665","Text":"twice tangent of minus Pi over 4"},{"Start":"03:34.665 ","End":"03:42.865","Text":"times 1 over cosine squared of minus Pi over 4."},{"Start":"03:42.865 ","End":"03:47.255","Text":"We\u0027ve already got tangent of minus Pi over 4 over here,"},{"Start":"03:47.255 ","End":"03:49.055","Text":"that was minus 1,"},{"Start":"03:49.055 ","End":"03:54.000","Text":"so no need to compute that again, times; now,"},{"Start":"03:54.000 ","End":"03:59.674","Text":"the cosine of minus Pi over 4 is the same as cosine of Pi over 4."},{"Start":"03:59.674 ","End":"04:05.930","Text":"Cosine of 45 degrees is 1 over the square root of 2,"},{"Start":"04:05.930 ","End":"04:13.759","Text":"so we get twice minus 1 and instead of putting 1 over,"},{"Start":"04:13.759 ","End":"04:16.760","Text":"I\u0027ll just put a dividing line here,"},{"Start":"04:16.760 ","End":"04:26.030","Text":"and cosine squared, and the cosine of minus Pi over 4 is 1 over the square root of 2."},{"Start":"04:26.030 ","End":"04:31.300","Text":"What we get basically is that this is equal"},{"Start":"04:31.300 ","End":"04:34.480","Text":"to 1 over the square root of 2 squared is"},{"Start":"04:34.480 ","End":"04:38.560","Text":"just a 1/2 and dividing by a 1/2 is like multiplying by 2,"},{"Start":"04:38.560 ","End":"04:44.020","Text":"so we get 2 times minus 1 times 2,"},{"Start":"04:44.020 ","End":"04:47.320","Text":"which is equal to minus 4."},{"Start":"04:47.320 ","End":"04:53.785","Text":"Now I can go back up here and write that this thing is minus 4."},{"Start":"04:53.785 ","End":"04:57.765","Text":"This is what I want to substitute in."},{"Start":"04:57.765 ","End":"04:59.670","Text":"I\u0027ve got my x_1,"},{"Start":"04:59.670 ","End":"05:01.905","Text":"I\u0027ve got my y_1,"},{"Start":"05:01.905 ","End":"05:05.144","Text":"and I\u0027ve got my f prime of x_1,"},{"Start":"05:05.144 ","End":"05:12.435","Text":"so I\u0027ve got y minus y_1 is 1 is"},{"Start":"05:12.435 ","End":"05:20.250","Text":"equal to f prime of x_1 we\u0027ve got is minus 4 times x minus;"},{"Start":"05:20.250 ","End":"05:23.610","Text":"and x_1 is minus Pi over 4."},{"Start":"05:23.610 ","End":"05:26.720","Text":"That\u0027s the equation of the tangent,"},{"Start":"05:26.720 ","End":"05:29.330","Text":"but I want to simplify it a bit."},{"Start":"05:29.330 ","End":"05:33.250","Text":"Y is equal to minus 4x."},{"Start":"05:33.250 ","End":"05:37.740","Text":"Now, here we have minus minus Pi over 4,"},{"Start":"05:37.740 ","End":"05:40.560","Text":"which is plus Pi over 4,"},{"Start":"05:40.560 ","End":"05:43.619","Text":"but there\u0027s also a minus 4,"},{"Start":"05:43.619 ","End":"05:48.465","Text":"so it\u0027s altogether minus Pi,"},{"Start":"05:48.465 ","End":"05:51.510","Text":"and finally, plus 1."},{"Start":"05:51.510 ","End":"05:54.120","Text":"We\u0027re bringing this over,"},{"Start":"05:54.120 ","End":"05:59.740","Text":"so this would be the equation of the tangent."},{"Start":"05:59.740 ","End":"06:03.510","Text":"That takes care of part a."},{"Start":"06:03.510 ","End":"06:05.995","Text":"Let\u0027s move on to part b."},{"Start":"06:05.995 ","End":"06:08.435","Text":"But here we\u0027re going to need a sketch."},{"Start":"06:08.435 ","End":"06:10.630","Text":"The graph of tangent squared,"},{"Start":"06:10.630 ","End":"06:14.495","Text":"obviously it\u0027s going to be positive or at least non-negative."},{"Start":"06:14.495 ","End":"06:18.555","Text":"I\u0027ll draw a rough sketch here."},{"Start":"06:18.555 ","End":"06:21.390","Text":"Mostly I need the negative. Let\u0027s see."},{"Start":"06:21.390 ","End":"06:24.450","Text":"I got from, I just take a look again,"},{"Start":"06:24.450 ","End":"06:30.720","Text":"between 0 above and minus Pi over 2 below."},{"Start":"06:30.720 ","End":"06:34.155","Text":"What I have is that here is 0,"},{"Start":"06:34.155 ","End":"06:39.760","Text":"here minus Pi over 2."},{"Start":"06:39.760 ","End":"06:43.295","Text":"Let me see now the tangent squared."},{"Start":"06:43.295 ","End":"06:47.810","Text":"Well, tangent of minus Pi over 2 is minus infinity so"},{"Start":"06:47.810 ","End":"06:53.195","Text":"here it goes to infinity and I better draw it more at the side."},{"Start":"06:53.195 ","End":"06:56.775","Text":"As I say, at minus Pi over 2,"},{"Start":"06:56.775 ","End":"06:59.055","Text":"I have, not that it really matters,"},{"Start":"06:59.055 ","End":"07:04.045","Text":"vertical asymptote that goes to plus infinity at 0,"},{"Start":"07:04.045 ","End":"07:06.695","Text":"tangent x is 0."},{"Start":"07:06.695 ","End":"07:14.960","Text":"I think the graph looks something like this minus Pi over 4, if I take a look,"},{"Start":"07:14.960 ","End":"07:24.445","Text":"we had that the y was equal to 1 and we had that the slope was equal to minus 4,"},{"Start":"07:24.445 ","End":"07:34.230","Text":"y equals or f of x is equal to tangent squared x."},{"Start":"07:34.230 ","End":"07:37.960","Text":"We have this tangent from part a."},{"Start":"07:37.960 ","End":"07:42.985","Text":"This line here is the tangent and that\u0027s its equation."},{"Start":"07:42.985 ","End":"07:45.580","Text":"We want the area bounded by the graph,"},{"Start":"07:45.580 ","End":"07:47.800","Text":"the tangent and the x-axis."},{"Start":"07:47.800 ","End":"07:51.550","Text":"There\u0027s the graph, there\u0027s the x-axis as the tangent."},{"Start":"07:51.550 ","End":"07:55.855","Text":"It looks like the small bit there."},{"Start":"07:55.855 ","End":"08:02.510","Text":"Let me see if I can just draw the outline of it from here, and from here."},{"Start":"08:02.970 ","End":"08:07.885","Text":"This bit here, and all that\u0027s in the middle."},{"Start":"08:07.885 ","End":"08:13.849","Text":"What I suggest is that we take the whole thing,"},{"Start":"08:13.920 ","End":"08:22.120","Text":"which would be including this bit here and here,"},{"Start":"08:22.120 ","End":"08:24.850","Text":"the yellow and here."},{"Start":"08:24.850 ","End":"08:30.280","Text":"We\u0027ll take the whole bit under this curve from here to here,"},{"Start":"08:30.280 ","End":"08:34.510","Text":"and then subtract this triangle here."},{"Start":"08:34.510 ","End":"08:37.285","Text":"Maybe I\u0027ll shade this 1 in."},{"Start":"08:37.285 ","End":"08:47.545","Text":"Also, I\u0027m going to take the integral from here, here and here."},{"Start":"08:47.545 ","End":"08:52.495","Text":"I\u0027m getting too many colors here we get the idea and subtract the yellow triangle,"},{"Start":"08:52.495 ","End":"08:56.540","Text":"the magenta minus the yellow equals the green."},{"Start":"08:57.660 ","End":"09:07.240","Text":"Let\u0027s start with the magenta will officially starting part b and we are going to get"},{"Start":"09:07.240 ","End":"09:12.220","Text":"the magenta bit is just the integral from minus Pi over"},{"Start":"09:12.220 ","End":"09:17.500","Text":"4 to 0 of this function here,"},{"Start":"09:17.500 ","End":"09:23.575","Text":"which is tangent squared of x dx."},{"Start":"09:23.575 ","End":"09:26.200","Text":"We\u0027ll compute this. When we\u0027re done with that,"},{"Start":"09:26.200 ","End":"09:29.300","Text":"we\u0027ll subtract the triangle part."},{"Start":"09:30.120 ","End":"09:34.495","Text":"We were supposed to show that this integral is equal to this."},{"Start":"09:34.495 ","End":"09:38.725","Text":"Let me just take a small break there."},{"Start":"09:38.725 ","End":"09:40.570","Text":"The way we show that this integral is this,"},{"Start":"09:40.570 ","End":"09:43.375","Text":"of course, is just by differentiating this."},{"Start":"09:43.375 ","End":"09:52.150","Text":"If I take tangent x minus x plus c and differentiate it,"},{"Start":"09:52.150 ","End":"10:02.805","Text":"what I get is the derivative of the tangent is 1 over cosine squared x."},{"Start":"10:02.805 ","End":"10:07.245","Text":"This bit is minus 1 and the c gives me nothing."},{"Start":"10:07.245 ","End":"10:14.004","Text":"This is equal to 1 over cosine squared x."},{"Start":"10:14.004 ","End":"10:21.849","Text":"If I write this 1 as cosine squared x over cosine squared x,"},{"Start":"10:21.849 ","End":"10:29.140","Text":"then what I get is 1 minus cosine squared x is sine"},{"Start":"10:29.140 ","End":"10:38.665","Text":"squared x on the denominator, cosine squared x."},{"Start":"10:38.665 ","End":"10:41.485","Text":"Because sine over cosine is tangent,"},{"Start":"10:41.485 ","End":"10:44.530","Text":"this is tangent squared x."},{"Start":"10:44.530 ","End":"10:47.215","Text":"If the derivative of this is this,"},{"Start":"10:47.215 ","End":"10:52.015","Text":"then the indefinite integral of this is this."},{"Start":"10:52.015 ","End":"10:55.540","Text":"Now I can continue and say that this is equal to,"},{"Start":"10:55.540 ","End":"10:58.000","Text":"for definite integrals we don\u0027t need the constant,"},{"Start":"10:58.000 ","End":"11:04.030","Text":"so it\u0027s tangent x minus x."},{"Start":"11:04.030 ","End":"11:10.195","Text":"Evaluate this between minus Pi over 4 and 0."},{"Start":"11:10.195 ","End":"11:12.460","Text":"Let\u0027s see what we get."},{"Start":"11:12.460 ","End":"11:15.370","Text":"We get, if we substitute 0,"},{"Start":"11:15.370 ","End":"11:23.470","Text":"we get tangent 0 is 0 minus 0 less because it\u0027s the lower limit,"},{"Start":"11:23.470 ","End":"11:28.315","Text":"tangent of minus Pi over 4,"},{"Start":"11:28.315 ","End":"11:33.520","Text":"minus, minus Pi over 4."},{"Start":"11:33.520 ","End":"11:37.555","Text":"What does this give us? The first bit gives us nothing."},{"Start":"11:37.555 ","End":"11:43.855","Text":"Tangent of minus Pi over 4 is minus 1."},{"Start":"11:43.855 ","End":"11:50.905","Text":"We\u0027ve got minus of minus 1 is 1."},{"Start":"11:50.905 ","End":"11:53.785","Text":"Here we have minus, minus,"},{"Start":"11:53.785 ","End":"11:55.795","Text":"minus Pi over 4,"},{"Start":"11:55.795 ","End":"11:59.375","Text":"so it\u0027s minus Pi over 4."},{"Start":"11:59.375 ","End":"12:02.355","Text":"That\u0027s the magenta bit."},{"Start":"12:02.355 ","End":"12:04.925","Text":"Now we\u0027re going to go for the yellow."},{"Start":"12:04.925 ","End":"12:07.735","Text":"We just need to find this point also."},{"Start":"12:07.735 ","End":"12:10.435","Text":"That\u0027s another piece of information we need to find."},{"Start":"12:10.435 ","End":"12:11.920","Text":"To find this point,"},{"Start":"12:11.920 ","End":"12:15.520","Text":"we need to see where the tangent hits the x-axis."},{"Start":"12:15.520 ","End":"12:20.740","Text":"Now the equation of the tangent is here and where it hits the x-axis."},{"Start":"12:20.740 ","End":"12:22.105","Text":"To get this point,"},{"Start":"12:22.105 ","End":"12:24.340","Text":"I need to set y equals 0,"},{"Start":"12:24.340 ","End":"12:30.400","Text":"so I get 0 equals minus 4x minus Pi plus 1."},{"Start":"12:30.400 ","End":"12:35.250","Text":"Bring the 4x to the other side and then divide by 4,"},{"Start":"12:35.250 ","End":"12:39.750","Text":"so x equals minus Pi plus 1 over 4."},{"Start":"12:39.750 ","End":"12:42.135","Text":"That\u0027s what this point is here."},{"Start":"12:42.135 ","End":"12:45.180","Text":"What we have here is a triangle."},{"Start":"12:45.180 ","End":"12:48.705","Text":"We get a triangle, this yellow thing,"},{"Start":"12:48.705 ","End":"12:52.499","Text":"where this part here,"},{"Start":"12:52.499 ","End":"12:58.235","Text":"this is the base, let\u0027s call it and this is the height."},{"Start":"12:58.235 ","End":"13:03.895","Text":"The base is the difference, this minus this."},{"Start":"13:03.895 ","End":"13:10.210","Text":"The base is minus Pi over 4 plus a quarter,"},{"Start":"13:10.210 ","End":"13:12.970","Text":"minus Pi over 4,"},{"Start":"13:12.970 ","End":"13:17.875","Text":"plus 1 quarter less minus Pi over 4."},{"Start":"13:17.875 ","End":"13:23.890","Text":"The Pi over 4 cancels and we\u0027re left with 1 quarter."},{"Start":"13:23.890 ","End":"13:29.845","Text":"The height is just the y of this, which is 1."},{"Start":"13:29.845 ","End":"13:32.560","Text":"We have the base and this."},{"Start":"13:32.560 ","End":"13:35.300","Text":"For the yellow bit,"},{"Start":"13:35.640 ","End":"13:43.345","Text":"that what we need is a half base times height,"},{"Start":"13:43.345 ","End":"13:52.015","Text":"and that is equal to 1 half times 1 quarter times 1,"},{"Start":"13:52.015 ","End":"13:55.450","Text":"which is equal to 1 eighth."},{"Start":"13:55.450 ","End":"14:01.525","Text":"Now what I have to do is take this and subtract this."},{"Start":"14:01.525 ","End":"14:06.010","Text":"Then I\u0027ve got the green bit."},{"Start":"14:06.010 ","End":"14:10.015","Text":"That will equal, that\u0027s what we\u0027re looking for,"},{"Start":"14:10.015 ","End":"14:16.375","Text":"1 minus Pi over 4 minus 1 eighth,"},{"Start":"14:16.375 ","End":"14:24.190","Text":"which equals 7 eighths minus Pi over 4."},{"Start":"14:24.190 ","End":"14:29.845","Text":"That is our answer unless you want a decimal approximation."},{"Start":"14:29.845 ","End":"14:37.040","Text":"On the calculator, I make it approximately equal to 0.09."},{"Start":"14:37.410 ","End":"14:42.715","Text":"This is the answer for part b."},{"Start":"14:42.715 ","End":"14:46.460","Text":"Finally done with this question."}],"Thumbnail":null,"ID":4711},{"Watched":false,"Name":"Exercise 16","Duration":"16m 15s","ChapterTopicVideoID":4704,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"I have a word problem involving computations of"},{"Start":"00:03.945 ","End":"00:10.230","Text":"area by means of definite integrals and tangents and so forth."},{"Start":"00:10.230 ","End":"00:14.100","Text":"I\u0027m really missing a picture, a diagram."},{"Start":"00:14.100 ","End":"00:17.610","Text":"Let\u0027s see what we can do before we start sketching."},{"Start":"00:17.610 ","End":"00:21.180","Text":"What I notice is the main thing is that we are given a parabola,"},{"Start":"00:21.180 ","End":"00:25.005","Text":"which is x squared minus 10x plus 25."},{"Start":"00:25.005 ","End":"00:29.865","Text":"Now most of you will probably recognize that this is a perfect square."},{"Start":"00:29.865 ","End":"00:35.745","Text":"That this is equal to x minus 5 all squared."},{"Start":"00:35.745 ","End":"00:38.700","Text":"Because when half this coefficient squared is this,"},{"Start":"00:38.700 ","End":"00:41.770","Text":"then we can make it something squared."},{"Start":"00:41.770 ","End":"00:45.140","Text":"Now we know that it only has 1 root,"},{"Start":"00:45.140 ","End":"00:48.185","Text":"meaning it only touches the x-axis once."},{"Start":"00:48.185 ","End":"00:51.050","Text":"It\u0027s a tangent at the x-axis,"},{"Start":"00:51.050 ","End":"00:53.540","Text":"and also it\u0027s an upward facing parabola."},{"Start":"00:53.540 ","End":"00:59.010","Text":"All in all, we can get the general shape to be something like this,"},{"Start":"00:59.010 ","End":"01:01.230","Text":"and here it is."},{"Start":"01:01.230 ","End":"01:03.675","Text":"This is a general shape of the parabola,"},{"Start":"01:03.675 ","End":"01:10.565","Text":"but it has a vertex on the x-axis and the x-axis is tangent to it here,"},{"Start":"01:10.565 ","End":"01:15.900","Text":"and that is the point where x is equal to 5,"},{"Start":"01:15.900 ","End":"01:17.390","Text":"that\u0027s the 5 here."},{"Start":"01:17.390 ","End":"01:19.520","Text":"This is going to help us a lot."},{"Start":"01:19.520 ","End":"01:23.180","Text":"Now we\u0027re talking about another point A,"},{"Start":"01:23.180 ","End":"01:25.910","Text":"also on the x-axis where x is 8,"},{"Start":"01:25.910 ","End":"01:29.310","Text":"so let\u0027s draw here 8."},{"Start":"01:29.310 ","End":"01:32.190","Text":"This is actually the point A,"},{"Start":"01:32.190 ","End":"01:35.030","Text":"and matters we\u0027ll label this one also."},{"Start":"01:35.030 ","End":"01:36.290","Text":"Let\u0027s call this one B,"},{"Start":"01:36.290 ","End":"01:38.600","Text":"which is the vertex of the parabola."},{"Start":"01:38.600 ","End":"01:43.990","Text":"We\u0027re told that through this point A tangents,"},{"Start":"01:43.990 ","End":"01:46.715","Text":"plural, are drawn to the parabola,"},{"Start":"01:46.715 ","End":"01:49.385","Text":"which means there must be more than 1 tangent."},{"Start":"01:49.385 ","End":"01:51.095","Text":"Well, of course there is,"},{"Start":"01:51.095 ","End":"01:55.280","Text":"because I\u0027ve already given it away that the x-axis is 1 tangent."},{"Start":"01:55.280 ","End":"01:57.185","Text":"Here\u0027s 1 tangent."},{"Start":"01:57.185 ","End":"02:00.350","Text":"Notice that we can also draw a line somewhere here"},{"Start":"02:00.350 ","End":"02:03.820","Text":"that will just graze the parabola and I\u0027ll draw that one,"},{"Start":"02:03.820 ","End":"02:12.050","Text":"and here\u0027s my second tangent to the parabola from this point A."},{"Start":"02:12.050 ","End":"02:17.180","Text":"What we have to do now is to find the equations of the tangents."},{"Start":"02:17.180 ","End":"02:20.915","Text":"I\u0027d like to mark in some other interesting points,"},{"Start":"02:20.915 ","End":"02:23.510","Text":"the tangent makes a contact at some point"},{"Start":"02:23.510 ","End":"02:26.180","Text":"it\u0027s hard to see exactly where it doesn\u0027t matter."},{"Start":"02:26.180 ","End":"02:28.585","Text":"Let\u0027s say it\u0027s here."},{"Start":"02:28.585 ","End":"02:32.700","Text":"That will be another point, call it C,"},{"Start":"02:32.700 ","End":"02:40.535","Text":"and the other interesting point will be the point on the x-axis just below it,"},{"Start":"02:40.535 ","End":"02:45.855","Text":"and I\u0027ll call that point D, there it is."},{"Start":"02:45.855 ","End":"02:49.400","Text":"In part A, I\u0027ll start with the easier ones,"},{"Start":"02:49.400 ","End":"02:51.785","Text":"the green horizontal one."},{"Start":"02:51.785 ","End":"02:54.695","Text":"There\u0027s really nothing to say here except that"},{"Start":"02:54.695 ","End":"02:59.045","Text":"the equation of the tangent is just y equals 0."},{"Start":"02:59.045 ","End":"03:01.845","Text":"Y equals 0 is the x-axis,"},{"Start":"03:01.845 ","End":"03:08.555","Text":"and that is the solution for the first tangent of part A."},{"Start":"03:08.555 ","End":"03:10.850","Text":"Now let\u0027s go for the other one,"},{"Start":"03:10.850 ","End":"03:15.680","Text":"which I will draw just to indicate which I\u0027m going for."},{"Start":"03:15.680 ","End":"03:17.450","Text":"In order to get that,"},{"Start":"03:17.450 ","End":"03:25.310","Text":"what I\u0027m going to do is find the equation of this tangent in terms of the point C,"},{"Start":"03:25.310 ","End":"03:27.370","Text":"and let\u0027s label it."},{"Start":"03:27.370 ","End":"03:29.640","Text":"Let\u0027s call this point t,"},{"Start":"03:29.640 ","End":"03:31.260","Text":"which is an unknown,"},{"Start":"03:31.260 ","End":"03:35.219","Text":"and C will be the point t,"},{"Start":"03:35.219 ","End":"03:43.230","Text":"and then t squared minus 10t plus 25."},{"Start":"03:43.230 ","End":"03:47.875","Text":"That will give us C, it\u0027ll also give us D, it\u0027s t, 0."},{"Start":"03:47.875 ","End":"03:53.800","Text":"The general equation for tangent to a curve at a point goes like this,"},{"Start":"03:53.800 ","End":"04:03.375","Text":"y minus y_1 equals f prime of x_1 times x minus x_1."},{"Start":"04:03.375 ","End":"04:06.615","Text":"That\u0027s one of the basic formulas."},{"Start":"04:06.615 ","End":"04:09.570","Text":"In our case, the x_1,"},{"Start":"04:09.570 ","End":"04:14.205","Text":"y_1 is the point C. This will be my x_1,"},{"Start":"04:14.205 ","End":"04:16.335","Text":"this will be my y_1,"},{"Start":"04:16.335 ","End":"04:19.540","Text":"the only thing I\u0027m missing is f prime of x_1,"},{"Start":"04:19.540 ","End":"04:23.480","Text":"which is the slope at the point C. We need to do a bit of"},{"Start":"04:23.480 ","End":"04:27.530","Text":"differentiation and we don\u0027t even see an f here,"},{"Start":"04:27.530 ","End":"04:29.450","Text":"well of course when y is this,"},{"Start":"04:29.450 ","End":"04:36.690","Text":"we\u0027ll take this to equal also f of x. I\u0027ll copy it,"},{"Start":"04:36.690 ","End":"04:42.200","Text":"f of x is equal to x squared minus 10x plus 25,"},{"Start":"04:42.200 ","End":"04:47.600","Text":"which gives us that f prime of x in general is 2x minus 10,"},{"Start":"04:47.600 ","End":"04:50.810","Text":"which means that f prime of our x_1,"},{"Start":"04:50.810 ","End":"04:52.780","Text":"but our x_1 is t,"},{"Start":"04:52.780 ","End":"05:00.595","Text":"f prime of t is equal to just 2t minus 10,"},{"Start":"05:00.595 ","End":"05:04.505","Text":"and f prime of t is the f prime of x_1 here."},{"Start":"05:04.505 ","End":"05:08.150","Text":"I have x_1 which is t,"},{"Start":"05:08.150 ","End":"05:11.765","Text":"I have y_1 which is this,"},{"Start":"05:11.765 ","End":"05:17.639","Text":"and I have f prime of x_1 which is this,"},{"Start":"05:17.639 ","End":"05:21.479","Text":"so y minus y_1,"},{"Start":"05:21.479 ","End":"05:28.920","Text":"which is t squared minus 10t plus 25 is equal to f prime of x_1,"},{"Start":"05:28.920 ","End":"05:31.110","Text":"and we said that x_1 is t,"},{"Start":"05:31.110 ","End":"05:40.140","Text":"so it\u0027s just this 2t minus 10 times x minus,"},{"Start":"05:40.140 ","End":"05:42.225","Text":"and our x_1 is t,"},{"Start":"05:42.225 ","End":"05:46.085","Text":"and this gives us the equation of a straight line."},{"Start":"05:46.085 ","End":"05:47.840","Text":"We\u0027ve got x and y here,"},{"Start":"05:47.840 ","End":"05:54.170","Text":"and it\u0027s linear but with a parameter t. Let\u0027s bring it to more normal form,"},{"Start":"05:54.170 ","End":"05:59.210","Text":"will say that y equals and put everything on the other side."},{"Start":"05:59.210 ","End":"06:03.800","Text":"Now, the coefficient of x is only going to come from here,"},{"Start":"06:03.800 ","End":"06:07.945","Text":"so we\u0027re going to get 2t minus 10x,"},{"Start":"06:07.945 ","End":"06:13.260","Text":"and then we\u0027re going to have 2t minus 10 times minus t,"},{"Start":"06:13.260 ","End":"06:18.240","Text":"so it\u0027s minus 2t squared,"},{"Start":"06:18.240 ","End":"06:20.835","Text":"and then plus 10t."},{"Start":"06:20.835 ","End":"06:25.195","Text":"Then what I get from bringing this over to the other side with a plus,"},{"Start":"06:25.195 ","End":"06:30.850","Text":"plus t squared minus 10t plus 25."},{"Start":"06:30.850 ","End":"06:33.310","Text":"Let\u0027s see if anything cancels, yes,"},{"Start":"06:33.310 ","End":"06:37.525","Text":"this 10t and this minus 10t cancel."},{"Start":"06:37.525 ","End":"06:42.730","Text":"Also minus 2t squared plus t squared is just minus t squared,"},{"Start":"06:42.730 ","End":"06:47.520","Text":"so I\u0027ll cancel this with the 2 so to speak."},{"Start":"06:47.520 ","End":"06:51.435","Text":"It\u0027s correct, it\u0027s not orthodox and you\u0027ll forgive me."},{"Start":"06:51.435 ","End":"06:58.980","Text":"What we get is that y equals 2t minus 10x,"},{"Start":"06:58.980 ","End":"07:05.110","Text":"and all that we\u0027re left with here is minus t squared plus 25."},{"Start":"07:05.110 ","End":"07:12.130","Text":"This would be the equation of the tangent whenever the x of the point of contact is"},{"Start":"07:12.130 ","End":"07:19.630","Text":"t. Now I\u0027m looking for the intersection of the tangent with the x-axis,"},{"Start":"07:19.630 ","End":"07:22.420","Text":"which means that y equals 0."},{"Start":"07:22.420 ","End":"07:27.855","Text":"Then I get that 0 equals the same thing,"},{"Start":"07:27.855 ","End":"07:32.740","Text":"and then I can bring this stuff to the other side and leave this here,"},{"Start":"07:32.740 ","End":"07:43.835","Text":"so I\u0027ve got 2t minus 10x equals t squared minus 25."},{"Start":"07:43.835 ","End":"07:47.085","Text":"Notice that this thing is 0 when t is 5."},{"Start":"07:47.085 ","End":"07:50.325","Text":"Well, let\u0027s just assume that t is not equal to 5."},{"Start":"07:50.325 ","End":"07:54.525","Text":"We can assume this t is not 5 because we\u0027ve got 5 covered already,"},{"Start":"07:54.525 ","End":"07:57.780","Text":"and that\u0027s point B and that\u0027s the green tangent."},{"Start":"07:57.780 ","End":"08:01.920","Text":"Now we\u0027re looking for the magenta tangent, so it\u0027s fine,"},{"Start":"08:01.920 ","End":"08:06.854","Text":"t not equal to 5, and then I can take t minus 5."},{"Start":"08:06.854 ","End":"08:13.709","Text":"Basically what I\u0027m saying is this is 2 t minus 5 x and this is a difference of squares,"},{"Start":"08:13.709 ","End":"08:18.239","Text":"so it\u0027s t minus 5, t plus 5,"},{"Start":"08:18.239 ","End":"08:20.490","Text":"and because it\u0027s not 0,"},{"Start":"08:20.490 ","End":"08:22.290","Text":"this t minus 5,"},{"Start":"08:22.290 ","End":"08:25.260","Text":"I can cancel it on both sides."},{"Start":"08:25.260 ","End":"08:31.455","Text":"I\u0027ve got x is equal to t plus 5 over 2,"},{"Start":"08:31.455 ","End":"08:34.905","Text":"but I know that x is equal to 8."},{"Start":"08:34.905 ","End":"08:37.350","Text":"Actually, I could\u0027ve done it a bit better,"},{"Start":"08:37.350 ","End":"08:41.115","Text":"I should have substituted at the same time y 0,"},{"Start":"08:41.115 ","End":"08:45.930","Text":"I could have substituted x equals 8 already here,"},{"Start":"08:45.930 ","End":"08:47.565","Text":"pretty much the same."},{"Start":"08:47.565 ","End":"08:53.415","Text":"In any event, now I get the equation that this is going to equal to 8,"},{"Start":"08:53.415 ","End":"08:56.175","Text":"and then t plus 5 over 2 is 8,"},{"Start":"08:56.175 ","End":"08:59.055","Text":"t plus 5 is 16,"},{"Start":"08:59.055 ","End":"09:02.940","Text":"and so t is equal to 11,"},{"Start":"09:02.940 ","End":"09:06.705","Text":"so I get t is equal to 11."},{"Start":"09:06.705 ","End":"09:10.230","Text":"But I have the equation of the tangent in terms of t,"},{"Start":"09:10.230 ","End":"09:11.790","Text":"and this line here,"},{"Start":"09:11.790 ","End":"09:13.560","Text":"this is after I\u0027ve tidied it up."},{"Start":"09:13.560 ","End":"09:17.265","Text":"All I have to do is put t equals 11 here,"},{"Start":"09:17.265 ","End":"09:26.220","Text":"and then I\u0027ll get that y equals 12x minus 96,"},{"Start":"09:26.220 ","End":"09:28.380","Text":"so I\u0027m going to highlight this."},{"Start":"09:28.380 ","End":"09:31.920","Text":"This is my magenta tangent,"},{"Start":"09:31.920 ","End":"09:35.310","Text":"and just to be consistent,"},{"Start":"09:35.310 ","End":"09:40.770","Text":"green tangent gets the green equation and the magenta tangent gets the magenta equation,"},{"Start":"09:40.770 ","End":"09:42.765","Text":"and isn\u0027t that nice."},{"Start":"09:42.765 ","End":"09:46.365","Text":"Now we have the equation of the 2 tangents,"},{"Start":"09:46.365 ","End":"09:48.915","Text":"and now we get on to part b."},{"Start":"09:48.915 ","End":"09:51.300","Text":"In part b, if we read it,"},{"Start":"09:51.300 ","End":"09:56.385","Text":"it\u0027s to compute the area bounded by the tangents on the parabola."},{"Start":"09:56.385 ","End":"09:59.805","Text":"The tangents, there\u0027s 2 of them,"},{"Start":"09:59.805 ","End":"10:01.920","Text":"and the parabola, that\u0027s this bit,"},{"Start":"10:01.920 ","End":"10:10.515","Text":"so we\u0027re talking about here up to the point C and then along the tangent,"},{"Start":"10:10.515 ","End":"10:14.220","Text":"and then along the x-axis."},{"Start":"10:14.220 ","End":"10:16.410","Text":"I\u0027ll draw a little s here,"},{"Start":"10:16.410 ","End":"10:20.730","Text":"and this is the area that we have to compute."},{"Start":"10:20.730 ","End":"10:29.100","Text":"Now what I suggest as a strategy is that we take the whole bit,"},{"Start":"10:29.100 ","End":"10:30.630","Text":"including, in other words,"},{"Start":"10:30.630 ","End":"10:33.000","Text":"we\u0027ll take B, C, D,"},{"Start":"10:33.000 ","End":"10:35.670","Text":"and then subtract from it A,"},{"Start":"10:35.670 ","End":"10:38.820","Text":"C, D. I want to take, first of all,"},{"Start":"10:38.820 ","End":"10:41.580","Text":"the larger bit from here,"},{"Start":"10:41.580 ","End":"10:44.460","Text":"and then to here,"},{"Start":"10:44.460 ","End":"10:49.095","Text":"and then along here,"},{"Start":"10:49.095 ","End":"10:54.645","Text":"then subtract the bit that\u0027s in the triangle,"},{"Start":"10:54.645 ","End":"11:01.215","Text":"and that will leave me this s. I think the idea is clear,"},{"Start":"11:01.215 ","End":"11:05.760","Text":"we get the smaller area by taking a larger one and subtracting a triangle,"},{"Start":"11:05.760 ","End":"11:09.300","Text":"and the triangle will be able to do with geometry without integration."},{"Start":"11:09.300 ","End":"11:12.210","Text":"The other one will require an integral."},{"Start":"11:12.210 ","End":"11:16.860","Text":"The whole yellow bit will just be the integral of"},{"Start":"11:16.860 ","End":"11:23.250","Text":"the parabola function between what and what?"},{"Start":"11:23.250 ","End":"11:27.000","Text":"Well, we\u0027ve already found that t is equal to 11."},{"Start":"11:27.000 ","End":"11:31.380","Text":"I need to know this height because I\u0027m going to be using geometry here,"},{"Start":"11:31.380 ","End":"11:37.125","Text":"so this height is the y coordinate of C,"},{"Start":"11:37.125 ","End":"11:39.225","Text":"and I\u0027ll need that,"},{"Start":"11:39.225 ","End":"11:47.310","Text":"so I\u0027ll have to plug it in to this expression and the answer comes out to be 36."},{"Start":"11:47.310 ","End":"11:51.915","Text":"In other words, this is the point 11, 36,"},{"Start":"11:51.915 ","End":"11:58.050","Text":"which means that the height is 36 and the base will be,"},{"Start":"11:58.050 ","End":"12:00.510","Text":"and I\u0027ll mark it on the inside, will be 3,"},{"Start":"12:00.510 ","End":"12:03.930","Text":"so when we do the area of this triangle that I shaded,"},{"Start":"12:03.930 ","End":"12:05.580","Text":"then we\u0027ll use that."},{"Start":"12:05.580 ","End":"12:07.830","Text":"Which actually brings me the point of,"},{"Start":"12:07.830 ","End":"12:10.570","Text":"why not do the triangle first?"},{"Start":"12:11.090 ","End":"12:15.120","Text":"The area of the triangle,"},{"Start":"12:15.120 ","End":"12:17.880","Text":"which I\u0027ll call A, C,"},{"Start":"12:17.880 ","End":"12:22.110","Text":"D, is equal to 1/2 base times height,"},{"Start":"12:22.110 ","End":"12:25.875","Text":"1/2 times 3 times 36."},{"Start":"12:25.875 ","End":"12:30.810","Text":"Let\u0027s see, 1/2 of 36 is 18 times 3 is 54,"},{"Start":"12:30.810 ","End":"12:34.950","Text":"so we must remember to subtract this at the end when we"},{"Start":"12:34.950 ","End":"12:39.630","Text":"take the integral of the parabola from 5-11."},{"Start":"12:39.630 ","End":"12:41.940","Text":"Now what I really need,"},{"Start":"12:41.940 ","End":"12:43.785","Text":"it\u0027s not a triangle,"},{"Start":"12:43.785 ","End":"12:47.100","Text":"but it\u0027s certainly B, C,"},{"Start":"12:47.100 ","End":"12:50.999","Text":"D, which is the outer yellow."},{"Start":"12:50.999 ","End":"12:55.675","Text":"This is equal to the integral from 5-11."},{"Start":"12:55.675 ","End":"13:04.165","Text":"This function here, which I can take as x squared minus 10x plus 25,"},{"Start":"13:04.165 ","End":"13:08.790","Text":"and then dx, so let\u0027s see,"},{"Start":"13:08.790 ","End":"13:10.560","Text":"it\u0027s an easy enough integral."},{"Start":"13:10.560 ","End":"13:18.270","Text":"What I get is x cubed over 3 minus 10x squared over 2,"},{"Start":"13:18.270 ","End":"13:21.720","Text":"which makes it 5x squared plus 25x,"},{"Start":"13:21.720 ","End":"13:28.900","Text":"and this has got to be taken between 5 and 11."},{"Start":"13:29.450 ","End":"13:33.030","Text":"Let\u0027s first of all put 11 in."},{"Start":"13:33.030 ","End":"13:37.559","Text":"11, makes it 11 cubed over 3,"},{"Start":"13:37.559 ","End":"13:43.170","Text":"which is 1,331 over 3,"},{"Start":"13:43.170 ","End":"13:46.620","Text":"minus 5 times 11 squared,"},{"Start":"13:46.620 ","End":"13:55.230","Text":"5 times 121 plus 25 times 11."},{"Start":"13:55.230 ","End":"13:59.070","Text":"That\u0027s that, and then I\u0027ll do the other 1 and"},{"Start":"13:59.070 ","End":"14:03.720","Text":"subtract or I\u0027ll just continue on the next line."},{"Start":"14:03.720 ","End":"14:06.180","Text":"With the 5, we\u0027re going to subtract,"},{"Start":"14:06.180 ","End":"14:08.955","Text":"so we have 5 cubed over 3,"},{"Start":"14:08.955 ","End":"14:12.420","Text":"which is 125 over 3."},{"Start":"14:12.420 ","End":"14:14.715","Text":"But I\u0027m going to subtract that."},{"Start":"14:14.715 ","End":"14:18.480","Text":"Everything\u0027s going to be reversed in sign when I\u0027m subtracting."},{"Start":"14:18.480 ","End":"14:28.080","Text":"Then with the 5, I\u0027m going to get a plus 5 times 5 squared is 5 cubed is 125,"},{"Start":"14:28.080 ","End":"14:36.390","Text":"and plus, and here the plus becomes a minus 25 times 5, again 125."},{"Start":"14:36.390 ","End":"14:39.630","Text":"Let\u0027s see what we can get from here,"},{"Start":"14:39.630 ","End":"14:41.085","Text":"it\u0027s a lot of arithmetic,"},{"Start":"14:41.085 ","End":"14:44.460","Text":"but it\u0027s tedious but not difficult."},{"Start":"14:44.460 ","End":"14:47.715","Text":"Let\u0027s see, if I subtract these 2,"},{"Start":"14:47.715 ","End":"14:48.420","Text":"I will"},{"Start":"14:48.420 ","End":"14:50.080","Text":"get"},{"Start":"14:57.230 ","End":"14:59.250","Text":"1,206"},{"Start":"14:59.250 ","End":"15:07.335","Text":"over 3 minus 5 times 121 is 605,"},{"Start":"15:07.335 ","End":"15:14.175","Text":"25 times 11 is 275."},{"Start":"15:14.175 ","End":"15:19.575","Text":"The 125s cancel, so we get"},{"Start":"15:19.575 ","End":"15:26.580","Text":"402 plus 275 minus 605."},{"Start":"15:26.580 ","End":"15:37.290","Text":"This plus this will give me 677 minus 605,"},{"Start":"15:37.290 ","End":"15:43.480","Text":"which equals 77 minus 5 is 72."},{"Start":"15:43.520 ","End":"15:47.885","Text":"That\u0027s the area of this BCD."},{"Start":"15:47.885 ","End":"15:50.104","Text":"But we\u0027re not quite done."},{"Start":"15:50.104 ","End":"15:56.765","Text":"Because remember, we have to subtract this 54, so ultimately,"},{"Start":"15:56.765 ","End":"16:07.625","Text":"what I get is 72 minus 54 is equal to 18."},{"Start":"16:07.625 ","End":"16:13.580","Text":"This will be the area and the end of part B and D,"},{"Start":"16:13.580 ","End":"16:16.440","Text":"and of the question."}],"Thumbnail":null,"ID":4712},{"Watched":false,"Name":"Exercise 17","Duration":"14m 3s","ChapterTopicVideoID":4705,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"Here we have another 1 of these word problems with"},{"Start":"00:02.940 ","End":"00:07.605","Text":"graphs and tangents and areas and so on."},{"Start":"00:07.605 ","End":"00:10.260","Text":"Let\u0027s see what\u0027s going on here."},{"Start":"00:10.260 ","End":"00:14.790","Text":"We\u0027re given a function f of x and this is its equation."},{"Start":"00:14.790 ","End":"00:17.550","Text":"It\u0027s on the domain x is bigger or equal to 0,"},{"Start":"00:17.550 ","End":"00:20.070","Text":"so only on the right side of the y-axis."},{"Start":"00:20.070 ","End":"00:25.890","Text":"This makes sense because the square root of x is only defined for x bigger or equal to 0,"},{"Start":"00:25.890 ","End":"00:27.974","Text":"so that\u0027s only natural."},{"Start":"00:27.974 ","End":"00:33.570","Text":"In part a, we have to find the equation of the line passing through the origin."},{"Start":"00:33.570 ","End":"00:37.680","Text":"Let\u0027s first of all, mark the origin that\u0027s here at 0,0"},{"Start":"00:37.680 ","End":"00:42.495","Text":"and it\u0027s got to be tangent to the graph here."},{"Start":"00:42.495 ","End":"00:46.850","Text":"I\u0027ve got to somehow draw a line here which is just going to graze it,"},{"Start":"00:46.850 ","End":"00:48.500","Text":"touch it at 1 point."},{"Start":"00:48.500 ","End":"00:51.775","Text":"Let\u0027s see if I can do that."},{"Start":"00:51.775 ","End":"00:59.510","Text":"Here\u0027s my sketch of the tangent and it has a point of contact, let\u0027s say here."},{"Start":"00:59.510 ","End":"01:05.960","Text":"This is a point we\u0027re going to need and we\u0027re also going to need for later on,"},{"Start":"01:05.960 ","End":"01:08.285","Text":"the point directly below."},{"Start":"01:08.285 ","End":"01:11.580","Text":"We\u0027ll give it a name later if need be."},{"Start":"01:11.830 ","End":"01:16.220","Text":"That\u0027s the picture and for part a we have to"},{"Start":"01:16.220 ","End":"01:20.390","Text":"find the equation of the tangent of the red line here."},{"Start":"01:20.390 ","End":"01:23.090","Text":"In part b, we\u0027re going to compute an area."},{"Start":"01:23.090 ","End":"01:25.060","Text":"Let\u0027s see which area we\u0027re talking about."},{"Start":"01:25.060 ","End":"01:26.860","Text":"It\u0027s bounded by the graph of the function,"},{"Start":"01:26.860 ","End":"01:28.630","Text":"the tangent and the y-axis."},{"Start":"01:28.630 ","End":"01:30.250","Text":"Here\u0027s the y-axis."},{"Start":"01:30.250 ","End":"01:32.775","Text":"Here\u0027s the tangent, here\u0027s the graph."},{"Start":"01:32.775 ","End":"01:36.285","Text":"Let me shade the area I think they mean."},{"Start":"01:36.285 ","End":"01:43.720","Text":"This is the area we have to compute and I\u0027ll call it S. I\u0027ll let you know now that"},{"Start":"01:43.720 ","End":"01:51.335","Text":"the strategy will be to compute the area below this curve between these 2 limits."},{"Start":"01:51.335 ","End":"01:53.805","Text":"That will give us something larger,"},{"Start":"01:53.805 ","End":"01:55.720","Text":"but then we\u0027ll subtract the triangle."},{"Start":"01:55.720 ","End":"01:58.525","Text":"It\u0027s a standard trick and then we\u0027ll get this."},{"Start":"01:58.525 ","End":"02:00.550","Text":"This we\u0027ll do with an integration."},{"Start":"02:00.550 ","End":"02:03.555","Text":"The triangle we\u0027ll do with regular geometry and so on."},{"Start":"02:03.555 ","End":"02:08.180","Text":"Let\u0027s do first things first and let\u0027s get on to part a and find"},{"Start":"02:08.180 ","End":"02:10.160","Text":"the equation of this line passing through"},{"Start":"02:10.160 ","End":"02:14.420","Text":"the origin and it\u0027s tangent to the given function."},{"Start":"02:14.420 ","End":"02:18.715","Text":"What we need is to label this point."},{"Start":"02:18.715 ","End":"02:22.340","Text":"Let\u0027s say that the first coordinate,"},{"Start":"02:22.340 ","End":"02:28.040","Text":"the x-coordinate of this point is t. Then this point will"},{"Start":"02:28.040 ","End":"02:35.325","Text":"be because of the equation t,t square root of t plus 4,"},{"Start":"02:35.325 ","End":"02:40.170","Text":"t root t plus 4."},{"Start":"02:40.170 ","End":"02:44.900","Text":"Any line that\u0027s passing through the origin is just going to be"},{"Start":"02:44.900 ","End":"02:50.000","Text":"a line of the form y equals mx,"},{"Start":"02:50.000 ","End":"02:53.434","Text":"where m is the slope of the line."},{"Start":"02:53.434 ","End":"02:56.600","Text":"Other words, m is y over x."},{"Start":"02:56.600 ","End":"02:59.905","Text":"Basically, we can get the slope very easily."},{"Start":"02:59.905 ","End":"03:03.680","Text":"I\u0027ll just write that, that this slope,"},{"Start":"03:03.680 ","End":"03:05.620","Text":"which I\u0027ll call m,"},{"Start":"03:05.620 ","End":"03:10.590","Text":"is going to be just the y of this point minus the x at this point."},{"Start":"03:10.590 ","End":"03:15.930","Text":"It\u0027s going to be t root t plus 4 over"},{"Start":"03:15.930 ","End":"03:23.440","Text":"t. I\u0027ll leave it like that for the moment though later we\u0027ll possibly simplify it."},{"Start":"03:23.440 ","End":"03:26.115","Text":"That\u0027s the slope on the 1 hand."},{"Start":"03:26.115 ","End":"03:28.310","Text":"Of course, you\u0027d get the same result if you use"},{"Start":"03:28.310 ","End":"03:31.550","Text":"the formula for slope between 2 given points."},{"Start":"03:31.550 ","End":"03:33.800","Text":"In fact, you\u0027d get exactly this,"},{"Start":"03:33.800 ","End":"03:39.110","Text":"but with a minus 0 here and a minus 0 on the denominator."},{"Start":"03:39.110 ","End":"03:44.060","Text":"If you use the formula for the y minus the y of the point and you call this 1,"},{"Start":"03:44.060 ","End":"03:46.055","Text":"0,0 and so on and so on."},{"Start":"03:46.055 ","End":"03:48.200","Text":"But there was not necessarily and in fact,"},{"Start":"03:48.200 ","End":"03:51.295","Text":"I\u0027m going to just erase that."},{"Start":"03:51.295 ","End":"03:54.710","Text":"I\u0027m going to compute the slope another way and then we\u0027ll get"},{"Start":"03:54.710 ","End":"03:57.500","Text":"an equation in t. The slope,"},{"Start":"03:57.500 ","End":"04:04.530","Text":"as you remember, of a tangent is also the derivative of the function at the point."},{"Start":"04:04.530 ","End":"04:08.955","Text":"What I\u0027m going to do is differentiate f of x,"},{"Start":"04:08.955 ","End":"04:12.870","Text":"so f of x is equal to,"},{"Start":"04:12.870 ","End":"04:15.080","Text":"for the purposes of differentiation,"},{"Start":"04:15.080 ","End":"04:19.640","Text":"I\u0027d rather not leave it as x root x. Root x is x to the 1/2 or 0.5."},{"Start":"04:19.640 ","End":"04:21.770","Text":"I slightly prefer fractions,"},{"Start":"04:21.770 ","End":"04:26.680","Text":"so I\u0027ll write it as x to the 1 and 1/2 plus 4,"},{"Start":"04:26.680 ","End":"04:30.590","Text":"so f prime of x is equal to,"},{"Start":"04:30.590 ","End":"04:32.660","Text":"put the coefficient in front,"},{"Start":"04:32.660 ","End":"04:35.570","Text":"reduce the coefficient by 1,"},{"Start":"04:35.570 ","End":"04:38.945","Text":"and the constant comes out to be 0."},{"Start":"04:38.945 ","End":"04:42.200","Text":"Now, at the point where x equals t,"},{"Start":"04:42.200 ","End":"04:46.820","Text":"I know that the slope is the same as the slope of this tangent."},{"Start":"04:46.820 ","End":"04:54.675","Text":"We know that f prime of t is equal to the same slope m,"},{"Start":"04:54.675 ","End":"05:01.050","Text":"but it\u0027s also equal to 1 and 1/2 t to the power of a 1/2,"},{"Start":"05:01.050 ","End":"05:05.115","Text":"and t to the power of a half is 3 over 2,"},{"Start":"05:05.115 ","End":"05:09.290","Text":"so 3 over 2 times the square root of t. Because now I\u0027m going to"},{"Start":"05:09.290 ","End":"05:13.490","Text":"equate these to each of them is equal to the slope m and that"},{"Start":"05:13.490 ","End":"05:18.480","Text":"gives me t square root of t plus 4 over"},{"Start":"05:18.480 ","End":"05:25.335","Text":"t is equal to 3 square root of t over 2,"},{"Start":"05:25.335 ","End":"05:27.680","Text":"and when we have an equation like this,"},{"Start":"05:27.680 ","End":"05:30.035","Text":"we can cross multiply."},{"Start":"05:30.035 ","End":"05:36.815","Text":"We get 2t square root of t plus 8 from this diagonal,"},{"Start":"05:36.815 ","End":"05:38.450","Text":"and from this diagonal,"},{"Start":"05:38.450 ","End":"05:43.470","Text":"we get equals t times 3 root t is just"},{"Start":"05:43.470 ","End":"05:51.750","Text":"3t root t. If I collect like terms together,"},{"Start":"05:51.750 ","End":"05:54.740","Text":"3 of these minus 2 of these is 1 of these,"},{"Start":"05:54.740 ","End":"06:00.800","Text":"we get just that t root t is equal"},{"Start":"06:00.800 ","End":"06:08.990","Text":"to 8 and t root t is actually t to the 3 over 2,"},{"Start":"06:08.990 ","End":"06:15.130","Text":"and then you write its root t cubed because t is also root t times root t,"},{"Start":"06:15.130 ","End":"06:18.245","Text":"or I can write it to the power of 1 and 1/2 or 3 over 2."},{"Start":"06:18.245 ","End":"06:20.930","Text":"Any event, this is what we have,"},{"Start":"06:20.930 ","End":"06:26.480","Text":"is equal to 8 and 8 I can write as 2 cubed."},{"Start":"06:26.480 ","End":"06:30.860","Text":"Of course you could probably do all this in the calculator with logarithms and so on,"},{"Start":"06:30.860 ","End":"06:33.440","Text":"but I just want to show you a simple way,"},{"Start":"06:33.440 ","End":"06:37.460","Text":"and that means that the square root of t is equal to"},{"Start":"06:37.460 ","End":"06:41.420","Text":"2 because with cube roots or even powered,"},{"Start":"06:41.420 ","End":"06:44.105","Text":"we don\u0027t get the plus or minus phenomenon."},{"Start":"06:44.105 ","End":"06:45.875","Text":"Square root of t is 2,"},{"Start":"06:45.875 ","End":"06:50.549","Text":"that means that t is going to be equal to 4,"},{"Start":"06:50.549 ","End":"06:56.720","Text":"so t equals 4 and that\u0027s an important intermediate result."},{"Start":"06:56.720 ","End":"07:00.200","Text":"What it means is that we can go to the picture."},{"Start":"07:00.200 ","End":"07:03.680","Text":"What we\u0027re trying to get actually are 3 things, we need,"},{"Start":"07:03.680 ","End":"07:06.545","Text":"the x and y of the point,"},{"Start":"07:06.545 ","End":"07:08.405","Text":"as well as the slope."},{"Start":"07:08.405 ","End":"07:10.490","Text":"Let\u0027s see what\u0027s missing."},{"Start":"07:10.490 ","End":"07:14.300","Text":"We have that t is equal to 4."},{"Start":"07:14.300 ","End":"07:16.790","Text":"What this thing is equal to,"},{"Start":"07:16.790 ","End":"07:19.265","Text":"this is like my x_1, y_1."},{"Start":"07:19.265 ","End":"07:25.370","Text":"Let\u0027s call this x_1 and then y_1 is the y of this point is"},{"Start":"07:25.370 ","End":"07:33.755","Text":"t square root of t plus 4 and this is equal to 4 root 4 plus 4,"},{"Start":"07:33.755 ","End":"07:42.965","Text":"as 4 times 2 is 8 plus 4 is 12 and the third quantity I need is f prime of x_1,"},{"Start":"07:42.965 ","End":"07:48.335","Text":"which is f prime of t and let\u0027s see what that equals,"},{"Start":"07:48.335 ","End":"07:51.005","Text":"which t is equal to 4."},{"Start":"07:51.005 ","End":"07:54.020","Text":"Let\u0027s just change that into a 4."},{"Start":"07:54.020 ","End":"07:58.460","Text":"Now I have f prime and it\u0027s written over here."},{"Start":"07:58.460 ","End":"08:05.810","Text":"I just have to put 4 into that and that gives me 1.5 or 3"},{"Start":"08:05.810 ","End":"08:14.045","Text":"over 2 times 4 to the 0.5 or square root of 4."},{"Start":"08:14.045 ","End":"08:17.045","Text":"Square root of 4 is 2,"},{"Start":"08:17.045 ","End":"08:19.175","Text":"2 with 2 cancels,"},{"Start":"08:19.175 ","End":"08:21.710","Text":"and this is just equal to 3."},{"Start":"08:21.710 ","End":"08:23.300","Text":"I have everything I want."},{"Start":"08:23.300 ","End":"08:26.825","Text":"I have x_1, I have y_1,"},{"Start":"08:26.825 ","End":"08:29.135","Text":"and I have f prime of x_1,"},{"Start":"08:29.135 ","End":"08:34.580","Text":"and they are respectively 4, 12, and 3."},{"Start":"08:34.580 ","End":"08:39.470","Text":"Now I can pull out the standard formula which says that"},{"Start":"08:39.470 ","End":"08:44.945","Text":"the equation of the tangent is as follows,"},{"Start":"08:44.945 ","End":"08:54.065","Text":"y minus y_1 equals f prime of x_1 times x minus x_1."},{"Start":"08:54.065 ","End":"08:58.445","Text":"If I use this formula and continue here,"},{"Start":"08:58.445 ","End":"09:03.845","Text":"I will get that y minus 12 is"},{"Start":"09:03.845 ","End":"09:10.175","Text":"equal to 3 times x minus 4."},{"Start":"09:10.175 ","End":"09:13.354","Text":"If I bring stuff to the other side,"},{"Start":"09:13.354 ","End":"09:21.200","Text":"I just get that y equals 3x minus 12 plus 12y equals 3x."},{"Start":"09:21.200 ","End":"09:24.845","Text":"Of course, I could have done it more simply because like I said,"},{"Start":"09:24.845 ","End":"09:28.610","Text":"the equation of a line through the origin"},{"Start":"09:28.610 ","End":"09:32.855","Text":"is just of the form y equals mx and we already had the slope."},{"Start":"09:32.855 ","End":"09:36.710","Text":"I could have just said straight away from here to here."},{"Start":"09:36.710 ","End":"09:39.530","Text":"But practice using the formula because it\u0027s not"},{"Start":"09:39.530 ","End":"09:42.335","Text":"always going to go through the origin. Very well."},{"Start":"09:42.335 ","End":"09:45.170","Text":"Now that is the answer to part A,"},{"Start":"09:45.170 ","End":"09:47.990","Text":"so it\u0027s worthy of highlighting,"},{"Start":"09:47.990 ","End":"09:50.870","Text":"and now on to part B."},{"Start":"09:50.870 ","End":"09:59.510","Text":"To remind you of the strategy and now that I know also that t is equal to 4 and in fact,"},{"Start":"09:59.510 ","End":"10:01.340","Text":"we know this point also."},{"Start":"10:01.340 ","End":"10:06.320","Text":"I\u0027m going to rewrite it as 4,12."},{"Start":"10:06.320 ","End":"10:08.210","Text":"We know everything."},{"Start":"10:08.210 ","End":"10:14.210","Text":"What we\u0027re going to do is take the integral from 0-4 of this bit here,"},{"Start":"10:14.210 ","End":"10:15.785","Text":"I\u0027ll just emphasize it."},{"Start":"10:15.785 ","End":"10:19.700","Text":"Yeah. I\u0027ll take the integral of this bit from 0-4"},{"Start":"10:19.700 ","End":"10:23.870","Text":"and I\u0027ll get the whole bit under this curve and then subtract the triangle."},{"Start":"10:23.870 ","End":"10:31.175","Text":"I want the integral from 0 to 4 of this curve,"},{"Start":"10:31.175 ","End":"10:37.190","Text":"which is x root x plus 4 and I\u0027ll choose this form,"},{"Start":"10:37.190 ","End":"10:43.940","Text":"x to the power of 1.5 plus 4 dx."},{"Start":"10:43.940 ","End":"10:46.160","Text":"Straightforward integral."},{"Start":"10:46.160 ","End":"10:56.570","Text":"Increase the exponent by 1 and then I\u0027ve got x to the power of 2.5."},{"Start":"10:56.570 ","End":"11:01.850","Text":"Then I have to divide by 2.5 and 1 over"},{"Start":"11:01.850 ","End":"11:07.925","Text":"2.5 is 0.4 if I\u0027m working with decimal."},{"Start":"11:07.925 ","End":"11:11.510","Text":"Then the integral of 4 is 4x."},{"Start":"11:11.510 ","End":"11:17.240","Text":"All this we have to evaluate between 0 and 4."},{"Start":"11:17.240 ","End":"11:23.315","Text":"If we put in 4, we get 4 to the power of 2.5,"},{"Start":"11:23.315 ","End":"11:29.945","Text":"which is 4 times 4 times square root of 4, which is 32."},{"Start":"11:29.945 ","End":"11:32.600","Text":"Or you could take, let\u0027s say it\u0027s 5 over 2,"},{"Start":"11:32.600 ","End":"11:35.270","Text":"square root of 4 is 2 to the fifth is 32."},{"Start":"11:35.270 ","End":"11:40.970","Text":"Anyway, 32 times 0.4 is"},{"Start":"11:40.970 ","End":"11:49.535","Text":"12.8 and 4 times 4 is 16 less,"},{"Start":"11:49.535 ","End":"11:51.095","Text":"what happens when we put in 0,"},{"Start":"11:51.095 ","End":"11:52.940","Text":"0 to this is 0,"},{"Start":"11:52.940 ","End":"11:55.610","Text":"0 times 4 is 0."},{"Start":"11:55.610 ","End":"11:58.670","Text":"So we don\u0027t get anything else from the 0."},{"Start":"11:58.670 ","End":"12:05.210","Text":"12.8 plus 16 is just 28.8,"},{"Start":"12:05.210 ","End":"12:08.570","Text":"and 28.8 is for the outer bit."},{"Start":"12:08.570 ","End":"12:11.690","Text":"Now the triangle, let\u0027s see what the base is."},{"Start":"12:11.690 ","End":"12:16.370","Text":"We have the base of the triangle is 4,"},{"Start":"12:16.370 ","End":"12:21.590","Text":"from 0 to 4, the height is equal to 12."},{"Start":"12:21.590 ","End":"12:24.995","Text":"I need 0.5 base times height."},{"Start":"12:24.995 ","End":"12:33.065","Text":"Let\u0027s just say the area of the triangle is 0.5 times the base 4,"},{"Start":"12:33.065 ","End":"12:35.510","Text":"times the height is 12."},{"Start":"12:35.510 ","End":"12:40.100","Text":"So 2 times 12 is 24."},{"Start":"12:40.100 ","End":"12:50.510","Text":"Finally, the S that we want will just be 28.8 minus the 24 and so we"},{"Start":"12:50.510 ","End":"12:56.525","Text":"get just 4.8 and"},{"Start":"12:56.525 ","End":"13:02.585","Text":"that would be the answer for part B and I\u0027ll highlight it."},{"Start":"13:02.585 ","End":"13:10.220","Text":"The answer to part A was the equation of the tangent y equals 3x and the answer to"},{"Start":"13:10.220 ","End":"13:18.095","Text":"part B is that the area that was shaded there is 4.8 and we are done."},{"Start":"13:18.095 ","End":"13:21.920","Text":"But wait, I thought I should just let you know what would be another way of"},{"Start":"13:21.920 ","End":"13:25.865","Text":"doing it and it\u0027s but of equal difficulty."},{"Start":"13:25.865 ","End":"13:29.660","Text":"I didn\u0027t have to do any tricks with subtracting triangles."},{"Start":"13:29.660 ","End":"13:32.300","Text":"I could have just gone and done the integral from"},{"Start":"13:32.300 ","End":"13:36.035","Text":"0-4 of the difference of the curve minus the tangent."},{"Start":"13:36.035 ","End":"13:40.400","Text":"As if I was breaking it up into vertical slices."},{"Start":"13:40.400 ","End":"13:47.840","Text":"Then I would have got the integral from 0-4 of the function here minus the 3x,"},{"Start":"13:47.840 ","End":"13:50.045","Text":"which is the function here."},{"Start":"13:50.045 ","End":"13:53.315","Text":"Integrals slightly harder to evaluate,"},{"Start":"13:53.315 ","End":"13:55.549","Text":"but it\u0027s about equal difficulty."},{"Start":"13:55.549 ","End":"13:59.360","Text":"But I thought I should mention in case you have a preference to doing it this"},{"Start":"13:59.360 ","End":"14:04.230","Text":"way and you don\u0027t like the geometry part. Now I\u0027m really done."}],"Thumbnail":null,"ID":4713},{"Watched":false,"Name":"Exercise 18","Duration":"10m 31s","ChapterTopicVideoID":4706,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"Here\u0027s another word problem involving graphs,"},{"Start":"00:03.690 ","End":"00:05.805","Text":"and areas, and so on."},{"Start":"00:05.805 ","End":"00:08.475","Text":"4In part a, I didn\u0027t really mean it."},{"Start":"00:08.475 ","End":"00:09.660","Text":"It says optional."},{"Start":"00:09.660 ","End":"00:14.070","Text":"I just meant that I knew everyone was the strongest students,"},{"Start":"00:14.070 ","End":"00:16.440","Text":"and I wouldn\u0027t have given part a which is really"},{"Start":"00:16.440 ","End":"00:19.905","Text":"just a hint for how to do the integral in part b."},{"Start":"00:19.905 ","End":"00:21.660","Text":"But it doesn\u0027t hurt,"},{"Start":"00:21.660 ","End":"00:23.340","Text":"we\u0027ll do it in all cases,"},{"Start":"00:23.340 ","End":"00:25.260","Text":"whatever your level is."},{"Start":"00:25.260 ","End":"00:28.150","Text":"Here we have to just differentiate this,"},{"Start":"00:28.150 ","End":"00:31.250","Text":"and we\u0027ll get something similar but not the same as this,"},{"Start":"00:31.250 ","End":"00:34.900","Text":"and that will be helpful in finding the integral of this,"},{"Start":"00:34.900 ","End":"00:38.090","Text":"and then we\u0027ll be able to solve this graph here and"},{"Start":"00:38.090 ","End":"00:41.150","Text":"find the area using definite integrals."},{"Start":"00:41.150 ","End":"00:43.475","Text":"Let\u0027s start with part a."},{"Start":"00:43.475 ","End":"00:46.665","Text":"We\u0027ll just differentiate this."},{"Start":"00:46.665 ","End":"00:53.510","Text":"In a, we have f of x equals cosine cubed of x,"},{"Start":"00:53.510 ","End":"00:57.860","Text":"and so f-prime of x is equal to,"},{"Start":"00:57.860 ","End":"01:00.035","Text":"first of all, we see something cubed."},{"Start":"01:00.035 ","End":"01:03.990","Text":"So it\u0027s 3 cosine of that thing squared,"},{"Start":"01:03.990 ","End":"01:07.655","Text":"and then we have the matter of an inner derivative because it\u0027s cosines."},{"Start":"01:07.655 ","End":"01:11.510","Text":"We need to multiply by minus sine x."},{"Start":"01:11.510 ","End":"01:15.910","Text":"Just rewriting this is put the minus in front,"},{"Start":"01:15.910 ","End":"01:22.605","Text":"minus 3 cosine squared x sine x."},{"Start":"01:22.605 ","End":"01:24.605","Text":"If I just look at this bit,"},{"Start":"01:24.605 ","End":"01:28.655","Text":"I see, it looks very much like this here."},{"Start":"01:28.655 ","End":"01:32.540","Text":"That\u0027s going to come in handy. That\u0027s a."},{"Start":"01:32.540 ","End":"01:34.540","Text":"Now, on to b."},{"Start":"01:34.540 ","End":"01:41.210","Text":"Now, we have this function: y equals cosine squared x sine x."},{"Start":"01:41.210 ","End":"01:42.740","Text":"In case I need the letter,"},{"Start":"01:42.740 ","End":"01:46.540","Text":"I\u0027ll call it g of x because f is already taken."},{"Start":"01:46.540 ","End":"01:52.140","Text":"On the domain from a 1/2 Pi to 3/2 Pi,"},{"Start":"01:52.140 ","End":"01:56.955","Text":"which is from 90-270 degrees."},{"Start":"01:56.955 ","End":"02:01.000","Text":"Let\u0027s see what this graph looks like."},{"Start":"02:01.000 ","End":"02:05.885","Text":"I\u0027d like to explain what\u0027s the conceptual difficulty in part b."},{"Start":"02:05.885 ","End":"02:10.265","Text":"The thing is that the area is not exactly an integral."},{"Start":"02:10.265 ","End":"02:12.860","Text":"There\u0027s a matter of absolute value."},{"Start":"02:12.860 ","End":"02:14.660","Text":"I\u0027ll show you what I mean with an example."},{"Start":"02:14.660 ","End":"02:20.975","Text":"Suppose I have a graph and there\u0027s a function that goes something like this."},{"Start":"02:20.975 ","End":"02:23.780","Text":"Then it goes into the negative part."},{"Start":"02:23.780 ","End":"02:26.735","Text":"Then there\u0027s a bit of a positive part,"},{"Start":"02:26.735 ","End":"02:32.630","Text":"and I want the area bounded by the x-axis and the graph of the function."},{"Start":"02:32.630 ","End":"02:36.660","Text":"In that case, I have to split it up into 3 bits."},{"Start":"02:36.660 ","End":"02:40.685","Text":"First of all, I find where the function is 0."},{"Start":"02:40.685 ","End":"02:45.130","Text":"Then I find this area separately,"},{"Start":"02:45.130 ","End":"02:49.510","Text":"this area separately, and this area separately."},{"Start":"02:49.510 ","End":"02:51.240","Text":"See, this is a function,"},{"Start":"02:51.240 ","End":"02:54.005","Text":"and I just take the integral from here to here,"},{"Start":"02:54.005 ","End":"02:57.485","Text":"this bit is going to contribute negatively,"},{"Start":"02:57.485 ","End":"03:01.240","Text":"and it\u0027s going to counteract the positive parts."},{"Start":"03:01.240 ","End":"03:04.130","Text":"What we do is we take the integral from here to here,"},{"Start":"03:04.130 ","End":"03:05.810","Text":"then from here to here,"},{"Start":"03:05.810 ","End":"03:07.295","Text":"and then from here to here."},{"Start":"03:07.295 ","End":"03:09.920","Text":"But in each case, we take the absolute value."},{"Start":"03:09.920 ","End":"03:13.730","Text":"The positive functions, the definite integral is the same as the area."},{"Start":"03:13.730 ","End":"03:16.070","Text":"But if the function is below the axis,"},{"Start":"03:16.070 ","End":"03:17.900","Text":"the definite integral is negative,"},{"Start":"03:17.900 ","End":"03:19.310","Text":"but the area is positive."},{"Start":"03:19.310 ","End":"03:25.805","Text":"So we have to take the absolute value in order to make the area always positive."},{"Start":"03:25.805 ","End":"03:27.230","Text":"That\u0027s what we\u0027re going to do here."},{"Start":"03:27.230 ","End":"03:35.375","Text":"Let\u0027s find out that between these limits where the function g of x could be 0."},{"Start":"03:35.375 ","End":"03:40.530","Text":"What I\u0027m going to do is say, solve the equation."},{"Start":"03:45.320 ","End":"03:52.145","Text":"Oops, I made a mistake and I wasn\u0027t recording while I was continuing to write."},{"Start":"03:52.145 ","End":"03:54.680","Text":"But never mind, I\u0027ll just go over it again."},{"Start":"03:54.680 ","End":"03:55.940","Text":"I hope you\u0027ll forgive me."},{"Start":"03:55.940 ","End":"04:00.110","Text":"We just about to say that g of x is 0,"},{"Start":"04:00.110 ","End":"04:01.790","Text":"we\u0027re looking for solutions."},{"Start":"04:01.790 ","End":"04:06.320","Text":"But between Pi over 2 and 3 Pi over 2,"},{"Start":"04:06.320 ","End":"04:10.370","Text":"which is between 90 and 270 degrees."},{"Start":"04:10.370 ","End":"04:11.990","Text":"We want to know what the function is 0,"},{"Start":"04:11.990 ","End":"04:13.880","Text":"just like in the diagram here,"},{"Start":"04:13.880 ","End":"04:18.100","Text":"we needed to find where the function is plus and where it\u0027s minus."},{"Start":"04:18.100 ","End":"04:20.465","Text":"If we look at g of x,"},{"Start":"04:20.465 ","End":"04:23.165","Text":"and g of x is the bit that\u0027s written here,"},{"Start":"04:23.165 ","End":"04:25.130","Text":"cosine squared sine x."},{"Start":"04:25.130 ","End":"04:27.695","Text":"That\u0027s what we defined it as originally."},{"Start":"04:27.695 ","End":"04:30.365","Text":"There\u0027s only 2 ways that this could be 0,"},{"Start":"04:30.365 ","End":"04:33.925","Text":"either the cosine is 0 or the sine is 0."},{"Start":"04:33.925 ","End":"04:35.500","Text":"Now, we\u0027ll follow each 1."},{"Start":"04:35.500 ","End":"04:36.920","Text":"If cosine is 0,"},{"Start":"04:36.920 ","End":"04:42.060","Text":"the general solution for the cosine 0 is 90 degrees"},{"Start":"04:42.060 ","End":"04:47.705","Text":"plus multiples of 180 degrees or in terms of Pi, that\u0027s what it is."},{"Start":"04:47.705 ","End":"04:52.190","Text":"The only values of k that are in our range are 0 and 1,"},{"Start":"04:52.190 ","End":"04:56.580","Text":"and they happen to give us exactly the endpoints of our interval."},{"Start":"04:56.580 ","End":"04:59.120","Text":"With our endpoints, the function is 0,"},{"Start":"04:59.120 ","End":"05:04.030","Text":"but we really need to know what happens in the middle of the interval."},{"Start":"05:04.030 ","End":"05:06.520","Text":"That\u0027s where the sine might come in."},{"Start":"05:06.520 ","End":"05:08.270","Text":"If sine x is 0,"},{"Start":"05:08.270 ","End":"05:13.044","Text":"the general solution is multiples of 180 degrees or Pi,"},{"Start":"05:13.044 ","End":"05:23.770","Text":"and there is only 1 such possibility for this range and that is that k is equal to 1,"},{"Start":"05:23.770 ","End":"05:28.410","Text":"and then we get to Pi and Pi is in this interval."},{"Start":"05:28.430 ","End":"05:31.715","Text":"What we have to do, in this case,"},{"Start":"05:31.715 ","End":"05:33.335","Text":"this here is Pi,"},{"Start":"05:33.335 ","End":"05:35.210","Text":"just like in the picture."},{"Start":"05:35.210 ","End":"05:39.965","Text":"We have to break our integral up into 2 pieces,"},{"Start":"05:39.965 ","End":"05:43.755","Text":"from Pi over 2 to Pi of g,"},{"Start":"05:43.755 ","End":"05:47.550","Text":"and then from Pi to 3 Pi over 2 of g,"},{"Start":"05:47.550 ","End":"05:49.605","Text":"but with absolute values."},{"Start":"05:49.605 ","End":"05:51.410","Text":"In the first part is positive anyway,"},{"Start":"05:51.410 ","End":"05:53.330","Text":"so the absolute value just doesn\u0027t hurt."},{"Start":"05:53.330 ","End":"05:54.940","Text":"The second part is negative,"},{"Start":"05:54.940 ","End":"06:00.300","Text":"and the absolute value will correct it and make this area positive and not negative."},{"Start":"06:01.250 ","End":"06:03.670","Text":"What we have to do is,"},{"Start":"06:03.670 ","End":"06:06.035","Text":"first of all, since g appears in both,"},{"Start":"06:06.035 ","End":"06:08.670","Text":"a minus will find the indefinite integral of g,"},{"Start":"06:08.670 ","End":"06:12.125","Text":"so I can use it in both these definite integrals."},{"Start":"06:12.125 ","End":"06:19.500","Text":"Now g of x is equal to cosine squared x sine x. I want to make use of part a,"},{"Start":"06:19.500 ","End":"06:21.710","Text":"where if I have cosine cubed,"},{"Start":"06:21.710 ","End":"06:23.255","Text":"the derivative is this."},{"Start":"06:23.255 ","End":"06:25.775","Text":"Now, I don\u0027t have the minus 3."},{"Start":"06:25.775 ","End":"06:36.585","Text":"I shall use the old trick of inserting a minus 3 and also multiplying by minus 1/3."},{"Start":"06:36.585 ","End":"06:42.240","Text":"Now, I can put the minus 3 here,"},{"Start":"06:42.240 ","End":"06:45.090","Text":"and maybe I need brackets."},{"Start":"06:45.090 ","End":"06:51.005","Text":"Then I compensate by putting minus 1/3 out here."},{"Start":"06:51.005 ","End":"06:56.390","Text":"Now, I can say that this is minus 1/3 and the integral"},{"Start":"06:56.390 ","End":"07:02.040","Text":"of this will now be exactly cosine cubed x."},{"Start":"07:02.040 ","End":"07:04.440","Text":"If it\u0027s an indefinite integral,"},{"Start":"07:04.440 ","End":"07:06.005","Text":"I\u0027ll put a constant."},{"Start":"07:06.005 ","End":"07:08.780","Text":"But now I\u0027m going to do definite integrals."},{"Start":"07:08.780 ","End":"07:12.125","Text":"So constant is not really helpful."},{"Start":"07:12.125 ","End":"07:19.849","Text":"What I get is minus 1/3 cosine cubed x."},{"Start":"07:19.849 ","End":"07:26.090","Text":"This is evaluated between Pi over 2 and Pi."},{"Start":"07:26.090 ","End":"07:29.210","Text":"Then I need the absolute value."},{"Start":"07:29.210 ","End":"07:31.745","Text":"Well, all of these bars is going to be confusing."},{"Start":"07:31.745 ","End":"07:34.940","Text":"I\u0027ll write it as a absolute value of this."},{"Start":"07:34.940 ","End":"07:36.650","Text":"Another way to write absolute value,"},{"Start":"07:36.650 ","End":"07:39.290","Text":"the bars are just going to be confusing."},{"Start":"07:39.290 ","End":"07:45.790","Text":"This plus the absolute value of the same thing"},{"Start":"07:45.790 ","End":"07:50.890","Text":"minus 1/3 cosine sine cubed x"},{"Start":"07:50.890 ","End":"07:57.115","Text":"evaluated between Pi and 3 Pi over 2."},{"Start":"07:57.115 ","End":"07:59.050","Text":"This is equal to this,"},{"Start":"07:59.050 ","End":"08:02.125","Text":"and then this is equal to, let\u0027s see now."},{"Start":"08:02.125 ","End":"08:05.690","Text":"We get the absolute value of."},{"Start":"08:05.690 ","End":"08:08.025","Text":"If I put in Pi,"},{"Start":"08:08.025 ","End":"08:14.610","Text":"I\u0027ve got minus 1/3 cosine cubed of Pi."},{"Start":"08:14.610 ","End":"08:18.510","Text":"The cosine of Pi is negative 1."},{"Start":"08:18.510 ","End":"08:19.890","Text":"So it\u0027s minus 3,"},{"Start":"08:19.890 ","End":"08:26.670","Text":"so minus 1 cubed, minus 1/3,"},{"Start":"08:26.670 ","End":"08:33.095","Text":"and the cosine of Pi over 2 is 0 cubed,"},{"Start":"08:33.095 ","End":"08:34.790","Text":"the absolute value of that,"},{"Start":"08:34.790 ","End":"08:44.105","Text":"plus absolute value of cosine of 3 Pi over 2 is actually the same as"},{"Start":"08:44.105 ","End":"08:47.960","Text":"cosine of Pi over 2 because"},{"Start":"08:47.960 ","End":"08:54.215","Text":"270 degrees and 90 degrees have the same cosine and cosine is an even function."},{"Start":"08:54.215 ","End":"08:57.545","Text":"That means that it\u0027s also 0."},{"Start":"08:57.545 ","End":"09:06.155","Text":"We get minus 1/3 0 cubed and then the cosine of Pi,"},{"Start":"09:06.155 ","End":"09:08.630","Text":"once again, we\u0027ve already had that,"},{"Start":"09:08.630 ","End":"09:10.930","Text":"that it\u0027s minus 1."},{"Start":"09:10.930 ","End":"09:12.764","Text":"We have to subtract,"},{"Start":"09:12.764 ","End":"09:14.625","Text":"but to subtract a minus."},{"Start":"09:14.625 ","End":"09:17.175","Text":"So it\u0027s plus 1/3."},{"Start":"09:17.175 ","End":"09:21.780","Text":"The cosine we said was minus 1 cubed."},{"Start":"09:21.780 ","End":"09:24.900","Text":"That\u0027s what we have there. This is equal to."},{"Start":"09:24.900 ","End":"09:28.085","Text":"Now, it can write a regular absolute value."},{"Start":"09:28.085 ","End":"09:30.740","Text":"It\u0027s the absolute value of,"},{"Start":"09:30.740 ","End":"09:32.195","Text":"what do we have here?"},{"Start":"09:32.195 ","End":"09:37.800","Text":"Minus 1 cubed is minus 1, so it\u0027s 1/3."},{"Start":"09:37.800 ","End":"09:41.970","Text":"Here we have the absolute value of, let\u0027s see,"},{"Start":"09:41.970 ","End":"09:47.700","Text":"minus 1 cubed is a minus 1/3."},{"Start":"09:47.700 ","End":"09:52.090","Text":"Each of these is equal to 1/3 and we get 2/3."},{"Start":"09:52.090 ","End":"09:55.085","Text":"But see how important the absolute values were."},{"Start":"09:55.085 ","End":"09:57.020","Text":"If we hadn\u0027t bothered with them,"},{"Start":"09:57.020 ","End":"10:00.620","Text":"the 1/3 and the minus 1/3 would have canceled each other out,"},{"Start":"10:00.620 ","End":"10:02.885","Text":"we would have just got 0."},{"Start":"10:02.885 ","End":"10:07.390","Text":"It would be the plus and the minus canceling each other out,"},{"Start":"10:07.390 ","End":"10:09.800","Text":"that when we say area between the curve,"},{"Start":"10:09.800 ","End":"10:11.795","Text":"and we say this and this,"},{"Start":"10:11.795 ","End":"10:15.140","Text":"we mean that this area is positive and this area is positive,"},{"Start":"10:15.140 ","End":"10:18.020","Text":"but the integral gives negative below the axis."},{"Start":"10:18.020 ","End":"10:21.110","Text":"So, natural fact, the area is 2/3,"},{"Start":"10:21.110 ","End":"10:25.280","Text":"but 1/3 above the axis and 1/3 below the axis."},{"Start":"10:25.280 ","End":"10:27.590","Text":"This is the answer for part b,"},{"Start":"10:27.590 ","End":"10:29.480","Text":"and we\u0027ve already done part a,"},{"Start":"10:29.480 ","End":"10:32.250","Text":"and so we are done."}],"Thumbnail":null,"ID":4714},{"Watched":false,"Name":"Exercise 19","Duration":"11m 14s","ChapterTopicVideoID":4707,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"Here we have another area problem of"},{"Start":"00:02.970 ","End":"00:08.100","Text":"some shape that\u0027s between a parabola and a line, hard to imagine."},{"Start":"00:08.100 ","End":"00:10.620","Text":"We really do need some kind of a sketch,"},{"Start":"00:10.620 ","End":"00:11.910","Text":"even a rough 1."},{"Start":"00:11.910 ","End":"00:15.510","Text":"Let me draw some axes. There we are."},{"Start":"00:15.510 ","End":"00:19.290","Text":"Now let\u0027s draw the graphs, 1 at a time."},{"Start":"00:19.290 ","End":"00:21.775","Text":"I\u0027ll go with the straight line first."},{"Start":"00:21.775 ","End":"00:26.315","Text":"A straight line, probably the easiest thing to do is make a little table,"},{"Start":"00:26.315 ","End":"00:30.290","Text":"compute the intersections with the axes perhaps."},{"Start":"00:30.290 ","End":"00:33.629","Text":"If I take x and y,"},{"Start":"00:33.629 ","End":"00:35.700","Text":"1 time I let x equals 0,"},{"Start":"00:35.700 ","End":"00:37.760","Text":"1 time I let y equal 0."},{"Start":"00:37.760 ","End":"00:41.150","Text":"If x is 0, then I get that y is 6,"},{"Start":"00:41.150 ","End":"00:44.120","Text":"and if y is 0, I get that x plus 6 is 0,"},{"Start":"00:44.120 ","End":"00:46.115","Text":"so x is minus 6."},{"Start":"00:46.115 ","End":"00:48.400","Text":"I meant to write 6 there."},{"Start":"00:48.400 ","End":"00:51.675","Text":"Here I have, say the point 0,"},{"Start":"00:51.675 ","End":"00:54.765","Text":"6, so y is 6 here."},{"Start":"00:54.765 ","End":"00:56.580","Text":"Then the point minus 6,"},{"Start":"00:56.580 ","End":"01:00.450","Text":"0 is somewhere, let\u0027s say here,"},{"Start":"01:00.450 ","End":"01:02.040","Text":"and this is minus 6,"},{"Start":"01:02.040 ","End":"01:04.425","Text":"and I\u0027ll just draw a line through them."},{"Start":"01:04.425 ","End":"01:06.765","Text":"As for the parabola,"},{"Start":"01:06.765 ","End":"01:12.005","Text":"it\u0027s actually a parabola on its side because if I take x in terms of y,"},{"Start":"01:12.005 ","End":"01:14.780","Text":"x is equal to minus y squared,"},{"Start":"01:14.780 ","End":"01:16.715","Text":"so it\u0027s a sideways parabola."},{"Start":"01:16.715 ","End":"01:20.910","Text":"Let\u0027s see, we could also make a table for that."},{"Start":"01:21.170 ","End":"01:26.720","Text":"If I take y to be 0,"},{"Start":"01:26.720 ","End":"01:31.475","Text":"then I have that x is 0."},{"Start":"01:31.475 ","End":"01:34.685","Text":"If I have y is 1,"},{"Start":"01:34.685 ","End":"01:38.650","Text":"then I get x is minus 1."},{"Start":"01:38.650 ","End":"01:42.875","Text":"Similarly, if I take y is minus 1,"},{"Start":"01:42.875 ","End":"01:46.630","Text":"then I also get x is minus 1."},{"Start":"01:46.630 ","End":"01:50.494","Text":"If I take y is equal to 2,"},{"Start":"01:50.494 ","End":"01:53.210","Text":"I get x is minus 4."},{"Start":"01:53.210 ","End":"01:54.980","Text":"Similarly, for minus 2,"},{"Start":"01:54.980 ","End":"01:56.510","Text":"I get minus 4."},{"Start":"01:56.510 ","End":"02:03.734","Text":"Let\u0027s see these points 0,0 will be here, minus 1,"},{"Start":"02:03.734 ","End":"02:08.520","Text":"1 say here, minus 1, minus 1,"},{"Start":"02:08.520 ","End":"02:12.780","Text":"here, minus 4, 2,"},{"Start":"02:12.780 ","End":"02:15.795","Text":"I actually think it lies on the line."},{"Start":"02:15.795 ","End":"02:18.150","Text":"Minus 4, minus 2,"},{"Start":"02:18.150 ","End":"02:20.745","Text":"that would be somewhere here."},{"Start":"02:20.745 ","End":"02:23.185","Text":"We have a parabola on its side,"},{"Start":"02:23.185 ","End":"02:26.360","Text":"and let\u0027s see if I can sketch this."},{"Start":"02:26.610 ","End":"02:30.675","Text":"I can see this is not really very important though."},{"Start":"02:30.675 ","End":"02:35.375","Text":"The area bounded by the parabola and the line,"},{"Start":"02:35.375 ","End":"02:39.670","Text":"I\u0027ll shade it is this area here, let me color."},{"Start":"02:39.670 ","End":"02:45.210","Text":"I\u0027ve shaded it, and this is the area that we want."},{"Start":"02:45.210 ","End":"02:48.840","Text":"There\u0027s 2 main ways of doing it,"},{"Start":"02:48.840 ","End":"02:51.390","Text":"and I\u0027m going to definitely prefer 1 of them."},{"Start":"02:51.390 ","End":"02:55.969","Text":"1 of them would be to divide the area"},{"Start":"02:55.969 ","End":"03:03.295","Text":"vertically by drawing a line here and then taking this bit plus this bit."},{"Start":"03:03.295 ","End":"03:06.245","Text":"In any event, this parabola,"},{"Start":"03:06.245 ","End":"03:08.390","Text":"this curve is not a function,"},{"Start":"03:08.390 ","End":"03:12.515","Text":"at least it\u0027s not a function of y in terms of x."},{"Start":"03:12.515 ","End":"03:18.500","Text":"What we would have to do would be to take 2 separate branches and to say"},{"Start":"03:18.500 ","End":"03:24.530","Text":"that the top bit is y equals the square root of minus x,"},{"Start":"03:24.530 ","End":"03:31.430","Text":"and the bottom bit would be y equals minus the square root of minus x."},{"Start":"03:31.430 ","End":"03:37.715","Text":"Then we would use the straight line and the lower curve from here to here,"},{"Start":"03:37.715 ","End":"03:40.205","Text":"we would have to figure out the x of this point."},{"Start":"03:40.205 ","End":"03:44.495","Text":"Then in the rest of it would take this curve minus this curve,"},{"Start":"03:44.495 ","End":"03:47.570","Text":"and it\u0027s quite messy."},{"Start":"03:47.570 ","End":"03:51.235","Text":"There is a better way of doing it."},{"Start":"03:51.235 ","End":"03:57.950","Text":"The other way of doing it is not to look at it as vertical slices as we usually do,"},{"Start":"03:57.950 ","End":"04:01.835","Text":"but this time to view the region horizontally,"},{"Start":"04:01.835 ","End":"04:09.730","Text":"sideways and think of it as broken up into strips this way."},{"Start":"04:09.770 ","End":"04:15.520","Text":"We look at x as a function of y and not y as a function of x."},{"Start":"04:15.520 ","End":"04:18.600","Text":"Before I continue, either of the method you"},{"Start":"04:18.600 ","End":"04:22.025","Text":"use the same computations we\u0027re going to have to do,"},{"Start":"04:22.025 ","End":"04:26.285","Text":"and that is we\u0027re going to have to compute this point here,"},{"Start":"04:26.285 ","End":"04:29.170","Text":"this point here, well,"},{"Start":"04:29.170 ","End":"04:32.540","Text":"and this point here, which is obviously the origin."},{"Start":"04:32.540 ","End":"04:35.630","Text":"We can find these 2 points by simply solving"},{"Start":"04:35.630 ","End":"04:40.220","Text":"2 equations and 2 unknowns to the parabola\u0027s equation and the line\u0027s equation."},{"Start":"04:40.220 ","End":"04:44.390","Text":"But just like we wrote the parabola as x in terms of y,"},{"Start":"04:44.390 ","End":"04:51.260","Text":"we can do the same thing with the line and easily see that x is equal to y minus 6."},{"Start":"04:51.260 ","End":"04:54.160","Text":"Now it\u0027s easy to solve these 2 equations."},{"Start":"04:54.160 ","End":"04:55.680","Text":"Let me just write it,"},{"Start":"04:55.680 ","End":"04:58.560","Text":"x equals minus y squared,"},{"Start":"04:58.560 ","End":"05:04.680","Text":"that\u0027s the parabola, x equals y minus 6, that\u0027s the line."},{"Start":"05:04.680 ","End":"05:06.470","Text":"Since they\u0027re both x equals,"},{"Start":"05:06.470 ","End":"05:16.535","Text":"I can compare the right-hand sides and get that minus y squared is equal to y minus 6."},{"Start":"05:16.535 ","End":"05:19.055","Text":"If I bring everything to 1 side,"},{"Start":"05:19.055 ","End":"05:25.545","Text":"I\u0027ll get y squared plus y minus 6 equals 0,"},{"Start":"05:25.545 ","End":"05:28.460","Text":"and quadratic equations, we don\u0027t waste time on."},{"Start":"05:28.460 ","End":"05:32.030","Text":"I\u0027m going to tell you the solutions right away."},{"Start":"05:32.030 ","End":"05:39.030","Text":"Y equals 2 or y equals minus 3."},{"Start":"05:39.030 ","End":"05:42.525","Text":"That\u0027s for y, and as for x,"},{"Start":"05:42.525 ","End":"05:45.365","Text":"x is got to be the same for both of them."},{"Start":"05:45.365 ","End":"05:47.749","Text":"Let\u0027s see which 1s we plug it into."},{"Start":"05:47.749 ","End":"05:51.095","Text":"If we plug it into here,"},{"Start":"05:51.095 ","End":"05:57.305","Text":"for x is 2, we get y is minus 4."},{"Start":"05:57.305 ","End":"05:59.340","Text":"Same thing here, if y is 2,"},{"Start":"05:59.340 ","End":"06:01.095","Text":"then x is minus 4,"},{"Start":"06:01.095 ","End":"06:04.140","Text":"so this is minus 4,"},{"Start":"06:04.140 ","End":"06:07.219","Text":"and if y is minus 3,"},{"Start":"06:07.219 ","End":"06:11.045","Text":"then we get minus 3 minus 6 is minus 9,"},{"Start":"06:11.045 ","End":"06:15.185","Text":"or alternatively minus of minus 3 squared which is also minus 9."},{"Start":"06:15.185 ","End":"06:17.690","Text":"By which I mean that this goes with this,"},{"Start":"06:17.690 ","End":"06:19.490","Text":"and this goes with this,"},{"Start":"06:19.490 ","End":"06:20.930","Text":"you can\u0027t mix and match."},{"Start":"06:20.930 ","End":"06:23.780","Text":"1 of the points is x is minus 4,"},{"Start":"06:23.780 ","End":"06:25.535","Text":"y is 2, minus 4,"},{"Start":"06:25.535 ","End":"06:31.125","Text":"2, that would be this 1, minus 4, 2."},{"Start":"06:31.125 ","End":"06:35.760","Text":"This 1 would be minus 9, minus 3."},{"Start":"06:35.760 ","End":"06:39.195","Text":"Minus 9, minus 3."},{"Start":"06:39.195 ","End":"06:45.065","Text":"For 1 thing, it means that this is the point where y is 2,"},{"Start":"06:45.065 ","End":"06:51.605","Text":"and this is the point where y is minus 3."},{"Start":"06:51.605 ","End":"06:59.135","Text":"Now what we have to do is take the integral used to be the upper minus the lower,"},{"Start":"06:59.135 ","End":"07:01.400","Text":"but on the case of horizontal,"},{"Start":"07:01.400 ","End":"07:05.700","Text":"it\u0027s the rightmost minus the leftmost."},{"Start":"07:05.700 ","End":"07:09.165","Text":"We do the parabola minus the line."},{"Start":"07:09.165 ","End":"07:12.260","Text":"What I get for the integral,"},{"Start":"07:12.260 ","End":"07:13.925","Text":"and let\u0027s just give it a name,"},{"Start":"07:13.925 ","End":"07:20.445","Text":"S. What I get is that S is the integral,"},{"Start":"07:20.445 ","End":"07:23.300","Text":"and this time we\u0027re going instead of from left to right,"},{"Start":"07:23.300 ","End":"07:24.755","Text":"from bottom to top,"},{"Start":"07:24.755 ","End":"07:29.870","Text":"from minus 3 to 2 of the upper 1,"},{"Start":"07:29.870 ","End":"07:31.700","Text":"which is the 1 where x is larger,"},{"Start":"07:31.700 ","End":"07:35.995","Text":"the parabola, which is minus y squared,"},{"Start":"07:35.995 ","End":"07:39.630","Text":"take away the lower 1,"},{"Start":"07:39.630 ","End":"07:42.480","Text":"which is y minus 6."},{"Start":"07:42.480 ","End":"07:46.290","Text":"All this is d_y, of course."},{"Start":"07:46.290 ","End":"07:51.805","Text":"Everything is switched, we\u0027re in the world of x in terms of y and not y in terms of x."},{"Start":"07:51.805 ","End":"07:54.655","Text":"As we saw, this makes things a lot simpler."},{"Start":"07:54.655 ","End":"07:57.520","Text":"Let\u0027s just do the integration."},{"Start":"07:57.520 ","End":"08:00.685","Text":"What we have for minus y squared,"},{"Start":"08:00.685 ","End":"08:05.875","Text":"we get minus y cubed over 3."},{"Start":"08:05.875 ","End":"08:09.445","Text":"Then we have minus y,"},{"Start":"08:09.445 ","End":"08:14.625","Text":"which becomes minus y squared over 2."},{"Start":"08:14.625 ","End":"08:16.860","Text":"Then we have plus 6,"},{"Start":"08:16.860 ","End":"08:20.200","Text":"which is just 6y."},{"Start":"08:20.210 ","End":"08:31.395","Text":"All this, we have to take between the limits minus 3 and 2 for y. Let\u0027s do that."},{"Start":"08:31.395 ","End":"08:35.145","Text":"Let\u0027s put in 2, and if we put in 2,"},{"Start":"08:35.145 ","End":"08:38.025","Text":"we get minus 8 over 3,"},{"Start":"08:38.025 ","End":"08:40.575","Text":"minus 4 over 2,"},{"Start":"08:40.575 ","End":"08:44.085","Text":"plus 6 times 2."},{"Start":"08:44.085 ","End":"08:46.505","Text":"If we put in minus 3,"},{"Start":"08:46.505 ","End":"08:52.805","Text":"we get minus 3 cubed over 3."},{"Start":"08:52.805 ","End":"08:55.865","Text":"Sorry, we need an extra minus here,"},{"Start":"08:55.865 ","End":"08:57.515","Text":"y is minus 3,"},{"Start":"08:57.515 ","End":"08:59.165","Text":"and there\u0027s a minus here,"},{"Start":"08:59.165 ","End":"09:01.715","Text":"and there\u0027s still a minus in front."},{"Start":"09:01.715 ","End":"09:04.415","Text":"I hope we don\u0027t get the minuses wrong."},{"Start":"09:04.415 ","End":"09:12.630","Text":"Minus, minus 3 squared over 2 plus 6 times minus 3."},{"Start":"09:12.680 ","End":"09:16.280","Text":"Let\u0027s see, minus 8 over 3,"},{"Start":"09:16.280 ","End":"09:17.870","Text":"I think I\u0027ll use mixed fractions,"},{"Start":"09:17.870 ","End":"09:26.950","Text":"that\u0027s minus 22/3, minus 4 over 2 is minus 2 plus 6 times 2 is plus 12."},{"Start":"09:26.950 ","End":"09:30.450","Text":"Then that\u0027s this 1, less."},{"Start":"09:30.450 ","End":"09:32.205","Text":"Let\u0027s see what this 1 is."},{"Start":"09:32.205 ","End":"09:35.060","Text":"Altogether, we have minus 4 times,"},{"Start":"09:35.060 ","End":"09:39.560","Text":"so it\u0027s a plus 3 cubed over 3 is like 3 squared,"},{"Start":"09:39.560 ","End":"09:41.570","Text":"so that\u0027s got to be 9."},{"Start":"09:41.570 ","End":"09:45.020","Text":"This has got to be a plus 9 over 2, but a minus,"},{"Start":"09:45.020 ","End":"09:47.320","Text":"so minus 9 over 2,"},{"Start":"09:47.320 ","End":"09:52.555","Text":"which is 41/2, and then plus 18."},{"Start":"09:52.555 ","End":"09:56.645","Text":"Let\u0027s see. Compute each 1 separately."},{"Start":"09:56.645 ","End":"09:58.489","Text":"What do I get?"},{"Start":"09:58.489 ","End":"10:01.220","Text":"12 minus"},{"Start":"10:01.220 ","End":"10:08.850","Text":"42/3 is 71/3."},{"Start":"10:08.850 ","End":"10:15.355","Text":"Here we have 27 minus 41/2, 221/2."},{"Start":"10:15.355 ","End":"10:24.065","Text":"But something wrong here because I was supposed to get something positive."},{"Start":"10:24.065 ","End":"10:26.745","Text":"I apologize, I copied the plus wrong,"},{"Start":"10:26.745 ","End":"10:28.425","Text":"this should have been plus,"},{"Start":"10:28.425 ","End":"10:30.125","Text":"and if this is plus,"},{"Start":"10:30.125 ","End":"10:34.550","Text":"then this makes this to be a minus."},{"Start":"10:34.550 ","End":"10:36.920","Text":"Therefore, what I get,"},{"Start":"10:36.920 ","End":"10:38.565","Text":"now I\u0027m going to erase this."},{"Start":"10:38.565 ","End":"10:40.020","Text":"Let\u0027s see what we do get."},{"Start":"10:40.020 ","End":"10:44.355","Text":"A 41/2 and 18 is 221/2."},{"Start":"10:44.355 ","End":"10:45.840","Text":"This with the minus,"},{"Start":"10:45.840 ","End":"10:50.715","Text":"it will be plus 221/2 minus 9,"},{"Start":"10:50.715 ","End":"10:54.255","Text":"and 221/2 minus 9 is 131/2,"},{"Start":"10:54.255 ","End":"10:58.080","Text":"so it\u0027s plus 131/2."},{"Start":"10:58.080 ","End":"11:02.655","Text":"We get 7 and 13 is 20,"},{"Start":"11:02.655 ","End":"11:06.375","Text":"1/3 and 1/2 is 5/6,"},{"Start":"11:06.375 ","End":"11:15.120","Text":"and so this should be the answer and I\u0027ll highlight it. That\u0027s all."}],"Thumbnail":null,"ID":4715},{"Watched":false,"Name":"Exercise 20","Duration":"9m 13s","ChapterTopicVideoID":4708,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"Previous exercise was just like this."},{"Start":"00:02.880 ","End":"00:08.790","Text":"It\u0027s another 1 of those parabola and line and it\u0027s a sideways parabola."},{"Start":"00:08.790 ","End":"00:11.250","Text":"We definitely need a sketch."},{"Start":"00:11.250 ","End":"00:18.825","Text":"I all ready started by drawing some axes and then we have a line and a parabola."},{"Start":"00:18.825 ","End":"00:27.855","Text":"Start with the line. What I like to do is make a little table of values and put x and y."},{"Start":"00:27.855 ","End":"00:31.935","Text":"Often it works out well if we do the intercepts,"},{"Start":"00:31.935 ","End":"00:34.695","Text":"that is the intersection with the axes."},{"Start":"00:34.695 ","End":"00:40.220","Text":"Let\u0027s try putting x equals 0 once and then putting y equals 0 and see what we get."},{"Start":"00:40.220 ","End":"00:44.810","Text":"If x is 0 we get y is minus"},{"Start":"00:44.810 ","End":"00:52.210","Text":"8 and if y is 0 then we get that x is plus 8,"},{"Start":"00:52.210 ","End":"00:53.570","Text":"and for a straight line,"},{"Start":"00:53.570 ","End":"00:55.595","Text":"2 points are enough."},{"Start":"00:55.595 ","End":"00:59.875","Text":"Let\u0027s mark them in 0, negative 8."},{"Start":"00:59.875 ","End":"01:03.240","Text":"Let\u0027s say that 8 is around here."},{"Start":"01:03.240 ","End":"01:09.840","Text":"This is minus 8 and here we have x is 8, y is 0."},{"Start":"01:09.840 ","End":"01:13.165","Text":"That would be somewhere here, 8,0."},{"Start":"01:13.165 ","End":"01:17.855","Text":"Now let me draw a line through that."},{"Start":"01:17.855 ","End":"01:21.770","Text":"Next is the parabola."},{"Start":"01:21.770 ","End":"01:25.570","Text":"For the parabola we\u0027ll also make a table,"},{"Start":"01:25.570 ","End":"01:31.685","Text":"and here we have x in terms of y."},{"Start":"01:31.685 ","End":"01:33.980","Text":"We\u0027ll try values of y."},{"Start":"01:33.980 ","End":"01:40.814","Text":"If y is 0 then x is 2 and"},{"Start":"01:40.814 ","End":"01:49.530","Text":"if y is 1 plus or minus 1 then x will equal 3,"},{"Start":"01:49.530 ","End":"01:55.500","Text":"and if y is plus or minus 2 squared is 4 plus 2 is 6."},{"Start":"01:55.500 ","End":"01:58.925","Text":"That\u0027s 5 values and that should be enough."},{"Start":"01:58.925 ","End":"02:02.690","Text":"Yeah, I wrote here also x in terms of why thinking"},{"Start":"02:02.690 ","End":"02:07.305","Text":"that we probably need that just like x is in terms of y here."},{"Start":"02:07.305 ","End":"02:09.990","Text":"We\u0027re actually looking at things sideways,"},{"Start":"02:09.990 ","End":"02:13.045","Text":"parabola on its side and so on."},{"Start":"02:13.045 ","End":"02:15.890","Text":"X is 2, y is 0,"},{"Start":"02:15.890 ","End":"02:17.915","Text":"so it\u0027s something like that,"},{"Start":"02:17.915 ","End":"02:23.190","Text":"2,0 and then 3 plus or minus 1,"},{"Start":"02:23.190 ","End":"02:28.425","Text":"so something like this and like this."},{"Start":"02:28.425 ","End":"02:32.280","Text":"Then 6 plus or minus 2,"},{"Start":"02:32.280 ","End":"02:39.810","Text":"so 6 is somewhere here and we have plus 1 that should be plus 2,"},{"Start":"02:39.810 ","End":"02:43.250","Text":"something this, 6 minus 2."},{"Start":"02:43.250 ","End":"02:48.355","Text":"It looks like it\u0027s going to be actually on the line and then let me just draw in"},{"Start":"02:48.355 ","End":"02:55.680","Text":"here and continue and here also carefully."},{"Start":"02:55.680 ","End":"03:00.575","Text":"Not such a great job around here but you get the idea."},{"Start":"03:00.575 ","End":"03:08.095","Text":"Okay, now the area we\u0027re talking about is this area here and I\u0027ll shade it."},{"Start":"03:08.095 ","End":"03:13.850","Text":"Once again we are going to do this integral where x is"},{"Start":"03:13.850 ","End":"03:19.040","Text":"a function of y and we\u0027re going to divide it up horizontally."},{"Start":"03:19.040 ","End":"03:22.055","Text":"I\u0027ll just remind you what was bad with doing it the other way."},{"Start":"03:22.055 ","End":"03:24.470","Text":"The other way we had to split it up into"},{"Start":"03:24.470 ","End":"03:28.520","Text":"2 bits and in this bit we have to take the top minus the bottom."},{"Start":"03:28.520 ","End":"03:34.355","Text":"Same here, only the here 2 functions are not really defined well."},{"Start":"03:34.355 ","End":"03:39.125","Text":"In 1 case we\u0027re taking y in terms of x and y squared is x minus 2."},{"Start":"03:39.125 ","End":"03:43.325","Text":"This will be plus the square root and this will be minus the square root."},{"Start":"03:43.325 ","End":"03:45.380","Text":"The integral will be hard to compute."},{"Start":"03:45.380 ","End":"03:46.940","Text":"We have to break it up."},{"Start":"03:46.940 ","End":"03:48.680","Text":"Not advisable."},{"Start":"03:48.680 ","End":"03:53.120","Text":"Easiest thing is to divide up horizontally."},{"Start":"03:53.120 ","End":"03:56.270","Text":"Think of it as the horizontal integral into"},{"Start":"03:56.270 ","End":"04:00.800","Text":"strips and that way all we need to do is, well,"},{"Start":"04:00.800 ","End":"04:04.670","Text":"first of all, we have to find out these 2 points and that\u0027s not hard to"},{"Start":"04:04.670 ","End":"04:09.990","Text":"do because we just intersect 2 equations and 2 unknowns."},{"Start":"04:10.640 ","End":"04:14.040","Text":"From here to here we already have"},{"Start":"04:14.040 ","End":"04:17.360","Text":"these 2 points as the y-coordinates of these and we\u0027ll take"},{"Start":"04:17.360 ","End":"04:22.130","Text":"the integral dy going from here to here"},{"Start":"04:22.130 ","End":"04:27.870","Text":"of the rightmost 1 which is the line minus the parabola."},{"Start":"04:27.870 ","End":"04:32.180","Text":"Let\u0027s do the first thing is to find the points of intersection."},{"Start":"04:32.180 ","End":"04:43.170","Text":"We have 2 equations and 2 unknowns and I\u0027m going to take this version that x is y plus 8."},{"Start":"04:43.170 ","End":"04:51.240","Text":"For the line and for the parabola it\u0027s all ready given that x equals y squared plus 2."},{"Start":"04:51.240 ","End":"04:56.275","Text":"To solve these 2 I equate the right-hand sides."},{"Start":"04:56.275 ","End":"05:02.795","Text":"I get that y squared plus 2 is equal to y plus 8."},{"Start":"05:02.795 ","End":"05:04.625","Text":"Bring everything to the left,"},{"Start":"05:04.625 ","End":"05:11.640","Text":"y squared minus y minus 6 equals 0."},{"Start":"05:11.640 ","End":"05:17.220","Text":"There\u0027s 2 solutions or if you like I could break it up into y plus"},{"Start":"05:17.220 ","End":"05:24.410","Text":"2 y minus 3 is 0 or just straightaway use the formula and get the answers."},{"Start":"05:24.410 ","End":"05:31.675","Text":"Either way we get that y is equal to minus 2 or 3."},{"Start":"05:31.675 ","End":"05:39.500","Text":"Now I can write that already here that this is minus 2 and this is 3."},{"Start":"05:39.500 ","End":"05:44.840","Text":"All we need is looking sideways is the integral."},{"Start":"05:44.840 ","End":"05:50.560","Text":"Now remember we\u0027re doing it as x as a function of y and it\u0027s going to be dy."},{"Start":"05:50.560 ","End":"06:01.500","Text":"The integral from minus 2 to 3 of the top 1 is really the rightmost 1 is the line,"},{"Start":"06:01.500 ","End":"06:07.755","Text":"and the straight line is that x equals y plus 8."},{"Start":"06:07.755 ","End":"06:13.575","Text":"It\u0027s y plus 8 for the top and for the bottom,"},{"Start":"06:13.575 ","End":"06:17.970","Text":"it\u0027s y squared plus 2, this difference."},{"Start":"06:17.970 ","End":"06:25.255","Text":"It will be positive so I don\u0027t need absolute value and all this with respect to y."},{"Start":"06:25.255 ","End":"06:27.780","Text":"Straightforward integral."},{"Start":"06:27.780 ","End":"06:34.844","Text":"We get minus y squared gives us minus y cubed over 3"},{"Start":"06:34.844 ","End":"06:43.785","Text":"and then we have a plus y which gives us plus y squared over 2."},{"Start":"06:43.785 ","End":"06:48.885","Text":"Then we have also a plus 6 from the 8 minus 2,"},{"Start":"06:48.885 ","End":"06:52.185","Text":"so this would be plus 6y."},{"Start":"06:52.185 ","End":"06:57.090","Text":"All this has to be taken between minus 2 and 3."},{"Start":"06:57.090 ","End":"06:59.910","Text":"Let\u0027s substitute 3 first."},{"Start":"06:59.910 ","End":"07:08.100","Text":"We get 3 cubed over 3 is just 3 squared so it\u0027s minus 9."},{"Start":"07:08.100 ","End":"07:10.920","Text":"Then y squared over 2"},{"Start":"07:10.920 ","End":"07:18.900","Text":"is 3 squared over"},{"Start":"07:18.900 ","End":"07:26.900","Text":"2 is 9 over 2 and then plus 6 times 3 that\u0027s the 3 bit."},{"Start":"07:26.900 ","End":"07:30.395","Text":"Now we have to take away the negative 2 bit."},{"Start":"07:30.395 ","End":"07:38.700","Text":"For negative 2 I get minus 2 cubed over"},{"Start":"07:38.700 ","End":"07:47.265","Text":"3 plus minus 2 squared over 2 plus 6 times minus 2."},{"Start":"07:47.265 ","End":"07:50.400","Text":"Let\u0027s see what we get."},{"Start":"07:50.400 ","End":"07:55.890","Text":"Here I have 18 minus 9 is 9,"},{"Start":"07:55.890 ","End":"08:03.060","Text":"9 plus 4 and 1/2 is 13 and 1/2."},{"Start":"08:03.060 ","End":"08:05.850","Text":"Let\u0027s see what we get in here,"},{"Start":"08:05.850 ","End":"08:10.815","Text":"8 over 3 or 2 and 2/3."},{"Start":"08:10.815 ","End":"08:20.370","Text":"This bit is 4 over 2 which is 2 and this bit is minus 12."},{"Start":"08:20.370 ","End":"08:24.840","Text":"Altogether we have, well,"},{"Start":"08:24.840 ","End":"08:27.105","Text":"it\u0027s got a minus here."},{"Start":"08:27.105 ","End":"08:30.650","Text":"Let me make it a plus and"},{"Start":"08:30.650 ","End":"08:34.160","Text":"we\u0027ll just reverse the signs of everything so it\u0027s in our heads."},{"Start":"08:34.160 ","End":"08:40.190","Text":"I was going to say this is 12 minus 2 minus this 12 minus 4"},{"Start":"08:40.190 ","End":"08:49.030","Text":"and 2/3 and that is equal to 7 and a 1/3."},{"Start":"08:49.030 ","End":"08:55.705","Text":"That is equal to 13 and 7 is 20."},{"Start":"08:55.705 ","End":"09:00.640","Text":"1/2 plus 1/3 is 5/6."},{"Start":"09:00.640 ","End":"09:05.000","Text":"This is what I get to be the area."},{"Start":"09:05.000 ","End":"09:14.909","Text":"This area S is this and unless I made a mistake that\u0027s the answer and I shall highlight."}],"Thumbnail":null,"ID":4716},{"Watched":false,"Name":"Exercise 21","Duration":"4m 26s","ChapterTopicVideoID":4709,"CourseChapterTopicPlaylistID":3985,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"This exercise is a backward exercise."},{"Start":"00:03.420 ","End":"00:05.190","Text":"I\u0027ll explain in what sense."},{"Start":"00:05.190 ","End":"00:10.260","Text":"Usually, we have to compute an area with the help of definite integrals."},{"Start":"00:10.260 ","End":"00:12.000","Text":"Here we\u0027re going to do the opposite."},{"Start":"00:12.000 ","End":"00:15.315","Text":"We\u0027re going to compute a definite integral with the help of areas."},{"Start":"00:15.315 ","End":"00:17.415","Text":"Let\u0027s start with part a,"},{"Start":"00:17.415 ","End":"00:21.630","Text":"and we need to draw a little sketch to see what\u0027s going on."},{"Start":"00:21.630 ","End":"00:31.890","Text":"But if you look at the function y equals the square root of a squared minus x squared,"},{"Start":"00:31.890 ","End":"00:36.140","Text":"you could actually if you switch sides and everything,"},{"Start":"00:36.140 ","End":"00:45.585","Text":"see that this is part of a curve x squared plus y squared equals a squared."},{"Start":"00:45.585 ","End":"00:48.270","Text":"You square this, you get y squared equals a squared minus x squared,"},{"Start":"00:48.270 ","End":"00:49.715","Text":"bring the x squared over,"},{"Start":"00:49.715 ","End":"00:53.515","Text":"which is the equation of a circle."},{"Start":"00:53.515 ","End":"00:55.655","Text":"Essentially, what we have,"},{"Start":"00:55.655 ","End":"00:57.110","Text":"and I\u0027ll just draw a quick sketch,"},{"Start":"00:57.110 ","End":"01:00.425","Text":"we don\u0027t need any great work of art here."},{"Start":"01:00.425 ","End":"01:03.265","Text":"We have a circle of radius a."},{"Start":"01:03.265 ","End":"01:06.870","Text":"This is a, this is a."},{"Start":"01:06.870 ","End":"01:11.280","Text":"Now, we are given from 0 to a,"},{"Start":"01:11.280 ","End":"01:13.215","Text":"the integral of the square root."},{"Start":"01:13.215 ","End":"01:16.079","Text":"We\u0027re not talking about the whole thing,"},{"Start":"01:16.079 ","End":"01:19.340","Text":"we\u0027re just talking about, first of all,"},{"Start":"01:19.340 ","End":"01:22.730","Text":"the positive square root and only from 0 to a,"},{"Start":"01:22.730 ","End":"01:29.220","Text":"so we\u0027re talking about this and the integral from 0 to a,"},{"Start":"01:29.220 ","End":"01:34.390","Text":"so essentially, we\u0027re looking at this area here."},{"Start":"01:34.390 ","End":"01:37.365","Text":"Now, this is quarter of the circle."},{"Start":"01:37.365 ","End":"01:40.145","Text":"The area of quarter of a circle,"},{"Start":"01:40.145 ","End":"01:47.765","Text":"we know it\u0027s simply Pi a squared is the area of the whole circle divided by 4."},{"Start":"01:47.765 ","End":"01:54.080","Text":"This is S. S is 1/4 of the circle,"},{"Start":"01:54.080 ","End":"01:59.700","Text":"which is 1/4 of Pi radius squared,"},{"Start":"01:59.700 ","End":"02:01.905","Text":"which is Pi a squared."},{"Start":"02:01.905 ","End":"02:08.190","Text":"This is equal to 1/4 Pi a squared."},{"Start":"02:08.190 ","End":"02:10.530","Text":"That\u0027s so much for a."},{"Start":"02:10.530 ","End":"02:15.770","Text":"Now, in b, we have an integral dy which means that instead of taking vertical,"},{"Start":"02:15.770 ","End":"02:17.615","Text":"we\u0027re going to have something horizontal."},{"Start":"02:17.615 ","End":"02:20.290","Text":"Let\u0027s see what happens in b."},{"Start":"02:20.290 ","End":"02:23.930","Text":"In b, we have this,"},{"Start":"02:23.930 ","End":"02:31.640","Text":"which is part of the function x equals the square root of a squared minus y squared,"},{"Start":"02:31.640 ","End":"02:33.290","Text":"and if you rearrange it,"},{"Start":"02:33.290 ","End":"02:40.790","Text":"you\u0027ll see that from this we get that also x squared plus a squared equals y squared."},{"Start":"02:40.790 ","End":"02:42.740","Text":"If we sketch it again,"},{"Start":"02:42.740 ","End":"02:45.275","Text":"we essentially get the same picture."},{"Start":"02:45.275 ","End":"02:54.240","Text":"We have a circle of radius a so that this is a and this is a."},{"Start":"02:54.240 ","End":"02:58.395","Text":"But this time, x is a function of y."},{"Start":"02:58.395 ","End":"03:02.520","Text":"Furthermore, it goes from minus a to a."},{"Start":"03:02.520 ","End":"03:07.520","Text":"Now it\u0027s not the whole circle because if x is equal to plus the square root"},{"Start":"03:07.520 ","End":"03:12.440","Text":"of this and x is on the right-hand side of the y-axis,"},{"Start":"03:12.440 ","End":"03:16.965","Text":"it\u0027s only the positive or non-zero x."},{"Start":"03:16.965 ","End":"03:21.975","Text":"What we get is from minus a to a,"},{"Start":"03:21.975 ","End":"03:28.160","Text":"and the area under this bit is x equals the square root."},{"Start":"03:28.160 ","End":"03:34.970","Text":"I\u0027ll just say that it\u0027s from here to here and all the inside,"},{"Start":"03:34.970 ","End":"03:37.220","Text":"I\u0027m not going to draw the whole thing."},{"Start":"03:37.220 ","End":"03:42.140","Text":"What we get is in this case a half circle."},{"Start":"03:42.140 ","End":"03:47.405","Text":"In this case, if I label it also as S in this case,"},{"Start":"03:47.405 ","End":"03:51.100","Text":"or maybe this should be S for the part a,"},{"Start":"03:51.100 ","End":"03:53.745","Text":"and this should be S for part b,"},{"Start":"03:53.745 ","End":"04:00.450","Text":"then S for part b is equal to 1/2 of the circle,"},{"Start":"04:00.450 ","End":"04:07.605","Text":"which is 1/2 same thing as here, Pi a squared."},{"Start":"04:07.605 ","End":"04:09.885","Text":"We\u0027ll just write it up here,"},{"Start":"04:09.885 ","End":"04:15.450","Text":"and this is equal to 1/2 Pi a squared."},{"Start":"04:15.450 ","End":"04:19.460","Text":"You can see that we can actually reverse the process of using"},{"Start":"04:19.460 ","End":"04:24.530","Text":"known areas such as circles from geometry to help us find definite integrals,"},{"Start":"04:24.530 ","End":"04:27.360","Text":"and we are done."}],"Thumbnail":null,"ID":4717}],"ID":3985},{"Name":"Curve Length","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Curve Length 1","Duration":"6m 6s","ChapterTopicVideoID":8443,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.400","Text":"In this clip, I\u0027ll be talking about 1 of the uses of"},{"Start":"00:03.400 ","End":"00:07.570","Text":"integration and that is in finding the length of a curve."},{"Start":"00:07.570 ","End":"00:10.570","Text":"For example, in this picture, we see we have a curve."},{"Start":"00:10.570 ","End":"00:16.975","Text":"1 might ask, what is the length of the curve from point A to point B?"},{"Start":"00:16.975 ","End":"00:18.640","Text":"I\u0027m giving an overview here."},{"Start":"00:18.640 ","End":"00:21.865","Text":"There are actually several subsections to this,"},{"Start":"00:21.865 ","End":"00:24.655","Text":"and it all depends on how the curve is given."},{"Start":"00:24.655 ","End":"00:26.290","Text":"In the very first 1,"},{"Start":"00:26.290 ","End":"00:30.005","Text":"y will be given explicitly as a function of x."},{"Start":"00:30.005 ","End":"00:32.155","Text":"We\u0027ll going to talk about first,"},{"Start":"00:32.155 ","End":"00:35.500","Text":"when y is y of x."},{"Start":"00:35.500 ","End":"00:37.090","Text":"In the next sections,"},{"Start":"00:37.090 ","End":"00:40.255","Text":"I\u0027ll be talking about possibility 2,"},{"Start":"00:40.255 ","End":"00:44.795","Text":"which is where x is given as a function of y."},{"Start":"00:44.795 ","End":"00:52.355","Text":"As a third possibility that both x and y are given in terms of a parameter t,"},{"Start":"00:52.355 ","End":"00:56.120","Text":"which is usually time in a physical problem."},{"Start":"00:56.120 ","End":"01:00.335","Text":"So x is x of t and y is y of"},{"Start":"01:00.335 ","End":"01:05.315","Text":"t. There is also a fourth 1 which I won\u0027t be getting into much,"},{"Start":"01:05.315 ","End":"01:09.710","Text":"just I\u0027ll say a few words about it and that is when we have an implicit function"},{"Start":"01:09.710 ","End":"01:14.670","Text":"that is something like f of x and y equals 0."},{"Start":"01:14.670 ","End":"01:18.620","Text":"In each case, the function will be given as well as 2 points,"},{"Start":"01:18.620 ","End":"01:20.630","Text":"A and B on the curve,"},{"Start":"01:20.630 ","End":"01:22.835","Text":"and we have to find the length of the curve."},{"Start":"01:22.835 ","End":"01:25.310","Text":"Let\u0027s begin with number 1."},{"Start":"01:25.310 ","End":"01:26.524","Text":"Here\u0027s the picture."},{"Start":"01:26.524 ","End":"01:28.490","Text":"We have y as a function of x."},{"Start":"01:28.490 ","End":"01:30.845","Text":"We\u0027ll call it y of x, could be f of x,"},{"Start":"01:30.845 ","End":"01:32.300","Text":"2 points A and B."},{"Start":"01:32.300 ","End":"01:35.720","Text":"Usually, we only need to know the x coordinates,"},{"Start":"01:35.720 ","End":"01:37.970","Text":"which is a and b."},{"Start":"01:37.970 ","End":"01:41.105","Text":"There is a formula and the formula is as follows;"},{"Start":"01:41.105 ","End":"01:46.910","Text":"l which is really a function of l from A to B is given by the formula,"},{"Start":"01:46.910 ","End":"01:54.680","Text":"the integral from a to b of the square root of 1 plus the derivative of y squared,"},{"Start":"01:54.680 ","End":"01:58.880","Text":"this function squared and the integral with respect to x."},{"Start":"01:58.880 ","End":"02:02.605","Text":"That\u0027s all that is to it and now for the example."},{"Start":"02:02.605 ","End":"02:10.400","Text":"Find the length of the curve y equals the square root of"},{"Start":"02:10.400 ","End":"02:15.260","Text":"1 minus x squared between"},{"Start":"02:15.260 ","End":"02:21.590","Text":"x equals minus 1 and x equals 1."},{"Start":"02:21.590 ","End":"02:25.550","Text":"The 3 pieces information we need to solve this,"},{"Start":"02:25.550 ","End":"02:29.210","Text":"is we need the function y of x,"},{"Start":"02:29.210 ","End":"02:34.220","Text":"we need the lower limit a and the upper limit b,"},{"Start":"02:34.220 ","End":"02:36.635","Text":"and if we have these 3 quantities,"},{"Start":"02:36.635 ","End":"02:39.005","Text":"then we can solve the problem."},{"Start":"02:39.005 ","End":"02:43.085","Text":"We\u0027re given the function of x, which is this."},{"Start":"02:43.085 ","End":"02:45.380","Text":"We are given the lower limit a,"},{"Start":"02:45.380 ","End":"02:48.845","Text":"which is this, and we\u0027re given b, which is this."},{"Start":"02:48.845 ","End":"02:53.375","Text":"That\u0027s all you need in order to plug into this formula here."},{"Start":"02:53.375 ","End":"02:55.145","Text":"Let\u0027s start."},{"Start":"02:55.145 ","End":"03:01.920","Text":"Now, notice that the formula needs the derivative of y with respect to x."},{"Start":"03:01.920 ","End":"03:08.570","Text":"The first thing we should do is differentiate y prime is equal to,"},{"Start":"03:08.570 ","End":"03:11.060","Text":"derivative of square root is 1"},{"Start":"03:11.060 ","End":"03:17.880","Text":"over twice the square root and the inner derivative at the top,"},{"Start":"03:17.880 ","End":"03:20.715","Text":"so it\u0027s minus 2x."},{"Start":"03:20.715 ","End":"03:23.070","Text":"We can cancel the 2,"},{"Start":"03:23.070 ","End":"03:26.735","Text":"this 2 disappears, and this 2 disappears."},{"Start":"03:26.735 ","End":"03:33.550","Text":"Next thing we need to figure out is 1 plus y prime squared."},{"Start":"03:33.550 ","End":"03:36.480","Text":"Let\u0027s, first of all, take y prime squared."},{"Start":"03:36.480 ","End":"03:42.455","Text":"We\u0027ll build it up. Y prime squared is equal to x squared,"},{"Start":"03:42.455 ","End":"03:46.580","Text":"because minus x all squared is x squared and the square root squared is just without"},{"Start":"03:46.580 ","End":"03:51.125","Text":"the square root over 1 minus x squared."},{"Start":"03:51.125 ","End":"03:56.755","Text":"Now what I need is 1 plus y prime squared."},{"Start":"03:56.755 ","End":"04:04.535","Text":"This is equal to 1 plus x squared over 1 minus x squared."},{"Start":"04:04.535 ","End":"04:06.380","Text":"Let me do some simplification."},{"Start":"04:06.380 ","End":"04:11.675","Text":"If I put it all over 1 minus x squared,"},{"Start":"04:11.675 ","End":"04:17.330","Text":"then here I get 1 minus x squared plus x squared."},{"Start":"04:17.330 ","End":"04:21.844","Text":"Once again, something cancels this x squared with this x squared,"},{"Start":"04:21.844 ","End":"04:24.095","Text":"and then I need to take the square root."},{"Start":"04:24.095 ","End":"04:29.885","Text":"Let\u0027s just do it all in 1 step and say that l from here,"},{"Start":"04:29.885 ","End":"04:32.045","Text":"we don\u0027t usually write the A and the B,"},{"Start":"04:32.045 ","End":"04:34.280","Text":"is equal to the integral."},{"Start":"04:34.280 ","End":"04:36.335","Text":"Now this a and this b,"},{"Start":"04:36.335 ","End":"04:45.020","Text":"were minus 1 and 1 and the square root of 1 plus y prime squared,"},{"Start":"04:45.020 ","End":"04:47.360","Text":"which we found out just here,"},{"Start":"04:47.360 ","End":"04:50.660","Text":"is 1 over 1 minus x squared."},{"Start":"04:50.660 ","End":"04:51.930","Text":"I could even complete it here,"},{"Start":"04:51.930 ","End":"04:54.375","Text":"it\u0027s 1 over 1 minus the x squared."},{"Start":"04:54.375 ","End":"04:57.760","Text":"The square root I can take separately for top and bottom."},{"Start":"04:57.760 ","End":"05:01.265","Text":"Let me rewrite that as square root of 1,"},{"Start":"05:01.265 ","End":"05:07.850","Text":"which is 1 over the square root of 1 minus x squared dx."},{"Start":"05:07.850 ","End":"05:09.560","Text":"This is an immediate integral."},{"Start":"05:09.560 ","End":"05:11.870","Text":"It could be done by substitution,"},{"Start":"05:11.870 ","End":"05:15.530","Text":"but I\u0027m looking at the table of integrals and I see that the"},{"Start":"05:15.530 ","End":"05:20.974","Text":"integral of 1 over the square root of 1 minus x squared is arcsine x,"},{"Start":"05:20.974 ","End":"05:25.625","Text":"which I have to take between minus 1 and 1."},{"Start":"05:25.625 ","End":"05:31.370","Text":"Now this is just equal to arcsine of the upper limit"},{"Start":"05:31.370 ","End":"05:37.245","Text":"of 1 minus arc sine of minus 1."},{"Start":"05:37.245 ","End":"05:41.360","Text":"Arcsine of 1 is the angle whose sine is 1,"},{"Start":"05:41.360 ","End":"05:43.235","Text":"which is 90 degrees."},{"Start":"05:43.235 ","End":"05:45.215","Text":"But we\u0027re working in radians,"},{"Start":"05:45.215 ","End":"05:49.340","Text":"so that\u0027s Pi over 2 and the angle whose sine is minus"},{"Start":"05:49.340 ","End":"05:53.945","Text":"1 is minus 90 degrees or minus Pi over 2."},{"Start":"05:53.945 ","End":"05:56.360","Text":"Of course, you could also do this on the calculator and then you\u0027ll"},{"Start":"05:56.360 ","End":"05:58.970","Text":"get a numerical estimation."},{"Start":"05:58.970 ","End":"06:06.550","Text":"This minus this is just equal to Pi and Pi is the answer."}],"Thumbnail":null,"ID":8640},{"Watched":false,"Name":"Curve Length 2","Duration":"2m 42s","ChapterTopicVideoID":8444,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.475","Text":"Now, we come to Part 2."},{"Start":"00:02.475 ","End":"00:05.910","Text":"Part 2 is when x is given as a function of y,"},{"Start":"00:05.910 ","End":"00:07.620","Text":"and here\u0027s the sketch."},{"Start":"00:07.620 ","End":"00:11.265","Text":"We want the length of the curve in red."},{"Start":"00:11.265 ","End":"00:14.100","Text":"We couldn\u0027t actually take the length of the curve if we"},{"Start":"00:14.100 ","End":"00:16.920","Text":"include the black bits because then it wouldn\u0027t be a function."},{"Start":"00:16.920 ","End":"00:19.920","Text":"A horizontal line could cut the curve in 2 places,"},{"Start":"00:19.920 ","End":"00:21.930","Text":"so each x would have more than 1 y,"},{"Start":"00:21.930 ","End":"00:23.190","Text":"but in the red part,"},{"Start":"00:23.190 ","End":"00:25.200","Text":"x is a function of y."},{"Start":"00:25.200 ","End":"00:29.085","Text":"It\u0027s very similar to the case where y is a function of x."},{"Start":"00:29.085 ","End":"00:30.730","Text":"The formula is almost the same,"},{"Start":"00:30.730 ","End":"00:35.450","Text":"except that we reverse x and y instead of derivative of y with respect to x,"},{"Start":"00:35.450 ","End":"00:37.950","Text":"it\u0027s derivative of x with respect to y."},{"Start":"00:37.950 ","End":"00:42.035","Text":"The points are labeled c and d, not a and b,"},{"Start":"00:42.035 ","End":"00:44.300","Text":"and it\u0027s dy and not dx,"},{"Start":"00:44.300 ","End":"00:46.910","Text":"but otherwise, it\u0027s remarkably similar."},{"Start":"00:46.910 ","End":"00:48.680","Text":"L is the same as l ab."},{"Start":"00:48.680 ","End":"00:50.840","Text":"I could write this here."},{"Start":"00:50.840 ","End":"00:53.870","Text":"The curve, I\u0027m going to give is x as a function of y,"},{"Start":"00:53.870 ","End":"00:59.750","Text":"so I\u0027ll have it x equals minus 3/4y plus 3,"},{"Start":"00:59.750 ","End":"01:07.445","Text":"and between y equals 0 and y equals 4."},{"Start":"01:07.445 ","End":"01:09.950","Text":"Let\u0027s continue on a fresh page."},{"Start":"01:09.950 ","End":"01:12.815","Text":"The 3 things that are important are the function,"},{"Start":"01:12.815 ","End":"01:15.500","Text":"this time, the x of y,"},{"Start":"01:15.500 ","End":"01:17.200","Text":"and also c and d,"},{"Start":"01:17.200 ","End":"01:21.275","Text":"that\u0027s c and that\u0027s d. Let\u0027s get to figuring this thing out."},{"Start":"01:21.275 ","End":"01:23.510","Text":"First of all, we want to know x prime of y,"},{"Start":"01:23.510 ","End":"01:24.680","Text":"then we square it, add 1,"},{"Start":"01:24.680 ","End":"01:27.730","Text":"take the square root, take the integral. Let\u0027s build it up."},{"Start":"01:27.730 ","End":"01:31.010","Text":"If x is this, and that\u0027s x and y,"},{"Start":"01:31.010 ","End":"01:39.180","Text":"then x prime with respect to y is equal to minus 3/4 because it\u0027s just the coefficient,"},{"Start":"01:39.180 ","End":"01:40.520","Text":"and this goes to nothing."},{"Start":"01:40.520 ","End":"01:42.725","Text":"This should be easy to compute."},{"Start":"01:42.725 ","End":"01:45.710","Text":"We get that l is equal to,"},{"Start":"01:45.710 ","End":"01:47.990","Text":"I\u0027ll straightaway get to this,"},{"Start":"01:47.990 ","End":"01:57.195","Text":"the integral from 0-4 of the square root of 1 plus,"},{"Start":"01:57.195 ","End":"02:02.385","Text":"it doesn\u0027t matter that\u0027s a minus here, 3/4 squared dy."},{"Start":"02:02.385 ","End":"02:05.320","Text":"This is a function of y, it\u0027s a constant function,"},{"Start":"02:05.320 ","End":"02:09.380","Text":"so this equals the integral from 0-4."},{"Start":"02:09.380 ","End":"02:11.870","Text":"Let me just simplify it before I integrate."},{"Start":"02:11.870 ","End":"02:16.100","Text":"This comes out to be 5 over 4,"},{"Start":"02:16.100 ","End":"02:20.290","Text":"and I\u0027ll let you do the computation, dy."},{"Start":"02:20.300 ","End":"02:22.500","Text":"This is just a constant,"},{"Start":"02:22.500 ","End":"02:26.560","Text":"so it\u0027s 5 over 4y,"},{"Start":"02:26.560 ","End":"02:29.420","Text":"but taken between 0 and 4."},{"Start":"02:29.420 ","End":"02:31.610","Text":"Now, if I put in y equals 4,"},{"Start":"02:31.610 ","End":"02:34.640","Text":"I get 5 times 4 over 4 is 5."},{"Start":"02:34.640 ","End":"02:36.934","Text":"If I put in 0, I get 0,"},{"Start":"02:36.934 ","End":"02:39.755","Text":"so the answer is just 5."},{"Start":"02:39.755 ","End":"02:43.320","Text":"That illustrates the second case."}],"Thumbnail":null,"ID":8641},{"Watched":false,"Name":"Curve Length 3","Duration":"3m 29s","ChapterTopicVideoID":8445,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:04.605","Text":"Now, we come to Part 3 of Length of a Curve."},{"Start":"00:04.605 ","End":"00:08.430","Text":"This time, the curve is given in parametric form,"},{"Start":"00:08.430 ","End":"00:12.420","Text":"both x and y are given as functions of a parameter t."},{"Start":"00:12.420 ","End":"00:18.069","Text":"Often but not always this is expressed as a physical problem where t is time."},{"Start":"00:18.069 ","End":"00:25.040","Text":"The quantities that are important here are a function of t that defines x,"},{"Start":"00:25.040 ","End":"00:27.425","Text":"another function y of t,"},{"Start":"00:27.425 ","End":"00:35.010","Text":"and 2 constants, time a and time b that when we plug them into x of t,"},{"Start":"00:35.010 ","End":"00:37.100","Text":"they will give me a and b."},{"Start":"00:37.100 ","End":"00:41.690","Text":"So t_a actually corresponds to a and t_b corresponds to b."},{"Start":"00:41.690 ","End":"00:45.650","Text":"We have a formula again and this time it\u0027s a little bit different,"},{"Start":"00:45.650 ","End":"00:49.880","Text":"there\u0027s no 1 plus l is l from a to b,"},{"Start":"00:49.880 ","End":"00:51.335","Text":"you could write that here too."},{"Start":"00:51.335 ","End":"00:53.900","Text":"We need to know the derivative of x,"},{"Start":"00:53.900 ","End":"00:55.730","Text":"the derivative of y with respect to t,"},{"Start":"00:55.730 ","End":"00:57.155","Text":"plug into the formula,"},{"Start":"00:57.155 ","End":"01:00.065","Text":"substitute the limits, and we\u0027ll get the answer."},{"Start":"01:00.065 ","End":"01:02.780","Text":"Best thing to do is to give an example."},{"Start":"01:02.780 ","End":"01:06.635","Text":"So we have to find the length of the parameterized curve."},{"Start":"01:06.635 ","End":"01:10.190","Text":"This time I give you the functions x of t,"},{"Start":"01:10.190 ","End":"01:12.080","Text":"which in this case is cosine t."},{"Start":"01:12.080 ","End":"01:13.880","Text":"The other function which gives y,"},{"Start":"01:13.880 ","End":"01:15.385","Text":"which is sine t,"},{"Start":"01:15.385 ","End":"01:21.110","Text":"and we have the t_a and t_b which are 0 and 2 Pi."},{"Start":"01:21.110 ","End":"01:23.270","Text":"Now the formula\u0027s still up here,"},{"Start":"01:23.270 ","End":"01:26.015","Text":"but let\u0027s just write the derivatives."},{"Start":"01:26.015 ","End":"01:32.165","Text":"The x prime with respect to t is minus sine t"},{"Start":"01:32.165 ","End":"01:36.110","Text":"and y prime is cosine t."},{"Start":"01:36.110 ","End":"01:39.275","Text":"So if I substitute in here,"},{"Start":"01:39.275 ","End":"01:47.060","Text":"I will get that the length of curve is equal to the integral from 0 to 2 Pi"},{"Start":"01:47.060 ","End":"01:53.425","Text":"of the derivative with respect to x squared is sine squared t,"},{"Start":"01:53.425 ","End":"01:56.040","Text":"there is a square root also,"},{"Start":"01:56.040 ","End":"02:02.710","Text":"and y prime squared will be cosine squared t dt."},{"Start":"02:02.710 ","End":"02:05.990","Text":"If you remember your trigonometrical identities,"},{"Start":"02:05.990 ","End":"02:10.330","Text":"you\u0027ll recall that sine squared t plus cosine squared t is 1,"},{"Start":"02:10.330 ","End":"02:17.160","Text":"so this comes out to be the integral from 0 to 2 Pi of 1 dt."},{"Start":"02:17.160 ","End":"02:22.440","Text":"The integral of 1 is just t from 0 to 2 Pi,"},{"Start":"02:22.440 ","End":"02:25.260","Text":"that makes it 2 Pi minus 0,"},{"Start":"02:25.260 ","End":"02:28.395","Text":"so the answer is 2 Pi."},{"Start":"02:28.395 ","End":"02:31.105","Text":"We\u0027re done with Part 3."},{"Start":"02:31.105 ","End":"02:35.150","Text":"As for Part 4, I\u0027ll mention it just very briefly."},{"Start":"02:35.150 ","End":"02:38.810","Text":"I wasn\u0027t really going to talk about implicit differentiation,"},{"Start":"02:38.810 ","End":"02:40.400","Text":"but I\u0027ll just say a few words."},{"Start":"02:40.400 ","End":"02:43.280","Text":"When we have f of x and y equals 0,"},{"Start":"02:43.280 ","End":"02:45.535","Text":"that\u0027s an implicit function."},{"Start":"02:45.535 ","End":"02:50.120","Text":"If we have a length of curve problem involving this,"},{"Start":"02:50.120 ","End":"02:52.415","Text":"there\u0027s really several things you could do,"},{"Start":"02:52.415 ","End":"02:54.860","Text":"and in the problem itself you\u0027ll see you could"},{"Start":"02:54.860 ","End":"02:58.655","Text":"isolate y in terms of x if that\u0027s possible,"},{"Start":"02:58.655 ","End":"03:03.090","Text":"or you could isolate x in terms of y,"},{"Start":"03:03.090 ","End":"03:05.735","Text":"or you could do implicit differentiation."},{"Start":"03:05.735 ","End":"03:08.660","Text":"I\u0027m not going to go into this, at least 1 of the exercises contains"},{"Start":"03:08.660 ","End":"03:12.860","Text":"an implicit function and an implicit differentiation and you\u0027ll see it there."},{"Start":"03:12.860 ","End":"03:16.805","Text":"It\u0027s not that frequent that you get asked this kind of question anyway."},{"Start":"03:16.805 ","End":"03:19.175","Text":"Having said a few words about Part 4,"},{"Start":"03:19.175 ","End":"03:24.005","Text":"we\u0027re done with the subject of length of curve via integration."},{"Start":"03:24.005 ","End":"03:30.150","Text":"I leave you to solve the exercises that follow this tutorial."}],"Thumbnail":null,"ID":8642},{"Watched":false,"Name":"Exercise 1","Duration":"9m 44s","ChapterTopicVideoID":4549,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"In this exercise, we have to find the length of curve of the function,"},{"Start":"00:04.950 ","End":"00:11.955","Text":"y equals x^2/3 between x equals 1 and x equals 8."},{"Start":"00:11.955 ","End":"00:16.469","Text":"Now, I\u0027ve written here the formula together with the sketch."},{"Start":"00:16.469 ","End":"00:22.265","Text":"The curve length is given by the formula length equals this integral, and so on."},{"Start":"00:22.265 ","End":"00:24.300","Text":"Let\u0027s start computing it."},{"Start":"00:24.300 ","End":"00:28.770","Text":"The first thing we need is y prime, which is here."},{"Start":"00:28.770 ","End":"00:33.870","Text":"If y equals x^2/3,"},{"Start":"00:33.870 ","End":"00:37.350","Text":"then we get that y prime is equal to,"},{"Start":"00:37.350 ","End":"00:38.985","Text":"just using the exponent rule,"},{"Start":"00:38.985 ","End":"00:43.915","Text":"is 2/3 x to the power of minus 1/3."},{"Start":"00:43.915 ","End":"00:47.090","Text":"Now I can substitute it in here and get the"},{"Start":"00:47.090 ","End":"00:51.035","Text":"integral that L is equal to, and I\u0027m looking here,"},{"Start":"00:51.035 ","End":"00:59.160","Text":"the integral from 1-8 of the square root of 1 plus"},{"Start":"00:59.160 ","End":"01:07.950","Text":"y prime squared is 2/3 x to the minus 1/3 squared dx."},{"Start":"01:07.950 ","End":"01:11.770","Text":"Let\u0027s do some simplification."},{"Start":"01:11.950 ","End":"01:19.610","Text":"I would like to compute the square root at the side without taking the whole integral."},{"Start":"01:19.610 ","End":"01:21.290","Text":"I want to simplify it."},{"Start":"01:21.290 ","End":"01:30.370","Text":"What I get here is the square root of 1 plus now 2/3 squared is 4/9."},{"Start":"01:30.370 ","End":"01:37.605","Text":"X to the minus 1/3 squared is x to the minus 2/3."},{"Start":"01:37.605 ","End":"01:39.690","Text":"This is equal to,"},{"Start":"01:39.690 ","End":"01:41.510","Text":"I\u0027m going to do 2 steps in 1."},{"Start":"01:41.510 ","End":"01:43.670","Text":"First of all, I\u0027m going to make this thing to"},{"Start":"01:43.670 ","End":"01:47.165","Text":"the power of 1/2 instead of the square root."},{"Start":"01:47.165 ","End":"01:52.700","Text":"I\u0027m going to put the negative exponent as an exponent in the denominator."},{"Start":"01:52.700 ","End":"02:02.010","Text":"What I get is 1 plus 4 over 9x^2/3."},{"Start":"02:02.470 ","End":"02:07.280","Text":"Next I\u0027m going to make a common denominator."},{"Start":"02:07.280 ","End":"02:10.440","Text":"I\u0027m going to put it all over 9x^2/3."},{"Start":"02:10.440 ","End":"02:17.255","Text":"All I have to do is multiply this by 9 times x^2/3."},{"Start":"02:17.255 ","End":"02:20.110","Text":"Here I get 4."},{"Start":"02:20.110 ","End":"02:25.195","Text":"All this is to the power of 1/2."},{"Start":"02:25.195 ","End":"02:27.440","Text":"Now using the laws of exponents,"},{"Start":"02:27.440 ","End":"02:31.275","Text":"I can take top and bottom separately."},{"Start":"02:31.275 ","End":"02:35.085","Text":"The numerator to the power of 1/2,"},{"Start":"02:35.085 ","End":"02:39.555","Text":"I\u0027ll just write it as the square root,"},{"Start":"02:39.555 ","End":"02:43.750","Text":"Set it to the power of 1/2 of 9x^2/3,"},{"Start":"02:44.190 ","End":"02:51.950","Text":"plus 4 over the square root of 9x^2/3."},{"Start":"02:52.610 ","End":"02:56.390","Text":"Now that I\u0027m thinking, I don\u0027t know why I went to the power of 1/2,"},{"Start":"02:56.390 ","End":"02:58.465","Text":"I should have left it all at square root."},{"Start":"02:58.465 ","End":"03:02.435","Text":"What I want to do is simplify the denominator."},{"Start":"03:02.435 ","End":"03:04.490","Text":"When we have a square root of a product,"},{"Start":"03:04.490 ","End":"03:07.670","Text":"we take the square root of each of them."},{"Start":"03:07.670 ","End":"03:10.160","Text":"I\u0027m just talking about the denominator now."},{"Start":"03:10.160 ","End":"03:17.035","Text":"The square root of 9x^2/3 is"},{"Start":"03:17.035 ","End":"03:24.445","Text":"equal to the square root of 9 times the square root of x^2/3."},{"Start":"03:24.445 ","End":"03:28.125","Text":"This equals square root of 9 is 3."},{"Start":"03:28.125 ","End":"03:33.765","Text":"Square root, I can think of this as x^1/3 squared."},{"Start":"03:33.765 ","End":"03:39.875","Text":"This will give me the absolute value of x^1/3,"},{"Start":"03:39.875 ","End":"03:47.180","Text":"because square root in general of a squared is equal to absolute value of a."},{"Start":"03:47.180 ","End":"03:52.240","Text":"In any event, in our range x is going from 1-8,"},{"Start":"03:52.240 ","End":"03:54.180","Text":"which is all positive,"},{"Start":"03:54.180 ","End":"03:57.905","Text":"so I can get rid of the absolute value there."},{"Start":"03:57.905 ","End":"04:04.365","Text":"Continuing here, I get the square root"},{"Start":"04:04.365 ","End":"04:14.880","Text":"of 9x^2/3 plus 4 over 3x to the 1/3."},{"Start":"04:14.880 ","End":"04:17.235","Text":"Moving back up here."},{"Start":"04:17.235 ","End":"04:21.305","Text":"We just did the square root and we simplified it. Let\u0027s continue."},{"Start":"04:21.305 ","End":"04:29.795","Text":"L is equal to the integral from 1-8 of this thing here, which I simplified."},{"Start":"04:29.795 ","End":"04:37.310","Text":"It\u0027s the square root of 9x^2/3 plus"},{"Start":"04:37.310 ","End":"04:44.860","Text":"4 over 3x^1/3 dx."},{"Start":"04:44.860 ","End":"04:46.865","Text":"When I do a substitution,"},{"Start":"04:46.865 ","End":"04:52.340","Text":"the thing to try first is to let the square root be equal to"},{"Start":"04:52.340 ","End":"04:59.510","Text":"t. Let\u0027s try the square root of 9x^2/3 plus 4,"},{"Start":"04:59.510 ","End":"05:03.220","Text":"we\u0027ll substitute that as t. Of course,"},{"Start":"05:03.220 ","End":"05:05.960","Text":"I also need dx, but I don\u0027t like square roots,"},{"Start":"05:05.960 ","End":"05:08.705","Text":"so let\u0027s raise both sides to the power of 2."},{"Start":"05:08.705 ","End":"05:15.755","Text":"Just get 9x^2/3 plus 4 is equal to t squared."},{"Start":"05:15.755 ","End":"05:18.860","Text":"Now differentiate both sides."},{"Start":"05:18.860 ","End":"05:23.770","Text":"Let\u0027s see, the right-hand side is going to be easier 2t dt."},{"Start":"05:23.770 ","End":"05:25.230","Text":"Let\u0027s see, the left-hand side."},{"Start":"05:25.230 ","End":"05:26.990","Text":"Well, the 4 gives me nothing."},{"Start":"05:26.990 ","End":"05:29.690","Text":"I just have to differentiate this."},{"Start":"05:29.690 ","End":"05:35.150","Text":"Let\u0027s see, 2/3 times 9 is 6 and lower the exponent by 1,"},{"Start":"05:35.150 ","End":"05:38.160","Text":"that makes it x to the minus 1/3."},{"Start":"05:38.160 ","End":"05:44.054","Text":"I can cancel the 2 with the 6 and write 3 here,"},{"Start":"05:44.054 ","End":"05:48.195","Text":"might help and of course there was a dx here."},{"Start":"05:48.195 ","End":"05:53.190","Text":"Now I can say what dx is equal to."},{"Start":"05:53.190 ","End":"05:58.020","Text":"I get that dx equals"},{"Start":"05:58.020 ","End":"06:04.680","Text":"t dt divided by this 3 here,"},{"Start":"06:04.680 ","End":"06:09.430","Text":"and this x to the minus 1/3 here."},{"Start":"06:09.430 ","End":"06:16.190","Text":"First of all, I\u0027m going to replace this whole expression under the square root."},{"Start":"06:16.190 ","End":"06:22.480","Text":"This is going to be t. The dx here,"},{"Start":"06:22.480 ","End":"06:27.469","Text":"I\u0027m going to replace by this here."},{"Start":"06:27.469 ","End":"06:31.055","Text":"But there\u0027s also something else I\u0027m going to have to replace,"},{"Start":"06:31.055 ","End":"06:35.745","Text":"and that is the limits of integration 8 and 1."},{"Start":"06:35.745 ","End":"06:43.490","Text":"Because this really should be written as x equals 1-x equals 8."},{"Start":"06:43.490 ","End":"06:47.750","Text":"I have to know what happens when x equals 1,"},{"Start":"06:47.750 ","End":"06:49.225","Text":"what does t equal?"},{"Start":"06:49.225 ","End":"06:52.215","Text":"If x equals 1,"},{"Start":"06:52.215 ","End":"06:55.545","Text":"then t is, x equals 1,"},{"Start":"06:55.545 ","End":"06:57.960","Text":"that means 1^2/3 is 1,"},{"Start":"06:57.960 ","End":"07:00.330","Text":"that\u0027s 9 plus 4 is 13,"},{"Start":"07:00.330 ","End":"07:04.965","Text":"t equals square root of 13."},{"Start":"07:04.965 ","End":"07:09.795","Text":"If x equals 8,"},{"Start":"07:09.795 ","End":"07:14.760","Text":"then 8^2/3 is 4,"},{"Start":"07:14.760 ","End":"07:17.160","Text":"4 times 9 is 36,"},{"Start":"07:17.160 ","End":"07:19.320","Text":"plus 4 is 40,"},{"Start":"07:19.320 ","End":"07:23.520","Text":"so t equals square root of 40."},{"Start":"07:23.520 ","End":"07:33.355","Text":"That\u0027s what I\u0027m also going to substitute this 1 and this 1 instead of the x."},{"Start":"07:33.355 ","End":"07:37.700","Text":"We end up getting the following."},{"Start":"07:37.700 ","End":"07:41.595","Text":"L equals the integral,"},{"Start":"07:41.595 ","End":"07:47.275","Text":"and this time from square root of 13 to square root of 40,"},{"Start":"07:47.275 ","End":"07:50.425","Text":"or just for emphasis I\u0027ll write the t,"},{"Start":"07:50.425 ","End":"07:55.630","Text":"that this is either limits for t. What\u0027s here is t,"},{"Start":"07:55.630 ","End":"07:58.930","Text":"this I\u0027ll leave as is,"},{"Start":"07:58.930 ","End":"08:04.550","Text":"didn\u0027t highlight it, I\u0027m leaving it as is 3x^1/3,"},{"Start":"08:04.550 ","End":"08:07.340","Text":"and dx from here is"},{"Start":"08:07.340 ","End":"08:12.050","Text":"tdt over"},{"Start":"08:12.050 ","End":"08:18.600","Text":"3x^minus 1/3."},{"Start":"08:18.600 ","End":"08:20.270","Text":"Let\u0027s simplify this."},{"Start":"08:20.270 ","End":"08:22.910","Text":"We can cancel some things,"},{"Start":"08:22.910 ","End":"08:27.100","Text":"x^1/3 cancels with x to the power of minus 1/3."},{"Start":"08:27.100 ","End":"08:34.830","Text":"Then what I\u0027m going to do is combine t and t to get t squared and 3 and 3 to get 9."},{"Start":"08:34.830 ","End":"08:43.550","Text":"We\u0027re going to end up with L equals the integral from square root of"},{"Start":"08:43.550 ","End":"08:47.580","Text":"13 to square root of 40"},{"Start":"08:47.580 ","End":"08:54.530","Text":"of t squared over 9 dt."},{"Start":"08:54.530 ","End":"08:58.610","Text":"What we\u0027re going to get is the integral of this."},{"Start":"08:58.610 ","End":"09:02.785","Text":"I raise the power by 1, it\u0027s t^3,"},{"Start":"09:02.785 ","End":"09:05.159","Text":"and then I divide by 3,"},{"Start":"09:05.159 ","End":"09:08.925","Text":"but the 3 combines with the 9 to give 27."},{"Start":"09:08.925 ","End":"09:18.920","Text":"This, I need to evaluate between square root of 13 and the square root of 40."},{"Start":"09:18.920 ","End":"09:27.140","Text":"This equals the square root of 40^3."},{"Start":"09:27.140 ","End":"09:28.950","Text":"They\u0027re both going to be over 27,"},{"Start":"09:28.950 ","End":"09:32.370","Text":"so I\u0027ll just combine that and put the 27 here."},{"Start":"09:32.370 ","End":"09:38.670","Text":"Square root of 13^3 over 27."},{"Start":"09:38.670 ","End":"09:41.555","Text":"I\u0027m not going to use a calculator here."},{"Start":"09:41.555 ","End":"09:43.130","Text":"This is the answer,"},{"Start":"09:43.130 ","End":"09:45.750","Text":"and we are done."}],"Thumbnail":null,"ID":4558},{"Watched":false,"Name":"Exercise 2","Duration":"6m 11s","ChapterTopicVideoID":4550,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.850","Text":"Here, we have to find the arc length of the curve as given here."},{"Start":"00:05.850 ","End":"00:10.470","Text":"This is the function and it\u0027s from 1-2."},{"Start":"00:10.470 ","End":"00:15.600","Text":"I\u0027ve written down the formula under the illustration."},{"Start":"00:15.600 ","End":"00:18.480","Text":"We\u0027re going to use this formula here."},{"Start":"00:18.480 ","End":"00:21.480","Text":"We have y, we have a and b,"},{"Start":"00:21.480 ","End":"00:23.595","Text":"we need y prime,"},{"Start":"00:23.595 ","End":"00:26.440","Text":"so let\u0027s do that first."},{"Start":"00:26.570 ","End":"00:31.795","Text":"But first, I\u0027ll rewrite y in a more convenient form."},{"Start":"00:31.795 ","End":"00:37.230","Text":"I\u0027ll write it as x^4 over"},{"Start":"00:37.230 ","End":"00:45.000","Text":"8 plus x^ minus 2 over 4."},{"Start":"00:45.000 ","End":"00:53.250","Text":"Then I can easily say that y prime is 4x cubed over 8."},{"Start":"00:53.250 ","End":"00:55.590","Text":"But 4 over 8 is 1/2,"},{"Start":"00:55.590 ","End":"01:01.485","Text":"so it\u0027s x cubed over 2."},{"Start":"01:01.485 ","End":"01:05.030","Text":"Here, we multiply the power,"},{"Start":"01:05.030 ","End":"01:06.725","Text":"we get minus 2."},{"Start":"01:06.725 ","End":"01:12.105","Text":"But minus 2 over 4 is minus 1/2,"},{"Start":"01:12.105 ","End":"01:17.360","Text":"so we get minus x^ minus 3 over 2."},{"Start":"01:17.360 ","End":"01:19.250","Text":"That half is the 2 here."},{"Start":"01:19.250 ","End":"01:22.200","Text":"We lower the power by 1."},{"Start":"01:22.550 ","End":"01:26.915","Text":"Now I want to substitute that in here."},{"Start":"01:26.915 ","End":"01:28.190","Text":"Let\u0027s, first of all,"},{"Start":"01:28.190 ","End":"01:33.315","Text":"figure out what is 1 plus y prime squared,"},{"Start":"01:33.315 ","End":"01:36.155","Text":"then we\u0027ll take the square root and the integral."},{"Start":"01:36.155 ","End":"01:38.975","Text":"We get 1 plus,"},{"Start":"01:38.975 ","End":"01:42.765","Text":"they should have taken the 1/2 outside the brackets,"},{"Start":"01:42.765 ","End":"01:49.365","Text":"but it\u0027s 1/2 x cubed minus x^ minus 3,"},{"Start":"01:49.365 ","End":"01:52.755","Text":"this thing squared, That\u0027s the squared here,"},{"Start":"01:52.755 ","End":"01:55.920","Text":"which equals 1 plus,"},{"Start":"01:55.920 ","End":"01:59.040","Text":"1/2 squared is 1/4,"},{"Start":"01:59.040 ","End":"02:03.375","Text":"and then I\u0027m using the formula for a minus b squared,"},{"Start":"02:03.375 ","End":"02:07.725","Text":"is a squared minus 2ab."},{"Start":"02:07.725 ","End":"02:09.975","Text":"This time this is just 1,"},{"Start":"02:09.975 ","End":"02:18.690","Text":"so it\u0027s minus 2 plus this thing squared is x^ minus 6."},{"Start":"02:20.060 ","End":"02:27.059","Text":"If I take the 1/4 outside,"},{"Start":"02:27.059 ","End":"02:29.385","Text":"everything makes this 4."},{"Start":"02:29.385 ","End":"02:34.500","Text":"So we\u0027ve got x^6 minus 2 plus"},{"Start":"02:34.500 ","End":"02:41.145","Text":"4 is plus 2 plus x^ minus 6."},{"Start":"02:41.145 ","End":"02:44.315","Text":"Now I want the square root of that."},{"Start":"02:44.315 ","End":"02:47.690","Text":"I guess I\u0027m building up to this integral slowly."},{"Start":"02:47.690 ","End":"02:56.910","Text":"The square root of 1 plus y prime squared is the square root of this, which is 1/2."},{"Start":"02:57.020 ","End":"03:00.560","Text":"This thing is a perfect square."},{"Start":"03:00.560 ","End":"03:03.590","Text":"I mean, look, this is the same as this,"},{"Start":"03:03.590 ","End":"03:05.510","Text":"but instead of a minus it\u0027s a plus,"},{"Start":"03:05.510 ","End":"03:08.550","Text":"and if I started out with a plus here,"},{"Start":"03:08.570 ","End":"03:18.445","Text":"I\u0027m claiming that the square root of this is just x cubed plus x^ minus 3."},{"Start":"03:18.445 ","End":"03:20.970","Text":"If you square this, you\u0027ll see,"},{"Start":"03:20.970 ","End":"03:22.935","Text":"we get x cubed squared, it\u0027s this,"},{"Start":"03:22.935 ","End":"03:25.295","Text":"plus twice, this times this gives this,"},{"Start":"03:25.295 ","End":"03:27.750","Text":"and this squared is this."},{"Start":"03:28.330 ","End":"03:32.155","Text":"Now we can write the integral,"},{"Start":"03:32.155 ","End":"03:42.410","Text":"and we can get the arc length or curve length as L equals the integral from a to b,"},{"Start":"03:42.410 ","End":"03:45.360","Text":"and a to b was 1-2."},{"Start":"03:45.740 ","End":"03:49.860","Text":"This thing is 1/2 of this thing."},{"Start":"03:49.860 ","End":"03:53.985","Text":"I can put the 1/2 in front and I get"},{"Start":"03:53.985 ","End":"04:02.440","Text":"x cubed plus x^ minus 3 dx."},{"Start":"04:03.710 ","End":"04:07.200","Text":"Now this is an easy integral."},{"Start":"04:07.200 ","End":"04:10.455","Text":"This is equal to 1/2."},{"Start":"04:10.455 ","End":"04:17.115","Text":"Let\u0027s see, x cubed gives me x^4 over 4,"},{"Start":"04:17.115 ","End":"04:23.240","Text":"and this gives me x^ minus 2 over minus 2,"},{"Start":"04:23.240 ","End":"04:26.080","Text":"so I\u0027ll put the minus there."},{"Start":"04:26.080 ","End":"04:30.690","Text":"This I want to take between 1 and 2."},{"Start":"04:30.690 ","End":"04:33.015","Text":"See, next step."},{"Start":"04:33.015 ","End":"04:37.085","Text":"Plug in 2, what do I get?"},{"Start":"04:37.085 ","End":"04:40.025","Text":"Well, let\u0027s leave the half completely outside."},{"Start":"04:40.025 ","End":"04:43.625","Text":"If I plug in 2, I get 2^4,"},{"Start":"04:43.625 ","End":"04:48.525","Text":"which is 16 over 4, which is 4."},{"Start":"04:48.525 ","End":"04:51.120","Text":"Here, if I put in 2,"},{"Start":"04:51.120 ","End":"04:53.130","Text":"I get 2^ minus 2,"},{"Start":"04:53.130 ","End":"04:56.715","Text":"which is a 1/4 over 2,"},{"Start":"04:56.715 ","End":"05:00.185","Text":"which is 1/8 minus,"},{"Start":"05:00.185 ","End":"05:04.025","Text":"let\u0027s see what I get when I put in 1,"},{"Start":"05:04.025 ","End":"05:14.230","Text":"I get 1^4 over 4 is a 1/4 minus 1 to any power is 1 over 2."},{"Start":"05:14.230 ","End":"05:17.340","Text":"Let\u0027s see what we get."},{"Start":"05:17.480 ","End":"05:20.010","Text":"Maybe I\u0027ll, first of all,"},{"Start":"05:20.010 ","End":"05:25.185","Text":"multiply by 1/2; so 1/2 times 4 is 2."},{"Start":"05:25.185 ","End":"05:28.650","Text":"This is minus 1/16,"},{"Start":"05:28.650 ","End":"05:31.785","Text":"and this is minus 1/8,"},{"Start":"05:31.785 ","End":"05:36.330","Text":"but this is plus 1/4."},{"Start":"05:36.330 ","End":"05:38.175","Text":"If I put it all on 16,"},{"Start":"05:38.175 ","End":"05:40.440","Text":"I leave the 2, it\u0027s 2,"},{"Start":"05:40.440 ","End":"05:47.910","Text":"it\u0027s minus 1/16 minus 2/16 plus 4/16."},{"Start":"05:47.910 ","End":"05:50.070","Text":"Let\u0027s see how many 16ths do we have?"},{"Start":"05:50.070 ","End":"05:52.635","Text":"We have plus 4, minus 2,"},{"Start":"05:52.635 ","End":"05:57.030","Text":"minus 1; 4 minus 2 minus 1 is 1."},{"Start":"05:57.030 ","End":"06:02.100","Text":"So it\u0027s 2 and 1/16."},{"Start":"06:02.100 ","End":"06:06.710","Text":"Or if you don\u0027t like mixed fractions and prefer improper fractions,"},{"Start":"06:06.710 ","End":"06:09.320","Text":"then it\u0027s 33 over 16."},{"Start":"06:09.320 ","End":"06:12.630","Text":"Any event we are done here."}],"Thumbnail":null,"ID":4559},{"Watched":false,"Name":"Exercise 3","Duration":"7m 46s","ChapterTopicVideoID":4551,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"Here we have another exercise in length of curve."},{"Start":"00:03.540 ","End":"00:06.150","Text":"I\u0027ve shown the diagram,"},{"Start":"00:06.150 ","End":"00:13.335","Text":"and the formula that the length of curve is from A to B is given by this expression."},{"Start":"00:13.335 ","End":"00:20.975","Text":"You can see that it involves y prime, and A and B in our case are 1 and 2,"},{"Start":"00:20.975 ","End":"00:23.990","Text":"and the y is given by,"},{"Start":"00:23.990 ","End":"00:26.780","Text":"the function of x is given here."},{"Start":"00:26.780 ","End":"00:29.990","Text":"Let\u0027s find y prime first."},{"Start":"00:29.990 ","End":"00:35.975","Text":"I\u0027ll rewrite y slightly differently as x to the 5th"},{"Start":"00:35.975 ","End":"00:42.570","Text":"over 15 plus x to the minus 3, over 4."},{"Start":"00:42.570 ","End":"00:44.540","Text":"See, instead of putting x cubed on the bottom,"},{"Start":"00:44.540 ","End":"00:47.420","Text":"I put it negative exponent here."},{"Start":"00:47.420 ","End":"00:57.465","Text":"Y prime is equal to 5_x to the 4th over 15 plus,"},{"Start":"00:57.465 ","End":"01:03.240","Text":"minus 3_x to the minus 4, over 4."},{"Start":"01:03.240 ","End":"01:05.490","Text":"Let\u0027s cancel something."},{"Start":"01:05.490 ","End":"01:09.450","Text":"5 and 15 goes 3 times."},{"Start":"01:09.450 ","End":"01:12.385","Text":"If I put a common denominator,"},{"Start":"01:12.385 ","End":"01:16.510","Text":"I will get that y prime is equal to 3 times 4,"},{"Start":"01:16.510 ","End":"01:19.455","Text":"is 12, so it\u0027s going to be over 12."},{"Start":"01:19.455 ","End":"01:25.035","Text":"X to the 4th over 3 is 4_x to the 4th over 12."},{"Start":"01:25.035 ","End":"01:28.065","Text":"Here I have minus 9,"},{"Start":"01:28.065 ","End":"01:33.665","Text":"so 3 times 3_x to the minus 4. That\u0027s y prime."},{"Start":"01:33.665 ","End":"01:41.125","Text":"Next I want, what is 1 plus y prime squared equal to?"},{"Start":"01:41.125 ","End":"01:44.050","Text":"Let\u0027s do the y prime squared first."},{"Start":"01:44.050 ","End":"01:50.110","Text":"What I have is 144 because I\u0027m squaring both numerator and denominator."},{"Start":"01:50.110 ","End":"01:55.715","Text":"The numerator using the formula from basic algebra,"},{"Start":"01:55.715 ","End":"01:58.970","Text":"I get this 1 squared,"},{"Start":"01:58.970 ","End":"02:02.980","Text":"which is 16_x to the 8th,"},{"Start":"02:02.980 ","End":"02:06.270","Text":"then minus twice this times this."},{"Start":"02:06.270 ","End":"02:09.650","Text":"Twice 4 times 9, is 72."},{"Start":"02:09.650 ","End":"02:12.750","Text":"X to the 4th times x to the minus 4 is just 1."},{"Start":"02:12.750 ","End":"02:14.510","Text":"I don\u0027t need anything else."},{"Start":"02:14.510 ","End":"02:20.665","Text":"Then plus 81_x to the minus 8."},{"Start":"02:20.665 ","End":"02:23.540","Text":"Now, I have to put here 1,"},{"Start":"02:23.540 ","End":"02:30.630","Text":"and I saved it because I want to put it as 144 over 144. That\u0027s equal to 1."},{"Start":"02:30.630 ","End":"02:34.260","Text":"Then it\u0027s easier for me to get the common denominator."},{"Start":"02:34.460 ","End":"02:39.770","Text":"In fact, what I get is almost the same as this."},{"Start":"02:39.770 ","End":"02:45.290","Text":"I have 144, I have 16_x to the 8th."},{"Start":"02:45.290 ","End":"02:49.260","Text":"I have 81_x to the minus 8."},{"Start":"02:49.260 ","End":"02:51.795","Text":"But look, the minus 72,"},{"Start":"02:51.795 ","End":"02:55.740","Text":"plus the 144, let me just circle it,"},{"Start":"02:55.740 ","End":"03:04.490","Text":"is 144, and this minus 72 together are going to give me plus 72."},{"Start":"03:04.490 ","End":"03:06.740","Text":"The only difference between this expression and"},{"Start":"03:06.740 ","End":"03:09.760","Text":"this expression is the minus and the plus."},{"Start":"03:09.760 ","End":"03:12.215","Text":"Because that\u0027s the case,"},{"Start":"03:12.215 ","End":"03:15.320","Text":"I can write this instead of a minus, I have a plus."},{"Start":"03:15.320 ","End":"03:17.495","Text":"It\u0027s also a perfect square."},{"Start":"03:17.495 ","End":"03:20.465","Text":"What I get is like this, but with a plus."},{"Start":"03:20.465 ","End":"03:22.770","Text":"I have 4_x to"},{"Start":"03:22.770 ","End":"03:32.970","Text":"the 4th plus 9_x to the minus 4 squared over 144."},{"Start":"03:32.970 ","End":"03:36.785","Text":"Now, if I take the square root of this,"},{"Start":"03:36.785 ","End":"03:40.940","Text":"the square root of 1 plus y prime squared,"},{"Start":"03:40.940 ","End":"03:45.920","Text":"I just get the square root of this, which is, well,"},{"Start":"03:45.920 ","End":"03:47.300","Text":"it\u0027s the absolute value,"},{"Start":"03:47.300 ","End":"03:51.235","Text":"but it\u0027s positive, so I don\u0027t need the absolute value."},{"Start":"03:51.235 ","End":"04:01.290","Text":"It\u0027s just 4_x to the 4th plus 9_x to the minus 4, over 12."},{"Start":"04:01.290 ","End":"04:03.920","Text":"Now I need to do the integral."},{"Start":"04:03.920 ","End":"04:09.920","Text":"What I need is the integral from 1-2 of this thing here,"},{"Start":"04:09.920 ","End":"04:11.240","Text":"which is this here,"},{"Start":"04:11.240 ","End":"04:18.615","Text":"of 4_x to the 4th plus 9_x to the minus 4,"},{"Start":"04:18.615 ","End":"04:24.230","Text":"over 12, d_x, which equals,"},{"Start":"04:24.230 ","End":"04:26.345","Text":"now what is this integral?"},{"Start":"04:26.345 ","End":"04:28.250","Text":"If I keep it over 12,"},{"Start":"04:28.250 ","End":"04:30.460","Text":"which is a constant."},{"Start":"04:30.460 ","End":"04:33.825","Text":"The integral of this 1 is,"},{"Start":"04:33.825 ","End":"04:35.160","Text":"raise the power by 1,"},{"Start":"04:35.160 ","End":"04:39.855","Text":"which makes it 5, and divide by 5."},{"Start":"04:39.855 ","End":"04:42.315","Text":"I get 4/5,"},{"Start":"04:42.315 ","End":"04:44.460","Text":"x to the 5th."},{"Start":"04:44.460 ","End":"04:49.185","Text":"The second 1 gives me, let\u0027s see."},{"Start":"04:49.185 ","End":"04:51.005","Text":"I raise the power by 1,"},{"Start":"04:51.005 ","End":"04:56.770","Text":"it\u0027s minus 3, and I divide by minus 3."},{"Start":"04:56.770 ","End":"05:00.885","Text":"It\u0027s 9 over minus 3,"},{"Start":"05:00.885 ","End":"05:03.915","Text":"x to the minus 3."},{"Start":"05:03.915 ","End":"05:07.720","Text":"All this taken between 1 and 2."},{"Start":"05:07.720 ","End":"05:10.100","Text":"I want to simplify a bit."},{"Start":"05:10.100 ","End":"05:15.830","Text":"This 9 over minus 3 is just minus 3."},{"Start":"05:15.910 ","End":"05:19.730","Text":"I don\u0027t want the fraction in the numerator as well."},{"Start":"05:19.730 ","End":"05:24.815","Text":"If I multiply numerator and denominator by 5,"},{"Start":"05:24.815 ","End":"05:27.930","Text":"that should make it easier."},{"Start":"05:27.930 ","End":"05:31.835","Text":"I\u0027ll just write it on the same line here."},{"Start":"05:31.835 ","End":"05:38.270","Text":"If I multiply the bottom by 5, I get 60,"},{"Start":"05:38.270 ","End":"05:43.325","Text":"and if I rewrite the top,"},{"Start":"05:43.325 ","End":"05:49.350","Text":"I get just 4_x to the 5th because the 5 cancels."},{"Start":"05:49.350 ","End":"05:54.600","Text":"Here I\u0027ll get minus 15_x to the minus 3,"},{"Start":"05:54.600 ","End":"06:01.350","Text":"over 60, and this between 1 and 2."},{"Start":"06:01.350 ","End":"06:06.055","Text":"You know what? I\u0027d like to split it up into 2 separate bits."},{"Start":"06:06.055 ","End":"06:08.990","Text":"I know that this calculation is not very important,"},{"Start":"06:08.990 ","End":"06:10.310","Text":"but let\u0027s just do it."},{"Start":"06:10.310 ","End":"06:13.610","Text":"I want to take 4_x to the 5th over 60."},{"Start":"06:13.610 ","End":"06:17.815","Text":"That will be x to the 5th over 15."},{"Start":"06:17.815 ","End":"06:21.195","Text":"This I need between 1 and 2."},{"Start":"06:21.195 ","End":"06:24.645","Text":"Minus 15 over 60 is 4."},{"Start":"06:24.645 ","End":"06:31.435","Text":"I need x to the minus 3, over 4."},{"Start":"06:31.435 ","End":"06:34.240","Text":"I\u0027ll put brackets here just to be safe."},{"Start":"06:34.240 ","End":"06:38.145","Text":"Also, between 1 and 2."},{"Start":"06:38.145 ","End":"06:41.745","Text":"This equals 2 to the 5th,"},{"Start":"06:41.745 ","End":"06:46.995","Text":"minus 1 to the 5th over 15,"},{"Start":"06:46.995 ","End":"06:51.225","Text":"less 2 to the minus 3,"},{"Start":"06:51.225 ","End":"06:56.380","Text":"minus 1 to the minus 3, over 4."},{"Start":"06:56.380 ","End":"06:58.355","Text":"What does this give me?"},{"Start":"06:58.355 ","End":"07:05.165","Text":"32 minus 1 is 31 over 15,"},{"Start":"07:05.165 ","End":"07:13.495","Text":"less 1/8, minus 1 is minus 7/8."},{"Start":"07:13.495 ","End":"07:15.570","Text":"That\u0027s going to make it plus."},{"Start":"07:15.570 ","End":"07:22.785","Text":"7/8 over 4 is 7 over 32."},{"Start":"07:22.785 ","End":"07:25.370","Text":"Finally, if we compute this,"},{"Start":"07:25.370 ","End":"07:33.660","Text":"I make it 1,097 over 480."},{"Start":"07:33.660 ","End":"07:36.260","Text":"If we just take 15 times 32, it\u0027s 480."},{"Start":"07:36.260 ","End":"07:43.940","Text":"31 times 32 plus 15 times 7 should give you this."},{"Start":"07:43.940 ","End":"07:47.070","Text":"We are done."}],"Thumbnail":null,"ID":4560},{"Watched":false,"Name":"Exercise 4","Duration":"3m 43s","ChapterTopicVideoID":4552,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.545","Text":"Here we have yet another 1 of these exercises with length of curve."},{"Start":"00:04.545 ","End":"00:07.350","Text":"The same picture, the same formula,"},{"Start":"00:07.350 ","End":"00:10.095","Text":"the only thing that changes is the function,"},{"Start":"00:10.095 ","End":"00:14.595","Text":"and that\u0027s this and the a and the b from the limits."},{"Start":"00:14.595 ","End":"00:18.105","Text":"So we want to plug into this formula,"},{"Start":"00:18.105 ","End":"00:21.780","Text":"1 plus y prime squared."},{"Start":"00:21.780 ","End":"00:24.530","Text":"We\u0027ll just do it bit by bit."},{"Start":"00:24.530 ","End":"00:27.425","Text":"First of all, let\u0027s do y prime."},{"Start":"00:27.425 ","End":"00:30.230","Text":"I\u0027ll copy y over here,"},{"Start":"00:30.230 ","End":"00:37.475","Text":"so y equals 2/3 1 plus x squared to the 3/2,"},{"Start":"00:37.475 ","End":"00:44.270","Text":"which gives me that y prime is 2/3 times 3/2 is just 1,"},{"Start":"00:44.270 ","End":"00:47.835","Text":"that\u0027s clear, 2/3 of 3/2 inverse fractions."},{"Start":"00:47.835 ","End":"00:53.960","Text":"We just get 1 plus x squared and lower the power by 1,"},{"Start":"00:53.960 ","End":"00:56.839","Text":"that\u0027s a 1/2, but times the inner derivative."},{"Start":"00:56.839 ","End":"01:00.200","Text":"This is chain rule, so times 2x."},{"Start":"01:00.200 ","End":"01:05.664","Text":"Y prime squared, I have a product,"},{"Start":"01:05.664 ","End":"01:08.110","Text":"take each bit squared separately and"},{"Start":"01:08.110 ","End":"01:11.275","Text":"the square root squared just gets rid of the square root."},{"Start":"01:11.275 ","End":"01:12.850","Text":"If you like, to the power of a 1/2,"},{"Start":"01:12.850 ","End":"01:15.685","Text":"to the power of 2, is like to the power of 1."},{"Start":"01:15.685 ","End":"01:19.760","Text":"That\u0027s 1 plus x squared without the 1/2,"},{"Start":"01:19.760 ","End":"01:24.595","Text":"and the other bits squared is 2x all squared."},{"Start":"01:24.595 ","End":"01:29.100","Text":"Now this thing is 4x squared,"},{"Start":"01:29.100 ","End":"01:31.080","Text":"so if I expand it,"},{"Start":"01:31.080 ","End":"01:33.870","Text":"I get 4x squared times 1,"},{"Start":"01:33.870 ","End":"01:41.475","Text":"which is 4x squared, times x squared is 4x^4."},{"Start":"01:41.475 ","End":"01:44.865","Text":"Now I\u0027m going to 1 plus y prime squared,"},{"Start":"01:44.865 ","End":"01:54.225","Text":"and this is going to equal 1 plus 4x squared plus 4x^4."},{"Start":"01:54.225 ","End":"01:58.680","Text":"The 1, 4, 4 coefficients are somehow familiar to me."},{"Start":"01:58.680 ","End":"02:02.810","Text":"I remember that this is related to a perfect square."},{"Start":"02:02.810 ","End":"02:10.095","Text":"In fact, you can check that this is equal to 1 plus 2x all squared,"},{"Start":"02:10.095 ","End":"02:11.660","Text":"it is a perfect square."},{"Start":"02:11.660 ","End":"02:13.580","Text":"It had to be the square root of this"},{"Start":"02:13.580 ","End":"02:15.980","Text":"here and it had to be the square root here and we all"},{"Start":"02:15.980 ","End":"02:20.445","Text":"we need to check is twice this times this is the middle term a,"},{"Start":"02:20.445 ","End":"02:23.450","Text":"which is wrong, which is why it was 2x squared."},{"Start":"02:23.450 ","End":"02:27.010","Text":"Yes, expand this and you\u0027ll see that this is what we get."},{"Start":"02:27.010 ","End":"02:37.130","Text":"The square root of 1 plus y prime squared is equal to just 1 plus 2x squared,"},{"Start":"02:37.130 ","End":"02:39.770","Text":"and the square root of something squared is the absolute value."},{"Start":"02:39.770 ","End":"02:41.420","Text":"But since this is positive,"},{"Start":"02:41.420 ","End":"02:44.045","Text":"I don\u0027t need the absolute value."},{"Start":"02:44.045 ","End":"02:47.075","Text":"I think we finally get to the integral."},{"Start":"02:47.075 ","End":"02:52.225","Text":"The integral will be from 0-3."},{"Start":"02:52.225 ","End":"02:55.570","Text":"We get the l,"},{"Start":"02:55.570 ","End":"03:05.600","Text":"the length of curve is the integral from 0-3 of 1 plus 2x squared dx."},{"Start":"03:05.600 ","End":"03:09.830","Text":"This is a straightforward integral. Let\u0027s see."},{"Start":"03:09.830 ","End":"03:15.725","Text":"This would be x plus 2/3x cubed,"},{"Start":"03:15.725 ","End":"03:19.680","Text":"all this between 0-3."},{"Start":"03:20.540 ","End":"03:24.525","Text":"If we plug in 3,"},{"Start":"03:24.525 ","End":"03:34.365","Text":"we get 3 plus 2/3 of 27 is 18 minus,"},{"Start":"03:34.365 ","End":"03:38.145","Text":"plug in 0, 0 plus 0,"},{"Start":"03:38.145 ","End":"03:41.145","Text":"so the answer is 21,"},{"Start":"03:41.145 ","End":"03:44.050","Text":"and we are done."}],"Thumbnail":null,"ID":4561},{"Watched":false,"Name":"Exercise 5","Duration":"6m 35s","ChapterTopicVideoID":4553,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"Here, again, I have 1 of those arc length, curve length questions."},{"Start":"00:05.490 ","End":"00:07.470","Text":"The same formula, the same picture."},{"Start":"00:07.470 ","End":"00:11.010","Text":"The difference each time is there\u0027s a different function of y of"},{"Start":"00:11.010 ","End":"00:16.155","Text":"x and different lower limit and a different upper limit."},{"Start":"00:16.155 ","End":"00:19.305","Text":"I need to compute this integral."},{"Start":"00:19.305 ","End":"00:21.620","Text":"Let\u0027s start from inside out."},{"Start":"00:21.620 ","End":"00:23.995","Text":"Let\u0027s start with y prime."},{"Start":"00:23.995 ","End":"00:26.100","Text":"Before we get to y prime,"},{"Start":"00:26.100 ","End":"00:32.640","Text":"we\u0027d better start with y. I just want to rewrite it as 1 1/3."},{"Start":"00:32.640 ","End":"00:36.645","Text":"x square root is x to the power of a 1/2."},{"Start":"00:36.645 ","End":"00:43.230","Text":"So I have 3x to the power of a 1/2minus x,"},{"Start":"00:43.230 ","End":"00:46.965","Text":"x to the power of a 1/2 is x to the power of 1 and a 1/2."},{"Start":"00:46.965 ","End":"00:49.485","Text":"1 and a 1/2 is 3 over 2."},{"Start":"00:49.485 ","End":"00:52.595","Text":"That makes it easier for me to get to the derivative."},{"Start":"00:52.595 ","End":"00:56.510","Text":"So I get y prime is 1 third."},{"Start":"00:56.510 ","End":"01:01.825","Text":"Derivative of this, 1/2 times 3 is 3 over 2."},{"Start":"01:01.825 ","End":"01:04.980","Text":"Lower the power by 1, it\u0027s that."},{"Start":"01:04.980 ","End":"01:10.440","Text":"Here I have 3 over 2 x to the power of 1/2,"},{"Start":"01:10.440 ","End":"01:12.255","Text":"I lowered the power by 1."},{"Start":"01:12.255 ","End":"01:13.680","Text":"What I actually get,"},{"Start":"01:13.680 ","End":"01:16.500","Text":"if I take 3 over 2 outside the brackets,"},{"Start":"01:16.500 ","End":"01:25.365","Text":"I get 1 third times 3 over 2 times x to the minus a 1/2 minus x to the plus a 1/2,"},{"Start":"01:25.365 ","End":"01:29.330","Text":"and since the 3 above and 3 below cancel,"},{"Start":"01:29.330 ","End":"01:32.000","Text":"I just have here 1/2."},{"Start":"01:32.000 ","End":"01:38.390","Text":"So now I can say that y prime squared is equal to,"},{"Start":"01:38.390 ","End":"01:40.910","Text":"1/2 squared is 1/4."},{"Start":"01:40.910 ","End":"01:45.960","Text":"Using the formula of a plus b or minus b,"},{"Start":"01:45.960 ","End":"01:53.939","Text":"a plus or minus b squared is a squared plus or minus 2ab plus b squared."},{"Start":"01:53.939 ","End":"01:55.725","Text":"This is always plus."},{"Start":"01:55.725 ","End":"01:57.480","Text":"If this is a plus,"},{"Start":"01:57.480 ","End":"01:58.980","Text":"it\u0027s a plus. Minus, it\u0027s a minus."},{"Start":"01:58.980 ","End":"02:00.300","Text":"Here we have a minus,"},{"Start":"02:00.300 ","End":"02:02.745","Text":"so a squared is this squared,"},{"Start":"02:02.745 ","End":"02:07.575","Text":"and this is x to the minus 1."},{"Start":"02:07.575 ","End":"02:14.645","Text":"The last 1 squared is plus x to the 1, which is just x."},{"Start":"02:14.645 ","End":"02:18.155","Text":"The middle term is minus twice this times this."},{"Start":"02:18.155 ","End":"02:22.805","Text":"Now, this times this is 1 because the minus a 1/2 and a 1/2 cancel,"},{"Start":"02:22.805 ","End":"02:26.625","Text":"so we just get minus 2."},{"Start":"02:26.625 ","End":"02:28.895","Text":"Now finally, I want to add 1."},{"Start":"02:28.895 ","End":"02:35.250","Text":"So 1 plus y prime squared is 4 over 4."},{"Start":"02:35.250 ","End":"02:43.275","Text":"I like to write the 1 as 4 over 4 plus x minus 1 minus 2 plus x over 4."},{"Start":"02:43.275 ","End":"02:50.720","Text":"What I get is 4 here and the minus 2 here get added,"},{"Start":"02:50.720 ","End":"02:53.000","Text":"so it makes it a plus 2."},{"Start":"02:53.000 ","End":"02:58.260","Text":"So, basically, what I get is x minus"},{"Start":"02:58.260 ","End":"03:04.780","Text":"1 plus 2 plus x over 4."},{"Start":"03:04.780 ","End":"03:13.040","Text":"Now, if we look at what happened when we squared this, we got this."},{"Start":"03:13.040 ","End":"03:14.720","Text":"We used this formula."},{"Start":"03:14.720 ","End":"03:19.130","Text":"But this is almost the same as this except with a plus."},{"Start":"03:19.130 ","End":"03:22.520","Text":"So it makes sense. This is also a perfect square."},{"Start":"03:22.520 ","End":"03:24.050","Text":"I\u0027ll even write it over here."},{"Start":"03:24.050 ","End":"03:27.305","Text":"This time it will be x^ to the minus a 1/2,"},{"Start":"03:27.305 ","End":"03:29.150","Text":"but a plus here,"},{"Start":"03:29.150 ","End":"03:31.190","Text":"because I have the same thing."},{"Start":"03:31.190 ","End":"03:33.850","Text":"Instead of a minus, I got a plus."},{"Start":"03:33.850 ","End":"03:37.505","Text":"All this squared. That\u0027s just the numerator."},{"Start":"03:37.505 ","End":"03:41.990","Text":"The quarter here somehow has come down to the denominator, that\u0027s fine."},{"Start":"03:41.990 ","End":"03:48.320","Text":"Then what I need is the integral of the square root of 1 plus y squared."},{"Start":"03:48.320 ","End":"03:51.090","Text":"So let\u0027s just write the square root."},{"Start":"03:51.250 ","End":"03:54.260","Text":"Take square root of top and bottom."},{"Start":"03:54.260 ","End":"03:56.930","Text":"So this will be without square,"},{"Start":"03:56.930 ","End":"04:01.950","Text":"will be just x to the minus a 1/2 plus x to the 1/2."},{"Start":"04:01.950 ","End":"04:04.955","Text":"Now square root of a square is the absolute value,"},{"Start":"04:04.955 ","End":"04:07.490","Text":"but we\u0027re on the range from 2 to 3,"},{"Start":"04:07.490 ","End":"04:10.069","Text":"and these are all positive, so no problem."},{"Start":"04:10.069 ","End":"04:13.265","Text":"The square root of the denominator is 2."},{"Start":"04:13.265 ","End":"04:19.415","Text":"Now, all we have to do is to take the integral of this from 0 and 3."},{"Start":"04:19.415 ","End":"04:23.805","Text":"So we want the integral from 0 to 3,"},{"Start":"04:23.805 ","End":"04:26.920","Text":"that\u0027s the l, the length of curve."},{"Start":"04:27.130 ","End":"04:32.030","Text":"I\u0027ll take the 1/2 outside the integral."},{"Start":"04:32.030 ","End":"04:40.005","Text":"So it\u0027s the integral of x to the minus a 1/2 plus x to the 1/2 dx."},{"Start":"04:40.005 ","End":"04:44.655","Text":"This will equal 1/2."},{"Start":"04:44.655 ","End":"04:46.575","Text":"Now the integral of this,"},{"Start":"04:46.575 ","End":"04:49.560","Text":"I raise it by 1, it\u0027s x to the 1/2,"},{"Start":"04:49.560 ","End":"04:52.880","Text":"divide by 1/2 is like multiplying by 2,"},{"Start":"04:52.880 ","End":"04:54.695","Text":"2x to the 1/2."},{"Start":"04:54.695 ","End":"04:59.270","Text":"The other one, raised by 1 is 3 over 2, divide by it,"},{"Start":"04:59.270 ","End":"05:05.000","Text":"2/3 of x to the 3 over 2,"},{"Start":"05:05.000 ","End":"05:07.460","Text":"between 0 and 3."},{"Start":"05:07.460 ","End":"05:11.315","Text":"Let me just throw the 1/2 in here."},{"Start":"05:11.315 ","End":"05:21.855","Text":"So we get x to the power of a 1/2 plus 1/3 because the 1/2 is canceling with the 2,"},{"Start":"05:21.855 ","End":"05:24.360","Text":"x to the 3 over 2,"},{"Start":"05:24.360 ","End":"05:28.630","Text":"and all this between 3 and 0. If I put in x equals 3,"},{"Start":"05:30.680 ","End":"05:37.975","Text":"you know what, I think it will simplify if I take x to the 1/2 outside,"},{"Start":"05:37.975 ","End":"05:40.645","Text":"so I\u0027ll do 1 more simplification."},{"Start":"05:40.645 ","End":"05:43.250","Text":"x to the 1/2,"},{"Start":"05:43.250 ","End":"05:46.334","Text":"and what I\u0027m left with is"},{"Start":"05:46.334 ","End":"05:55.905","Text":"1 plus 1/3 x after I take x to the 1/2 out,"},{"Start":"05:55.905 ","End":"05:58.665","Text":"and all this between 0 and 3."},{"Start":"05:58.665 ","End":"06:01.010","Text":"So put in x equals 3,"},{"Start":"06:01.010 ","End":"06:02.915","Text":"we get 3 to the 1/2,"},{"Start":"06:02.915 ","End":"06:05.570","Text":"could write that as square root of 3,"},{"Start":"06:05.570 ","End":"06:10.559","Text":"and 1 plus 1/3 of 3,"},{"Start":"06:10.559 ","End":"06:14.070","Text":"1 plus 3 over 3."},{"Start":"06:14.070 ","End":"06:16.020","Text":"That\u0027s the part of 3."},{"Start":"06:16.020 ","End":"06:19.125","Text":"Now the part of 0 is just 0,"},{"Start":"06:19.125 ","End":"06:24.330","Text":"square root 1 plus 0 over 3."},{"Start":"06:24.330 ","End":"06:26.480","Text":"Clearly, the last term is 0."},{"Start":"06:26.480 ","End":"06:29.405","Text":"Here I get 1 plus 3 over 3 is 2."},{"Start":"06:29.405 ","End":"06:34.040","Text":"So the answer is twice the square root of 3."},{"Start":"06:34.040 ","End":"06:36.660","Text":"We are done."}],"Thumbnail":null,"ID":4562},{"Watched":false,"Name":"Exercise 6","Duration":"8m 40s","ChapterTopicVideoID":4554,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"Here we have another 1 of those length of curve questions."},{"Start":"00:03.960 ","End":"00:05.970","Text":"They\u0027re all very similar,"},{"Start":"00:05.970 ","End":"00:09.615","Text":"the same diagram, the same formula,"},{"Start":"00:09.615 ","End":"00:17.040","Text":"and the only real difference is that the a and b change and the function y changes."},{"Start":"00:17.040 ","End":"00:20.190","Text":"In this case, the function is natural log of x,"},{"Start":"00:20.190 ","End":"00:24.360","Text":"and the a is 1, and the b is 2."},{"Start":"00:24.360 ","End":"00:27.120","Text":"This was just phrased slightly differently."},{"Start":"00:27.120 ","End":"00:30.015","Text":"They gave us extra information that we don\u0027t need."},{"Start":"00:30.015 ","End":"00:31.560","Text":"I mean, when x is 1,"},{"Start":"00:31.560 ","End":"00:33.090","Text":"natural log of 1 is 0,"},{"Start":"00:33.090 ","End":"00:35.295","Text":"when x is 2, natural log of 2 is this."},{"Start":"00:35.295 ","End":"00:37.020","Text":"They gave us the full x, y."},{"Start":"00:37.020 ","End":"00:40.215","Text":"In other words, they gave us a and b."},{"Start":"00:40.215 ","End":"00:44.505","Text":"Anyway, so we can ignore the extraneous data."},{"Start":"00:44.505 ","End":"00:46.500","Text":"Let\u0027s start doing it."},{"Start":"00:46.500 ","End":"00:48.380","Text":"I like to build it up in bits."},{"Start":"00:48.380 ","End":"00:51.410","Text":"First of all, if y is natural log of x,"},{"Start":"00:51.410 ","End":"00:55.885","Text":"I\u0027m proceeding then y prime equals 1 over x."},{"Start":"00:55.885 ","End":"01:00.920","Text":"That means that 1 plus y prime"},{"Start":"01:00.920 ","End":"01:08.930","Text":"squared is 1 plus 1 over x squared."},{"Start":"01:08.930 ","End":"01:12.195","Text":"I can write this as 1 over x squared,"},{"Start":"01:12.195 ","End":"01:14.780","Text":"and then putting a common denominator,"},{"Start":"01:14.780 ","End":"01:19.335","Text":"we get x squared plus 1 over x squared."},{"Start":"01:19.335 ","End":"01:21.630","Text":"Then we need the square roots."},{"Start":"01:21.630 ","End":"01:23.685","Text":"Let\u0027s start with the integral."},{"Start":"01:23.685 ","End":"01:26.940","Text":"The integral from a to b,"},{"Start":"01:26.940 ","End":"01:31.560","Text":"which is 1 to 2, of x squared minus 1,"},{"Start":"01:31.560 ","End":"01:32.880","Text":"well, the square root of that."},{"Start":"01:32.880 ","End":"01:35.900","Text":"I\u0027ll take the square roots separately on the top and on the bottom."},{"Start":"01:35.900 ","End":"01:41.600","Text":"I have the square root of x squared plus 1 on the numerator."},{"Start":"01:41.600 ","End":"01:45.700","Text":"Here, the square root of x squared is the absolute value of x,"},{"Start":"01:45.700 ","End":"01:48.055","Text":"but we\u0027re going from 1-2,"},{"Start":"01:48.055 ","End":"01:52.340","Text":"so it\u0027s just x and dx."},{"Start":"01:52.340 ","End":"01:54.920","Text":"How are we going to do this?"},{"Start":"01:54.920 ","End":"01:58.255","Text":"I think a substitution at least is worth trying."},{"Start":"01:58.255 ","End":"02:01.180","Text":"Let\u0027s try substituting t as the square root."},{"Start":"02:01.180 ","End":"02:06.010","Text":"Let\u0027s say the x squared plus 1 root is t,"},{"Start":"02:06.010 ","End":"02:09.660","Text":"and then we need dt and dx."},{"Start":"02:09.660 ","End":"02:12.290","Text":"In order to not get messed up with square roots,"},{"Start":"02:12.290 ","End":"02:14.720","Text":"let\u0027s take both sides and square them."},{"Start":"02:14.720 ","End":"02:20.045","Text":"We get that x squared plus 1 is t squared,"},{"Start":"02:20.045 ","End":"02:24.305","Text":"and then differentiate it and get that 2x, from here,"},{"Start":"02:24.305 ","End":"02:29.605","Text":"dx is equal to 2t, dt."},{"Start":"02:29.605 ","End":"02:33.050","Text":"I don\u0027t need the 2 there, it just cancels."},{"Start":"02:33.050 ","End":"02:36.595","Text":"I can just erase the 2 here and here."},{"Start":"02:36.595 ","End":"02:41.355","Text":"What I\u0027d like is to isolate the dx,"},{"Start":"02:41.355 ","End":"02:48.705","Text":"so dx becomes t dt over x."},{"Start":"02:48.705 ","End":"02:51.245","Text":"Now, I\u0027m going to substitute things."},{"Start":"02:51.245 ","End":"02:58.624","Text":"The first thing I\u0027m going to substitute is the square root of x squared plus 1."},{"Start":"02:58.624 ","End":"03:01.125","Text":"I\u0027m going to substitute t for that."},{"Start":"03:01.125 ","End":"03:04.130","Text":"The x, I\u0027m going to leave alone meanwhile."},{"Start":"03:04.130 ","End":"03:09.470","Text":"The dx, I\u0027m going to substitute from here as"},{"Start":"03:09.470 ","End":"03:17.670","Text":"t dt over x. I also have to substitute the limits."},{"Start":"03:17.670 ","End":"03:19.920","Text":"I haven\u0027t done that yet,"},{"Start":"03:19.920 ","End":"03:21.630","Text":"so let\u0027s just see."},{"Start":"03:21.630 ","End":"03:26.175","Text":"When x equals 1,"},{"Start":"03:26.175 ","End":"03:30.360","Text":"then t equals, so let\u0027s see, x is 1,"},{"Start":"03:30.360 ","End":"03:33.630","Text":"1 squared plus 1 is 2,"},{"Start":"03:33.630 ","End":"03:36.640","Text":"so that\u0027s square root of 2."},{"Start":"03:36.950 ","End":"03:39.660","Text":"When x is 2,"},{"Start":"03:39.660 ","End":"03:42.165","Text":"2 squared plus 1 is 5,"},{"Start":"03:42.165 ","End":"03:47.025","Text":"so t equals the square root of 5."},{"Start":"03:47.025 ","End":"03:51.335","Text":"Instead of 2 and 1, what I\u0027m going to get is"},{"Start":"03:51.335 ","End":"03:56.420","Text":"the square root of 2 and the square root of 5, because, I mean,"},{"Start":"03:56.420 ","End":"04:00.380","Text":"this is for x equals 1 to x equals 2,"},{"Start":"04:00.380 ","End":"04:03.950","Text":"and then we\u0027re going to get t. What we\u0027re going to get"},{"Start":"04:03.950 ","End":"04:08.350","Text":"after all these transformations is the integral."},{"Start":"04:08.350 ","End":"04:11.560","Text":"I\u0027m just writing t equals for emphasis,"},{"Start":"04:11.560 ","End":"04:16.895","Text":"from square root of 2 to square root of 5."},{"Start":"04:16.895 ","End":"04:26.315","Text":"This we said is t. Dx is t dt over x."},{"Start":"04:26.315 ","End":"04:30.605","Text":"There was already an x here and I have to repeat that."},{"Start":"04:30.605 ","End":"04:34.595","Text":"Now, what happens here, this bit here,"},{"Start":"04:34.595 ","End":"04:38.840","Text":"I can simplify it because this is equal to,"},{"Start":"04:38.840 ","End":"04:40.340","Text":"I will just do it at the side,"},{"Start":"04:40.340 ","End":"04:47.145","Text":"so the bit that\u0027s in orange is t squared over x squared."},{"Start":"04:47.145 ","End":"04:49.190","Text":"I don\u0027t want x\u0027s here."},{"Start":"04:49.190 ","End":"04:53.675","Text":"I want everything in terms of t. How can I get rid of x squared?"},{"Start":"04:53.675 ","End":"04:58.730","Text":"That is easy because if we look over here,"},{"Start":"04:58.730 ","End":"05:07.110","Text":"we can see that x squared can be isolated as t squared minus 1."},{"Start":"05:07.120 ","End":"05:14.610","Text":"This we get is t squared over t squared minus 1."},{"Start":"05:14.610 ","End":"05:17.360","Text":"Continuing with our integral,"},{"Start":"05:17.360 ","End":"05:25.010","Text":"we get the integral from square root of 2 to square root of 5,"},{"Start":"05:25.010 ","End":"05:31.460","Text":"of t squared over t squared minus 1 dt."},{"Start":"05:31.460 ","End":"05:33.590","Text":"How do we do this?"},{"Start":"05:33.590 ","End":"05:36.395","Text":"We are going to use some standard tricks here."},{"Start":"05:36.395 ","End":"05:41.860","Text":"This is equal to the integral, same limits."},{"Start":"05:41.860 ","End":"05:47.420","Text":"I\u0027m going to write it as t squared minus 1 because,"},{"Start":"05:47.420 ","End":"05:48.860","Text":"I wanted to cancel,"},{"Start":"05:48.860 ","End":"05:50.330","Text":"but I can\u0027t just put a minus 1,"},{"Start":"05:50.330 ","End":"05:52.475","Text":"so I put a plus 1 and I\u0027m fine."},{"Start":"05:52.475 ","End":"05:58.260","Text":"Dt, which equals, now this over this is 1,"},{"Start":"05:58.260 ","End":"06:07.750","Text":"so it\u0027s the integral of 1 plus 1 over t squared minus 1 dt."},{"Start":"06:07.750 ","End":"06:13.430","Text":"This 1 here, I\u0027ll just circle it could be done with partial fractions."},{"Start":"06:13.430 ","End":"06:16.505","Text":"Well, I\u0027m going to give you the answer from the formula sheet."},{"Start":"06:16.505 ","End":"06:18.925","Text":"This is equal to,"},{"Start":"06:18.925 ","End":"06:21.975","Text":"first the 1 which gives me t,"},{"Start":"06:21.975 ","End":"06:25.625","Text":"and this from the formula sheet gives"},{"Start":"06:25.625 ","End":"06:33.310","Text":"1/2 natural log of t minus 1 over t plus 1."},{"Start":"06:33.310 ","End":"06:39.230","Text":"All this, between square root of 2, square root of 5."},{"Start":"06:39.230 ","End":"06:42.440","Text":"Of course, if they didn\u0027t give you a formula sheet,"},{"Start":"06:42.440 ","End":"06:44.190","Text":"you\u0027ll have to do this by partial fractions."},{"Start":"06:44.190 ","End":"06:46.280","Text":"I\u0027ll just give you the general idea."},{"Start":"06:46.280 ","End":"06:51.050","Text":"The general idea is to write 1 over t squared minus 1,"},{"Start":"06:51.050 ","End":"06:54.185","Text":"which is really 1 over t minus 1,"},{"Start":"06:54.185 ","End":"06:58.085","Text":"t plus 1 by the difference of squares formula."},{"Start":"06:58.085 ","End":"07:06.695","Text":"Then we write it as a over t minus 1 plus b over t plus 1."},{"Start":"07:06.695 ","End":"07:10.550","Text":"Eventually, we multiply out,"},{"Start":"07:10.550 ","End":"07:12.920","Text":"we substitute t equals 1,"},{"Start":"07:12.920 ","End":"07:17.000","Text":"and we get a, substitute t equals minus 1, we get b."},{"Start":"07:17.000 ","End":"07:23.945","Text":"In the end, we get that a is a half and b is minus a half."},{"Start":"07:23.945 ","End":"07:30.860","Text":"Then we write it as 1/2 of 1 over t minus 1,"},{"Start":"07:30.860 ","End":"07:34.560","Text":"minus 1 over t plus 1."},{"Start":"07:34.560 ","End":"07:36.420","Text":"I\u0027m just doing this very roughly."},{"Start":"07:36.420 ","End":"07:40.325","Text":"Each of these is natural logarithm and so on."},{"Start":"07:40.325 ","End":"07:43.835","Text":"Anyway, you\u0027ll get to this answer with partial fractions."},{"Start":"07:43.835 ","End":"07:47.960","Text":"The final thing we have left to do is just the substitution."},{"Start":"07:47.960 ","End":"07:50.810","Text":"Let\u0027s see what we get here."},{"Start":"07:50.810 ","End":"07:53.385","Text":"Substitute square root of 5,"},{"Start":"07:53.385 ","End":"08:02.895","Text":"and we get square root of 5 plus 1/2 natural log of square root of 5 minus 1,"},{"Start":"08:02.895 ","End":"08:05.730","Text":"over square root of 5 plus 1."},{"Start":"08:05.730 ","End":"08:11.465","Text":"I dropped the absolute value because everything\u0027s positive here, less,"},{"Start":"08:11.465 ","End":"08:15.485","Text":"same thing with square root of 2 plus"},{"Start":"08:15.485 ","End":"08:23.055","Text":"1/2 natural log of square root minus 1,"},{"Start":"08:23.055 ","End":"08:26.655","Text":"over the square root of 2 plus 1."},{"Start":"08:26.655 ","End":"08:28.670","Text":"If you feel like doing it on"},{"Start":"08:28.670 ","End":"08:33.260","Text":"the calculator or you want to try and simplify it, you\u0027re welcome to."},{"Start":"08:33.260 ","End":"08:36.860","Text":"As far as I\u0027m concerned, with curved length,"},{"Start":"08:36.860 ","End":"08:41.880","Text":"this is good enough as an answer. That\u0027s it."}],"Thumbnail":null,"ID":4563},{"Watched":false,"Name":"Exercise 7","Duration":"6m 7s","ChapterTopicVideoID":4555,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.290","Text":"In this exercise, we have to find the length of the curve of x squared between 1 and 2."},{"Start":"00:06.290 ","End":"00:10.755","Text":"As usual, I provide the little sketch with the formula."},{"Start":"00:10.755 ","End":"00:15.495","Text":"All that\u0027s different in each one is the particular function y of x,"},{"Start":"00:15.495 ","End":"00:17.160","Text":"here it\u0027s x squared,"},{"Start":"00:17.160 ","End":"00:21.105","Text":"and the limits x is 1 and x is 2."},{"Start":"00:21.105 ","End":"00:24.690","Text":"This one turns out to be a bit lengthy,"},{"Start":"00:24.690 ","End":"00:28.240","Text":"so there\u0027s a formula here that will help us later."},{"Start":"00:28.240 ","End":"00:30.800","Text":"Meanwhile, let\u0027s get started."},{"Start":"00:30.800 ","End":"00:36.000","Text":"We have to compute this integral. Let\u0027s see."},{"Start":"00:36.000 ","End":"00:45.440","Text":"L is equal to the integral from 1-2 of the square root of 1 plus, now y prime,"},{"Start":"00:45.440 ","End":"00:46.850","Text":"I\u0027ll just write that here, of course,"},{"Start":"00:46.850 ","End":"00:48.985","Text":"y prime is 2x,"},{"Start":"00:48.985 ","End":"00:52.890","Text":"so plus 2x, all squared,"},{"Start":"00:52.890 ","End":"01:03.270","Text":"dx, and this equals the integral from 1-2 of the square root of 1 plus 4x squared dx."},{"Start":"01:03.270 ","End":"01:05.530","Text":"Now, they\u0027ve given us a hint,"},{"Start":"01:05.530 ","End":"01:12.390","Text":"and we have to make this somehow look like x squared plus or minus a squared."},{"Start":"01:12.390 ","End":"01:14.250","Text":"We\u0027re going to go for the plus, of course."},{"Start":"01:14.250 ","End":"01:15.780","Text":"Here I have 4x squared,"},{"Start":"01:15.780 ","End":"01:17.430","Text":"I want 1x squared."},{"Start":"01:17.430 ","End":"01:21.470","Text":"I\u0027m going to take 4 outside the square root."},{"Start":"01:21.470 ","End":"01:25.295","Text":"It\u0027ll come out as 2, which will come out as in the front."},{"Start":"01:25.295 ","End":"01:27.350","Text":"I\u0027m not going to do it in every little detail,"},{"Start":"01:27.350 ","End":"01:28.370","Text":"you can do this thing."},{"Start":"01:28.370 ","End":"01:30.450","Text":"I take 4 out,"},{"Start":"01:30.450 ","End":"01:35.525","Text":"so what I\u0027m left with is the square root of 1/4"},{"Start":"01:35.525 ","End":"01:41.040","Text":"plus x squared dx."},{"Start":"01:41.040 ","End":"01:43.925","Text":"Just to make it really look like this,"},{"Start":"01:43.925 ","End":"01:46.250","Text":"I\u0027ll write it as twice the integral from"},{"Start":"01:46.250 ","End":"01:49.865","Text":"1-2 of the square root and I\u0027ll put the x squared first."},{"Start":"01:49.865 ","End":"01:52.165","Text":"Here I see a squared."},{"Start":"01:52.165 ","End":"01:57.315","Text":"I will write this as 1 over 2 squared, that\u0027s the 1/4."},{"Start":"01:57.315 ","End":"01:59.470","Text":"Basically, I have here,"},{"Start":"01:59.470 ","End":"02:03.830","Text":"is the integral of the square root of x squared plus"},{"Start":"02:03.830 ","End":"02:10.875","Text":"a squared dx with a equaling 1/2."},{"Start":"02:10.875 ","End":"02:13.785","Text":"Now, they\u0027ve given us the formula."},{"Start":"02:13.785 ","End":"02:16.390","Text":"I\u0027ve given you the formula, whose they?"},{"Start":"02:16.390 ","End":"02:19.310","Text":"But I got it from a formula sheet."},{"Start":"02:19.310 ","End":"02:21.740","Text":"I didn\u0027t do the integration myself."},{"Start":"02:21.740 ","End":"02:23.645","Text":"I trust the formula sheet."},{"Start":"02:23.645 ","End":"02:27.440","Text":"If you don\u0027t have a formula sheet,"},{"Start":"02:27.440 ","End":"02:29.045","Text":"it\u0027s going to be difficult."},{"Start":"02:29.045 ","End":"02:33.020","Text":"But I can tell you that the substitution you would try in"},{"Start":"02:33.020 ","End":"02:37.160","Text":"case you can\u0027t use a formula for this kind of"},{"Start":"02:37.160 ","End":"02:46.555","Text":"situation is x equals a tangent t. That\u0027s what we try and do."},{"Start":"02:46.555 ","End":"02:49.760","Text":"In our case, x equals 1/2 tangent t,"},{"Start":"02:49.760 ","End":"02:53.360","Text":"but it still comes out very messy and I don\u0027t want to get"},{"Start":"02:53.360 ","End":"02:58.205","Text":"bogged down with integrals when we\u0027re here to learn about curve length."},{"Start":"02:58.205 ","End":"03:03.120","Text":"Let me continue by using the formula here."},{"Start":"03:03.120 ","End":"03:08.085","Text":"What we get is the answer."},{"Start":"03:08.085 ","End":"03:10.100","Text":"This is the indefinite integral,"},{"Start":"03:10.100 ","End":"03:17.970","Text":"which is 1/2 x square root of x squared plus a squared."},{"Start":"03:17.970 ","End":"03:20.175","Text":"Now, the a squared, I\u0027ll put as a 1/4,"},{"Start":"03:20.175 ","End":"03:22.680","Text":"that\u0027s my a squared,"},{"Start":"03:22.680 ","End":"03:25.260","Text":"x squared plus 1/4,"},{"Start":"03:25.260 ","End":"03:33.485","Text":"and plus or minus 1/2 a squared is a 1/4, natural log."},{"Start":"03:33.485 ","End":"03:35.210","Text":"We\u0027re in the positive zone,"},{"Start":"03:35.210 ","End":"03:38.380","Text":"so I don\u0027t think I\u0027ll need the absolute value."},{"Start":"03:38.380 ","End":"03:45.815","Text":"Natural log of x plus the square root of x squared plus 1/4."},{"Start":"03:45.815 ","End":"03:49.130","Text":"All this, better put another bracket here,"},{"Start":"03:49.130 ","End":"03:54.250","Text":"between the limits 1 and 2."},{"Start":"03:54.250 ","End":"03:57.570","Text":"Now, I don\u0027t quite like the x squared plus 1/4"},{"Start":"03:57.570 ","End":"04:01.085","Text":"and I prefer to have the 1 plus 4x squared."},{"Start":"04:01.085 ","End":"04:04.280","Text":"Well, just like we took 2 outside and got this,"},{"Start":"04:04.280 ","End":"04:06.035","Text":"if I put 2 back in,"},{"Start":"04:06.035 ","End":"04:09.375","Text":"I\u0027ll get back to 1 plus 4 x squared."},{"Start":"04:09.375 ","End":"04:11.665","Text":"This is equal to,"},{"Start":"04:11.665 ","End":"04:13.670","Text":"I want to put 2 in here."},{"Start":"04:13.670 ","End":"04:15.695","Text":"I also have to divide by 2."},{"Start":"04:15.695 ","End":"04:18.145","Text":"It\u0027s going to be 1/4x,"},{"Start":"04:18.145 ","End":"04:21.075","Text":"and then 1 plus 4x squared,"},{"Start":"04:21.075 ","End":"04:22.845","Text":"or 4x squared plus 1,"},{"Start":"04:22.845 ","End":"04:25.050","Text":"plus or minus. Now, let\u0027s see."},{"Start":"04:25.050 ","End":"04:28.135","Text":"I have 1/2 times 1/4 is an 1/8,"},{"Start":"04:28.135 ","End":"04:31.770","Text":"and I\u0027ll take another 2. I\u0027m going to put a 2 in."},{"Start":"04:31.770 ","End":"04:33.465","Text":"I have to also multiply by 1/2,"},{"Start":"04:33.465 ","End":"04:38.910","Text":"so 1/2 times 1/4 times 1/2 is 1/16."},{"Start":"04:38.910 ","End":"04:40.970","Text":"Now, why am I putting plus or minus?"},{"Start":"04:40.970 ","End":"04:43.085","Text":"It\u0027s just a plus in our case, of course."},{"Start":"04:43.085 ","End":"04:45.440","Text":"Again, I have x plus,"},{"Start":"04:45.440 ","End":"04:49.195","Text":"and the square root will be 1 plus 4x squared."},{"Start":"04:49.195 ","End":"04:53.470","Text":"This, I have to take from 1-2."},{"Start":"04:53.470 ","End":"04:58.170","Text":"All that\u0027s left now is the substitution part. Let\u0027s see."},{"Start":"04:58.170 ","End":"05:01.725","Text":"Let\u0027s put in x equals 2."},{"Start":"05:01.725 ","End":"05:04.530","Text":"1 plus 4x squared,"},{"Start":"05:04.530 ","End":"05:10.035","Text":"x squared is 4, 4 times 4 is 16, plus 1 is 17."},{"Start":"05:10.035 ","End":"05:16.090","Text":"1/4 times 2, oh,"},{"Start":"05:16.090 ","End":"05:18.305","Text":"I forgot the square root here, sorry,"},{"Start":"05:18.305 ","End":"05:19.670","Text":"times square root of"},{"Start":"05:19.670 ","End":"05:26.299","Text":"17 plus 1/16 of"},{"Start":"05:26.299 ","End":"05:31.100","Text":"2 plus square root of 17."},{"Start":"05:31.100 ","End":"05:35.585","Text":"All this minus, if I put in 1,"},{"Start":"05:35.585 ","End":"05:43.040","Text":"I get 1/4 times"},{"Start":"05:43.040 ","End":"05:50.070","Text":"square root of 5 plus 1/16,"},{"Start":"05:50.070 ","End":"05:54.190","Text":"this will be 1 plus square root of 5."},{"Start":"05:57.170 ","End":"06:01.775","Text":"I don\u0027t have the patience to do this arithmetic."},{"Start":"06:01.775 ","End":"06:03.590","Text":"Let\u0027s just leave it at that."},{"Start":"06:03.590 ","End":"06:08.190","Text":"This is not interesting. We\u0027re done."}],"Thumbnail":null,"ID":4564},{"Watched":false,"Name":"Exercise 8","Duration":"5m 24s","ChapterTopicVideoID":4556,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In this exercise, we have to compute the length of"},{"Start":"00:02.490 ","End":"00:06.480","Text":"curve of this function between 1 and 8."},{"Start":"00:06.480 ","End":"00:11.670","Text":"The difference here, is that we have an implicit function,"},{"Start":"00:11.670 ","End":"00:16.950","Text":"y is not given explicitly in terms of x but implicitly."},{"Start":"00:16.950 ","End":"00:20.450","Text":"We still have A and B given and we have"},{"Start":"00:20.450 ","End":"00:24.810","Text":"this formula that we need the integral from a to b of this expression."},{"Start":"00:24.810 ","End":"00:30.180","Text":"The first thing we need is y prime and because it\u0027s an implicit function,"},{"Start":"00:30.180 ","End":"00:33.700","Text":"we\u0027ll do an implicit differentiation."},{"Start":"00:34.850 ","End":"00:38.325","Text":"Let me copy it first,"},{"Start":"00:38.325 ","End":"00:44.585","Text":"2/3 plus y^2/3 equals 4."},{"Start":"00:44.585 ","End":"00:48.440","Text":"Differentiate both sides using exponents."},{"Start":"00:48.440 ","End":"00:53.945","Text":"This is 2/3, x^minus 1/3, if I subtract 1."},{"Start":"00:53.945 ","End":"00:58.880","Text":"Similarly here, 2/3y^minus 1/3 but remember,"},{"Start":"00:58.880 ","End":"01:04.055","Text":"in implicit differentiation when we have an expression with y, we add a y prime,"},{"Start":"01:04.055 ","End":"01:07.610","Text":"multiply that is, equals 0."},{"Start":"01:07.610 ","End":"01:10.475","Text":"Now, let\u0027s isolate y prime."},{"Start":"01:10.475 ","End":"01:15.485","Text":"Well, we can also cancel this 2/3 with this 2/3 to make life easier."},{"Start":"01:15.485 ","End":"01:18.565","Text":"0 divided by 2/3 is still 0."},{"Start":"01:18.565 ","End":"01:24.140","Text":"What we get is bringing this to the other side,"},{"Start":"01:24.140 ","End":"01:29.330","Text":"we get minus x^minus 1/3,"},{"Start":"01:29.330 ","End":"01:31.520","Text":"and then divide by this."},{"Start":"01:31.520 ","End":"01:36.795","Text":"Well, dividing by minus 1/3 is like multiplying."},{"Start":"01:36.795 ","End":"01:42.735","Text":"It\u0027s y^1/3 in the numerator instead of y^minus 1/3 in the denominator."},{"Start":"01:42.735 ","End":"01:45.080","Text":"Now, what we need is not y prime,"},{"Start":"01:45.080 ","End":"01:49.835","Text":"we need 1 plus y prime squared. Let\u0027s see."},{"Start":"01:49.835 ","End":"01:56.165","Text":"1 plus y prime squared is equal to."},{"Start":"01:56.165 ","End":"01:58.730","Text":"Now, y prime squared,"},{"Start":"01:58.730 ","End":"02:07.220","Text":"the minus will disappear and we\u0027ll get x^minus 2/3, y^2/3."},{"Start":"02:07.220 ","End":"02:10.400","Text":"Because if I raise a product squared to the power of 2,"},{"Start":"02:10.400 ","End":"02:16.630","Text":"then each of the factors is raised to the power of 2 using the laws of exponents etc."},{"Start":"02:16.630 ","End":"02:21.585","Text":"Plus 1. Now, we have here both x and y."},{"Start":"02:21.585 ","End":"02:27.190","Text":"We wanted preferably just in terms of x."},{"Start":"02:27.190 ","End":"02:33.720","Text":"What I can see is that this y^2/3,"},{"Start":"02:33.720 ","End":"02:39.390","Text":"I can use the y^2/3 here and say that it\u0027s 4"},{"Start":"02:39.390 ","End":"02:46.950","Text":"minus x^2/3 and so I get, x^minus 2/3."},{"Start":"02:46.950 ","End":"02:51.935","Text":"From here, just by bringing this to the other side,"},{"Start":"02:51.935 ","End":"02:59.755","Text":"we get 4 minus x^2/3 and still plus 1."},{"Start":"02:59.755 ","End":"03:02.340","Text":"Let\u0027s open the brackets."},{"Start":"03:02.340 ","End":"03:08.610","Text":"Expand x^minus 2/3 times 4, 4x^minus 2/3."},{"Start":"03:08.610 ","End":"03:13.815","Text":"Look, x^plus 2/3 is just 1."},{"Start":"03:13.815 ","End":"03:15.765","Text":"It\u0027s x^0 which is 1."},{"Start":"03:15.765 ","End":"03:21.330","Text":"Here, this times this is minus 1 plus 1, and so"},{"Start":"03:21.330 ","End":"03:24.214","Text":"these minus 1 plus 1 will cancel."},{"Start":"03:24.214 ","End":"03:26.525","Text":"This is all we have left."},{"Start":"03:26.525 ","End":"03:30.275","Text":"What we want is the square root of that."},{"Start":"03:30.275 ","End":"03:37.850","Text":"The square root of 1 plus y prime squared is the square root to this."},{"Start":"03:37.850 ","End":"03:40.730","Text":"I take the square root of each piece separately."},{"Start":"03:40.730 ","End":"03:48.375","Text":"The square root of 4 is 2 and the square root of this is to the power of 1/2,"},{"Start":"03:48.375 ","End":"03:55.575","Text":"is x^minus 1/3, because 2/3 times a 1/2 is 1/3."},{"Start":"03:55.575 ","End":"03:58.460","Text":"Now, finally, we can get to the integral."},{"Start":"03:58.460 ","End":"04:02.855","Text":"We want the integral of this thing between 1 and 8. Let me do it over here."},{"Start":"04:02.855 ","End":"04:08.640","Text":"What I want is the integral from 1-8 of this thing,"},{"Start":"04:08.640 ","End":"04:11.805","Text":"I can put the 2 in front and it\u0027s"},{"Start":"04:11.805 ","End":"04:20.395","Text":"x^minus 1/3 dx and this is equal to,"},{"Start":"04:20.395 ","End":"04:24.965","Text":"if I raise the power by 1 and divide by the new power,"},{"Start":"04:24.965 ","End":"04:32.330","Text":"so I get 2 divided by 2/3."},{"Start":"04:32.330 ","End":"04:40.755","Text":"It\u0027s x^2/3 evaluated between 1 and 8."},{"Start":"04:40.755 ","End":"04:48.000","Text":"This equals, now, 2 divided by 2/3 is 2 times 3 over 2 which is just 3."},{"Start":"04:48.000 ","End":"04:50.520","Text":"Then, we have to substitute,"},{"Start":"04:50.520 ","End":"04:54.600","Text":"so it\u0027s 8^2/3 minus 1^2/3."},{"Start":"04:54.600 ","End":"05:01.560","Text":"Now, 8^2/3 is cube root of 8 squared,"},{"Start":"05:01.560 ","End":"05:04.850","Text":"or the cube root of 8 all squared anyway,"},{"Start":"05:04.850 ","End":"05:08.810","Text":"comes out to be 4 and 1 to the power of anything is 1,"},{"Start":"05:08.810 ","End":"05:12.909","Text":"3 times 4 minus 1,"},{"Start":"05:12.909 ","End":"05:21.170","Text":"and this is equal to 3 times 3, which equals 9."},{"Start":"05:21.170 ","End":"05:25.260","Text":"That is the answer. We\u0027re done."}],"Thumbnail":null,"ID":4565},{"Watched":false,"Name":"Exercise 9","Duration":"3m 51s","ChapterTopicVideoID":4557,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"Here we have another length of curve exercise."},{"Start":"00:02.820 ","End":"00:08.190","Text":"But this time we have x in terms of y and not y in terms of x."},{"Start":"00:08.190 ","End":"00:13.650","Text":"Fortunately, there is a formula for this case too,"},{"Start":"00:13.650 ","End":"00:15.945","Text":"when x is a function of y,"},{"Start":"00:15.945 ","End":"00:17.715","Text":"and here\u0027s the sketch,"},{"Start":"00:17.715 ","End":"00:21.765","Text":"we have a very similar formula to the 1 we had before."},{"Start":"00:21.765 ","End":"00:24.420","Text":"Previously we had y prime,"},{"Start":"00:24.420 ","End":"00:26.880","Text":"and now we have x prime."},{"Start":"00:26.880 ","End":"00:29.380","Text":"Of course, in this particular case,"},{"Start":"00:29.380 ","End":"00:32.630","Text":"I could isolate y in terms of x."},{"Start":"00:32.630 ","End":"00:35.090","Text":"I could multiply by 3 over 2"},{"Start":"00:35.090 ","End":"00:39.470","Text":"and then raise to the power of 2/3 and I could do it that way,"},{"Start":"00:39.470 ","End":"00:41.625","Text":"but it won\u0027t always work that way."},{"Start":"00:41.625 ","End":"00:43.570","Text":"You should know this technique."},{"Start":"00:43.570 ","End":"00:45.970","Text":"In fact, even a y is given in terms of x,"},{"Start":"00:45.970 ","End":"00:49.545","Text":"it might be convenient to switch it as x in terms of y,"},{"Start":"00:49.545 ","End":"00:52.640","Text":"you might get an easier integral, for example."},{"Start":"00:52.640 ","End":"00:55.280","Text":"That\u0027s just something I wanted to note."},{"Start":"00:55.280 ","End":"00:56.900","Text":"In our particular case,"},{"Start":"00:56.900 ","End":"01:05.040","Text":"the function x of y is 2/3 y to the power of 3 over 2,"},{"Start":"01:05.040 ","End":"01:10.770","Text":"and the limits c and d are 0 and 3."},{"Start":"01:11.170 ","End":"01:14.760","Text":"Let\u0027s see what x prime is first."},{"Start":"01:14.760 ","End":"01:16.500","Text":"Let me copy this."},{"Start":"01:16.500 ","End":"01:22.155","Text":"X equals 2/3 y to the 3 over 2,"},{"Start":"01:22.155 ","End":"01:24.810","Text":"but I need x prime."},{"Start":"01:24.810 ","End":"01:27.105","Text":"Let\u0027s see, we differentiate this."},{"Start":"01:27.105 ","End":"01:31.070","Text":"3 over 2 times 2/3 is equal to 1,"},{"Start":"01:31.070 ","End":"01:32.760","Text":"so this cancels with this,"},{"Start":"01:32.760 ","End":"01:34.700","Text":"so I got 2/3 of 3 over 2,"},{"Start":"01:34.700 ","End":"01:37.940","Text":"which is 1, and I reduce the power by 1,"},{"Start":"01:37.940 ","End":"01:43.520","Text":"so instead of if 3/2, it\u0027s 1/2."},{"Start":"01:43.520 ","End":"01:51.260","Text":"X prime squared and building up is equal to y to the power of a 1/2 squared."},{"Start":"01:51.260 ","End":"01:54.340","Text":"Square root of y squared is just y itself."},{"Start":"01:54.340 ","End":"01:56.340","Text":"How convenient."},{"Start":"01:56.340 ","End":"01:59.390","Text":"Now if I substitute x prime in here,"},{"Start":"01:59.390 ","End":"02:06.650","Text":"I\u0027ll get the length of curve is equal to the integral from c to d,"},{"Start":"02:06.650 ","End":"02:14.015","Text":"which is 0-3 of the square root of 1 plus."},{"Start":"02:14.015 ","End":"02:17.495","Text":"Now I\u0027m just going to substitute, I mean,"},{"Start":"02:17.495 ","End":"02:22.970","Text":"this bit here is the same as this x prime squared here,"},{"Start":"02:22.970 ","End":"02:30.890","Text":"which is just y, so I end up getting the square root of 1 plus y dy."},{"Start":"02:30.890 ","End":"02:33.709","Text":"This is a straightforward integral."},{"Start":"02:33.709 ","End":"02:36.380","Text":"It\u0027s almost as if it was just y."},{"Start":"02:36.380 ","End":"02:38.540","Text":"The y plus 1 hardly makes any difference,"},{"Start":"02:38.540 ","End":"02:42.080","Text":"but still I\u0027d like to write it as exponent,"},{"Start":"02:42.080 ","End":"02:47.070","Text":"so it\u0027s 1 plus y to the power of 1/2 dy."},{"Start":"02:47.690 ","End":"02:51.470","Text":"I raise the power by 1,"},{"Start":"02:51.470 ","End":"02:53.165","Text":"that\u0027s 3 over 2,"},{"Start":"02:53.165 ","End":"02:55.330","Text":"and divide by that."},{"Start":"02:55.330 ","End":"03:02.340","Text":"I get 1 over 3 over 2 is just 2/3,"},{"Start":"03:02.340 ","End":"03:05.805","Text":"and so we get 2/3,"},{"Start":"03:05.805 ","End":"03:09.855","Text":"1 plus y to the 3 over 2,"},{"Start":"03:09.855 ","End":"03:13.935","Text":"and this between 0 and 3."},{"Start":"03:13.935 ","End":"03:16.625","Text":"Well, I can take the 2/3 outside the brackets."},{"Start":"03:16.625 ","End":"03:18.715","Text":"Now, y equals 3,"},{"Start":"03:18.715 ","End":"03:22.195","Text":"then I get 3 plus 1 is 4,"},{"Start":"03:22.195 ","End":"03:26.330","Text":"4 to the power of 3 over 2 is 8."},{"Start":"03:26.330 ","End":"03:31.135","Text":"The reason is it\u0027s the square root of 4 cubed so I have 8."},{"Start":"03:31.135 ","End":"03:36.800","Text":"Then I put in 0 and I get 1 to the power of 3 over 2, which is 1,"},{"Start":"03:36.800 ","End":"03:40.474","Text":"because 1 to the power of anything is 1,"},{"Start":"03:40.474 ","End":"03:42.540","Text":"so 8 minus 1 is 7,"},{"Start":"03:42.540 ","End":"03:47.880","Text":"7 times 2/3 is 14 over 3."},{"Start":"03:47.880 ","End":"03:52.150","Text":"That is the answer. We\u0027re done."}],"Thumbnail":null,"ID":4566},{"Watched":false,"Name":"Exercise 10","Duration":"5m 30s","ChapterTopicVideoID":4558,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"Here, we have another length of curve problem,"},{"Start":"00:02.400 ","End":"00:05.564","Text":"this time with x in terms of y,"},{"Start":"00:05.564 ","End":"00:07.710","Text":"and we\u0027re given the limits for y."},{"Start":"00:07.710 ","End":"00:12.090","Text":"The important things are the function itself,"},{"Start":"00:12.090 ","End":"00:13.590","Text":"that x is a function of y,"},{"Start":"00:13.590 ","End":"00:15.525","Text":"so this is this."},{"Start":"00:15.525 ","End":"00:19.740","Text":"Then the other thing that\u0027s important is the limit c and d here,"},{"Start":"00:19.740 ","End":"00:22.765","Text":"which are 1 and 2."},{"Start":"00:22.765 ","End":"00:29.190","Text":"What we need is now the formula in our case to figure it out."},{"Start":"00:29.190 ","End":"00:39.540","Text":"First of all, I\u0027ll copy x is y^4 over 4."},{"Start":"00:39.540 ","End":"00:42.210","Text":"You know what? I\u0027ll rewrite this lightly."},{"Start":"00:42.210 ","End":"00:48.060","Text":"I\u0027ll put y^ minus 2 instead of y squared in the denominator. Let\u0027s see."},{"Start":"00:48.060 ","End":"00:51.495","Text":"That\u0027s x, x prime equals,"},{"Start":"00:51.495 ","End":"00:54.675","Text":"derivative of this is 4y cubed,"},{"Start":"00:54.675 ","End":"01:04.740","Text":"the 4 stays, and here we have minus 2y^ minus 3 over 8, which stays."},{"Start":"01:04.740 ","End":"01:06.945","Text":"Let\u0027s see. Can I simplify this a bit?"},{"Start":"01:06.945 ","End":"01:11.135","Text":"Yes, sure. 4 over 4 is 1 so that\u0027s y cubed."},{"Start":"01:11.135 ","End":"01:19.240","Text":"2 over 8 is 1/4 so I have minus 1/4 y^ minus 3."},{"Start":"01:19.240 ","End":"01:21.885","Text":"What\u0027s x prime squared?"},{"Start":"01:21.885 ","End":"01:23.840","Text":"x prime squared equals,"},{"Start":"01:23.840 ","End":"01:26.435","Text":"and I\u0027ll just remind you of the formula,"},{"Start":"01:26.435 ","End":"01:29.270","Text":"this is a and this is minus b,"},{"Start":"01:29.270 ","End":"01:30.680","Text":"so when it\u0027s minus here,"},{"Start":"01:30.680 ","End":"01:37.085","Text":"it\u0027s minus here, and so we get y cubed squared is y^6,"},{"Start":"01:37.085 ","End":"01:42.140","Text":"minus 2ab is minus twice this times this."},{"Start":"01:42.140 ","End":"01:47.895","Text":"But look, y^3 times y^ minus 3 is just 1."},{"Start":"01:47.895 ","End":"01:51.105","Text":"I have minus twice 1/4,"},{"Start":"01:51.105 ","End":"01:55.170","Text":"so that\u0027s minus, 2/4 is a 1/2,"},{"Start":"01:55.170 ","End":"01:58.410","Text":"and then plus b squared,"},{"Start":"01:58.410 ","End":"02:01.290","Text":"which is plus 1/16,"},{"Start":"02:01.290 ","End":"02:06.240","Text":"because this is 1/4 times 1/4, y^ minus 6."},{"Start":"02:06.240 ","End":"02:09.350","Text":"If I take, instead of this,"},{"Start":"02:09.350 ","End":"02:13.470","Text":"1 plus x prime squared."},{"Start":"02:13.470 ","End":"02:17.840","Text":"If I add 1, I get the same thing here."},{"Start":"02:17.840 ","End":"02:21.815","Text":"Now the minus 1/2 plus the 1 becomes plus 1/2,"},{"Start":"02:21.815 ","End":"02:24.040","Text":"and here also the same."},{"Start":"02:24.040 ","End":"02:27.080","Text":"What\u0027s just happened between here and here?"},{"Start":"02:27.080 ","End":"02:31.625","Text":"I\u0027ve changed the minus to a plus over here."},{"Start":"02:31.625 ","End":"02:36.715","Text":"This formula says we still have a perfect square if we change from minus to plus."},{"Start":"02:36.715 ","End":"02:44.855","Text":"This is just, instead of the square of y cubed minus 1/4 y^ minus 3,"},{"Start":"02:44.855 ","End":"02:51.375","Text":"we\u0027ll get y cubed plus 1/4 y^ minus 3."},{"Start":"02:51.375 ","End":"02:52.980","Text":"This thing is squared,"},{"Start":"02:52.980 ","End":"02:58.175","Text":"so when we take the square root of 1 plus x prime squared,"},{"Start":"02:58.175 ","End":"02:59.690","Text":"then it\u0027s without the 2,"},{"Start":"02:59.690 ","End":"03:03.885","Text":"it\u0027s y cubed plus 1/4 y^ minus 3."},{"Start":"03:03.885 ","End":"03:06.325","Text":"Next, we\u0027re needing to do a little integral."},{"Start":"03:06.325 ","End":"03:10.100","Text":"What we need is the integral from c to d,"},{"Start":"03:10.100 ","End":"03:12.320","Text":"which is from 1-2."},{"Start":"03:12.320 ","End":"03:20.025","Text":"What we need is the integral from 1-2 of this thing,"},{"Start":"03:20.025 ","End":"03:29.230","Text":"of y cubed plus 1/4 y^ minus 3 dy."},{"Start":"03:31.850 ","End":"03:34.680","Text":"It\u0027s an easy integral,"},{"Start":"03:34.680 ","End":"03:40.110","Text":"y^3 so y^4, and we divide by the 4."},{"Start":"03:40.110 ","End":"03:42.450","Text":"Here, we raise it by 1,"},{"Start":"03:42.450 ","End":"03:44.565","Text":"it\u0027s y^ minus 2,"},{"Start":"03:44.565 ","End":"03:46.760","Text":"and if we divide by minus 2,"},{"Start":"03:46.760 ","End":"03:48.880","Text":"we get minus an 8,"},{"Start":"03:48.880 ","End":"03:51.914","Text":"and I\u0027ll put the 8 in the bottom,"},{"Start":"03:51.914 ","End":"03:58.990","Text":"y^ minus 2, and all this from 1-2."},{"Start":"03:59.060 ","End":"04:02.190","Text":"Let\u0027s see. If we put 2,"},{"Start":"04:02.190 ","End":"04:07.940","Text":"2^4 is 16, we get 16 over 4 minus, well,"},{"Start":"04:07.940 ","End":"04:13.650","Text":"let\u0027s just say it\u0027s a 1/4 over 8, it is 1/32,"},{"Start":"04:13.650 ","End":"04:15.735","Text":"but I want to show how I got there,"},{"Start":"04:15.735 ","End":"04:19.010","Text":"and then minus same expression,"},{"Start":"04:19.010 ","End":"04:20.180","Text":"but this time with 1."},{"Start":"04:20.180 ","End":"04:22.250","Text":"1 to anything is 1,"},{"Start":"04:22.250 ","End":"04:26.165","Text":"so it\u0027s 1/4 minus 1/8."},{"Start":"04:26.165 ","End":"04:28.880","Text":"Maybe I\u0027ll take the quarters separately."},{"Start":"04:28.880 ","End":"04:33.540","Text":"16 minus 1/4 is 15/4."},{"Start":"04:33.540 ","End":"04:36.060","Text":"How many 8ths do I have?"},{"Start":"04:36.060 ","End":"04:41.970","Text":"I have minus 1/4 plus 1/8,"},{"Start":"04:41.970 ","End":"04:46.170","Text":"so that\u0027s minus 3/4 over 8."},{"Start":"04:46.170 ","End":"04:49.975","Text":"This second expression is just this thing,"},{"Start":"04:49.975 ","End":"04:52.430","Text":"is 3/4 times 8,"},{"Start":"04:52.430 ","End":"04:55.310","Text":"this is just 3/32."},{"Start":"04:55.310 ","End":"04:57.590","Text":"I just noticed a little mistake."},{"Start":"04:57.590 ","End":"04:58.670","Text":"This should be plus."},{"Start":"04:58.670 ","End":"05:01.300","Text":"We said that this was going to be plus."},{"Start":"05:01.300 ","End":"05:05.225","Text":"Continuing, this is equal to,"},{"Start":"05:05.225 ","End":"05:08.530","Text":"let\u0027s put it all over 32."},{"Start":"05:08.530 ","End":"05:13.580","Text":"So 15 I have to multiply it by 8, 15 times 8."},{"Start":"05:13.580 ","End":"05:17.940","Text":"Let\u0027 see, 15 times 8 is 120,"},{"Start":"05:17.940 ","End":"05:21.575","Text":"plus 3 over 32."},{"Start":"05:21.575 ","End":"05:28.060","Text":"So our answer will be 123/32,"},{"Start":"05:28.060 ","End":"05:30.970","Text":"and we are done."}],"Thumbnail":null,"ID":4567},{"Watched":false,"Name":"Exercise 11","Duration":"4m 17s","ChapterTopicVideoID":4559,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"This exercise talks about the distance traveled by a particle."},{"Start":"00:04.965 ","End":"00:09.550","Text":"We\u0027re given its x and y coordinates as functions of time."},{"Start":"00:09.550 ","End":"00:12.570","Text":"Also, we have the start time and the end time."},{"Start":"00:12.570 ","End":"00:19.080","Text":"This type of problem is actually a length of curve problem in the parametric dial."},{"Start":"00:19.080 ","End":"00:21.105","Text":"Basically what we need to know,"},{"Start":"00:21.105 ","End":"00:24.270","Text":"the x-coordinate as a function of time,"},{"Start":"00:24.270 ","End":"00:27.180","Text":"the y-coordinate to the function of time,"},{"Start":"00:27.180 ","End":"00:29.610","Text":"and also the start and end times."},{"Start":"00:29.610 ","End":"00:31.095","Text":"We have all these."},{"Start":"00:31.095 ","End":"00:33.930","Text":"This is our x, which is x of t. This is y,"},{"Start":"00:33.930 ","End":"00:37.795","Text":"which is y of t. This is the start time,"},{"Start":"00:37.795 ","End":"00:40.040","Text":"which is called here t_b."},{"Start":"00:40.040 ","End":"00:43.850","Text":"This is the end time which is called t_b."},{"Start":"00:43.850 ","End":"00:48.215","Text":"All we need to do is apply this formula here in red."},{"Start":"00:48.215 ","End":"00:51.580","Text":"I see that we need both derivatives."},{"Start":"00:51.580 ","End":"00:54.855","Text":"Let\u0027s start and differentiate."},{"Start":"00:54.855 ","End":"01:00.140","Text":"X prime is equal to cosine t as"},{"Start":"01:00.140 ","End":"01:06.215","Text":"derivative minus sine t. Here we have a product."},{"Start":"01:06.215 ","End":"01:09.815","Text":"This times this, so we use the product rule."},{"Start":"01:09.815 ","End":"01:13.190","Text":"It\u0027s the derivative of the first which is 1"},{"Start":"01:13.190 ","End":"01:16.460","Text":"times the second which is sine t. I\u0027ll write it as"},{"Start":"01:16.460 ","End":"01:23.690","Text":"1 times sine t. Then we have T as is and the derivative of sine t,"},{"Start":"01:23.690 ","End":"01:28.430","Text":"which is cosine t. Y prime,"},{"Start":"01:28.430 ","End":"01:31.985","Text":"is equal to derivative of sine t,"},{"Start":"01:31.985 ","End":"01:34.790","Text":"which is cosine t. Now,"},{"Start":"01:34.790 ","End":"01:37.505","Text":"here we have a product minus,"},{"Start":"01:37.505 ","End":"01:41.240","Text":"first of all, differentiate the t which is 1,"},{"Start":"01:41.240 ","End":"01:44.400","Text":"and leave cosine t alone."},{"Start":"01:44.400 ","End":"01:46.010","Text":"In the second one,"},{"Start":"01:46.010 ","End":"01:52.715","Text":"we leave the t as is and differentiate the cosine to get minus sign."},{"Start":"01:52.715 ","End":"01:55.700","Text":"Now, we have stuff that cancels."},{"Start":"01:55.700 ","End":"02:01.730","Text":"The minus sine t cancels with plus sine t. The 1 makes no difference and"},{"Start":"02:01.730 ","End":"02:07.520","Text":"the cosine of t cancels with the cosine t. These two minuses combine to give a plus."},{"Start":"02:07.520 ","End":"02:11.070","Text":"Then we can now compute what it says here,"},{"Start":"02:11.070 ","End":"02:13.360","Text":"that the length l,"},{"Start":"02:13.360 ","End":"02:23.905","Text":"which is equal to the square root from the integral from 0 to pi over 2."},{"Start":"02:23.905 ","End":"02:25.610","Text":"Now, what do we have here?"},{"Start":"02:25.610 ","End":"02:27.920","Text":"We have x prime squared,"},{"Start":"02:27.920 ","End":"02:30.830","Text":"which is t cosine of t squared,"},{"Start":"02:30.830 ","End":"02:36.740","Text":"which is just t squared cosine squared t. Then y"},{"Start":"02:36.740 ","End":"02:44.395","Text":"prime squared is just t squared sine squared t and all this dt."},{"Start":"02:44.395 ","End":"02:48.470","Text":"Now look, the cosine squared plus sine squared is"},{"Start":"02:48.470 ","End":"02:54.635","Text":"1 sine squared of Alpha plus cosine squared of Alpha equals 1,"},{"Start":"02:54.635 ","End":"02:58.270","Text":"because here I can take t squared outside the brackets."},{"Start":"02:58.270 ","End":"03:04.520","Text":"What we get is the integral from 0 to"},{"Start":"03:04.520 ","End":"03:10.700","Text":"pi over 2 of the square root of just t squared."},{"Start":"03:10.700 ","End":"03:13.940","Text":"Because after we\u0027ve taken the cosine squared plus"},{"Start":"03:13.940 ","End":"03:17.210","Text":"sine squared equals 1 and taken t squared outside the brackets,"},{"Start":"03:17.210 ","End":"03:18.635","Text":"that\u0027s what we\u0027re left with."},{"Start":"03:18.635 ","End":"03:27.020","Text":"Dt, which is just equal to the integral from 0 to pi over 2."},{"Start":"03:27.020 ","End":"03:29.900","Text":"Now, the square root of t squared is absolute value of t,"},{"Start":"03:29.900 ","End":"03:31.520","Text":"but t is positive,"},{"Start":"03:31.520 ","End":"03:33.900","Text":"so it\u0027s just tdt."},{"Start":"03:34.270 ","End":"03:37.565","Text":"That\u0027s a pretty straightforward integral."},{"Start":"03:37.565 ","End":"03:47.050","Text":"We get t squared over 2 between 0 and pi over 2."},{"Start":"03:47.060 ","End":"03:50.400","Text":"If we put t is pi over 2,"},{"Start":"03:50.400 ","End":"03:56.760","Text":"we get pi over 2 squared over 2."},{"Start":"03:56.760 ","End":"03:59.690","Text":"If we put 0 in,"},{"Start":"03:59.690 ","End":"04:03.685","Text":"we get 0 squared over 2."},{"Start":"04:03.685 ","End":"04:05.794","Text":"Now, this thing is 0."},{"Start":"04:05.794 ","End":"04:10.055","Text":"The numerator is pi squared over 4 divided by 2."},{"Start":"04:10.055 ","End":"04:14.165","Text":"We\u0027re just left with pi squared over 8."},{"Start":"04:14.165 ","End":"04:18.210","Text":"That is the answer. We are done."}],"Thumbnail":null,"ID":4568},{"Watched":false,"Name":"Exercise 12","Duration":"3m 14s","ChapterTopicVideoID":4560,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"This is one of those distance problems,"},{"Start":"00:02.400 ","End":"00:07.245","Text":"which is really a length of curve problem in the parametric form."},{"Start":"00:07.245 ","End":"00:11.220","Text":"The things we need to know in order to use the formula here,"},{"Start":"00:11.220 ","End":"00:15.450","Text":"we need to know what the function x of t is and this is what it is."},{"Start":"00:15.450 ","End":"00:23.355","Text":"Y of t is this and the start and endpoints for t are 0 and 4."},{"Start":"00:23.355 ","End":"00:25.440","Text":"Now, that we know all these,"},{"Start":"00:25.440 ","End":"00:28.920","Text":"we\u0027re going to start computing l from this formula."},{"Start":"00:28.920 ","End":"00:35.480","Text":"I see I need both x prime and y prime and derivatives with respect to t. Let\u0027s see,"},{"Start":"00:35.480 ","End":"00:41.105","Text":"x prime of t is equal to the derivative of t squared over 2"},{"Start":"00:41.105 ","End":"00:47.550","Text":"is just t. Y prime is equal to?"},{"Start":"00:47.550 ","End":"00:53.175","Text":"Let\u0027s see, 3 over 2 times 1/3 is 1/2."},{"Start":"00:53.175 ","End":"00:56.910","Text":"That\u0027s simple fraction problem."},{"Start":"00:56.910 ","End":"01:01.290","Text":"Now, we have to lower by 1 the exponent,"},{"Start":"01:01.290 ","End":"01:05.980","Text":"so we get 2t plus 1^1/2."},{"Start":"01:05.980 ","End":"01:11.685","Text":"We also have to multiply by the inner derivative, which is 2."},{"Start":"01:11.685 ","End":"01:17.660","Text":"Actually, we can cancel the 2 goes with the 1/2,"},{"Start":"01:17.660 ","End":"01:19.990","Text":"so that makes life simpler."},{"Start":"01:19.990 ","End":"01:22.730","Text":"Now, I want to see what this expression is."},{"Start":"01:22.730 ","End":"01:25.910","Text":"X prime squared plus y prime squared."},{"Start":"01:25.910 ","End":"01:32.825","Text":"X prime squared plus y prime squared is,"},{"Start":"01:32.825 ","End":"01:34.400","Text":"this part is easy,"},{"Start":"01:34.400 ","End":"01:37.980","Text":"that\u0027s t squared plus,"},{"Start":"01:37.980 ","End":"01:39.765","Text":"this is easy too,"},{"Start":"01:39.765 ","End":"01:45.350","Text":"because 1^1/2, when you square it is just to the power of 1."},{"Start":"01:45.350 ","End":"01:49.360","Text":"We get plus 2t plus 1."},{"Start":"01:49.360 ","End":"01:52.470","Text":"Notice that this is a perfect square,"},{"Start":"01:52.470 ","End":"01:55.945","Text":"it\u0027s so familiar, it should be."},{"Start":"01:55.945 ","End":"02:00.410","Text":"Anyway, you can always check by multiplying this and you\u0027ll get this."},{"Start":"02:00.410 ","End":"02:04.015","Text":"Now, we\u0027re ready to tackle the integral."},{"Start":"02:04.015 ","End":"02:09.130","Text":"L is the integral from t_a to t_b,"},{"Start":"02:09.130 ","End":"02:14.170","Text":"which is 0 to 4 of"},{"Start":"02:14.170 ","End":"02:20.305","Text":"the square root of x squared plus y squared is just the square root of t plus 1 squared,"},{"Start":"02:20.305 ","End":"02:22.780","Text":"which is just t plus 1."},{"Start":"02:22.780 ","End":"02:25.840","Text":"Normally, I would need absolute values because"},{"Start":"02:25.840 ","End":"02:29.320","Text":"the square root of a squared is absolute value of a,"},{"Start":"02:29.320 ","End":"02:31.090","Text":"but since t is from 0 to 4,"},{"Start":"02:31.090 ","End":"02:32.140","Text":"I don\u0027t need that."},{"Start":"02:32.140 ","End":"02:35.545","Text":"Just regular brackets and dt."},{"Start":"02:35.545 ","End":"02:38.460","Text":"This is a straightforward integral."},{"Start":"02:38.460 ","End":"02:48.200","Text":"This is equal to t squared over 2 plus t taken between 0 and 4."},{"Start":"02:48.200 ","End":"02:56.010","Text":"If I put in 4, I get 4 squared over 2 plus 4."},{"Start":"02:56.010 ","End":"02:59.070","Text":"If I put in 0, everything 0,"},{"Start":"02:59.070 ","End":"03:04.365","Text":"so, well, I could write minus 0 plus 0."},{"Start":"03:04.365 ","End":"03:13.550","Text":"In short, 4 squared over 2 is 16 over 2 is 8 plus 4 is 12 and 12 is our final answer."},{"Start":"03:13.550 ","End":"03:15.600","Text":"We are done."}],"Thumbnail":null,"ID":4569},{"Watched":false,"Name":"Exercise 13","Duration":"4m 5s","ChapterTopicVideoID":4561,"CourseChapterTopicPlaylistID":3986,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"Here, again, we have 1 of these distance problems,"},{"Start":"00:03.945 ","End":"00:06.135","Text":"and distance traveled by a particle,"},{"Start":"00:06.135 ","End":"00:12.029","Text":"which really is just the parametric form of a length of curve problem."},{"Start":"00:12.029 ","End":"00:15.270","Text":"The things we need to know in length of curve,"},{"Start":"00:15.270 ","End":"00:18.390","Text":"as you can see from this formula there\u0027s 4 quantities."},{"Start":"00:18.390 ","End":"00:20.520","Text":"We need to know x as a function of t,"},{"Start":"00:20.520 ","End":"00:22.200","Text":"that\u0027s the x of t,"},{"Start":"00:22.200 ","End":"00:25.140","Text":"we need to know y as a function of t,"},{"Start":"00:25.140 ","End":"00:28.500","Text":"and we need to know the limits of integration,"},{"Start":"00:28.500 ","End":"00:31.050","Text":"which is from 0-4 for"},{"Start":"00:31.050 ","End":"00:38.205","Text":"t. Let\u0027s go about computing l. We need each of the derivatives first."},{"Start":"00:38.205 ","End":"00:42.615","Text":"Let\u0027s see what is x prime, what\u0027s y prime."},{"Start":"00:42.615 ","End":"00:49.515","Text":"X prime equals, now here we have 3/2 times 1/3,"},{"Start":"00:49.515 ","End":"00:52.075","Text":"which is just 1/2."},{"Start":"00:52.075 ","End":"00:54.935","Text":"Then we reduce the exponent by 1,"},{"Start":"00:54.935 ","End":"00:59.040","Text":"so we get 2t plus 1^1/2."},{"Start":"00:59.170 ","End":"01:05.655","Text":"But there\u0027s also an internal derivative in a derivative which is 2."},{"Start":"01:05.655 ","End":"01:09.540","Text":"This 2 just cancels with this 1/2."},{"Start":"01:09.540 ","End":"01:13.615","Text":"Now y prime is equal to,"},{"Start":"01:13.615 ","End":"01:17.025","Text":"derivative of t squared/2 is just t,"},{"Start":"01:17.025 ","End":"01:21.110","Text":"derivative of t is 1."},{"Start":"01:21.110 ","End":"01:23.255","Text":"Now that we have both of these,"},{"Start":"01:23.255 ","End":"01:27.680","Text":"let\u0027s see what is x prime squared plus y prime squared."},{"Start":"01:27.680 ","End":"01:32.065","Text":"X prime squared plus y prime squared,"},{"Start":"01:32.065 ","End":"01:35.460","Text":"brackets here, is equal to."},{"Start":"01:35.460 ","End":"01:38.520","Text":"This 1 squared, the power of 2,"},{"Start":"01:38.520 ","End":"01:40.170","Text":"and the power of 1/2 cancels,"},{"Start":"01:40.170 ","End":"01:42.475","Text":"so it\u0027s just 2t plus 1."},{"Start":"01:42.475 ","End":"01:49.115","Text":"The second bit is this thing squared is just t plus 1 squared."},{"Start":"01:49.115 ","End":"01:51.050","Text":"What does this give us?"},{"Start":"01:51.050 ","End":"01:55.670","Text":"This is 2 squared plus 2t plus 1."},{"Start":"01:55.670 ","End":"02:05.860","Text":"That makes it, t squared plus 2t plus 2t is 4t plus 1 plus 1 is plus 2."},{"Start":"02:05.860 ","End":"02:10.250","Text":"At this point, I was really expecting to get a perfect square,"},{"Start":"02:10.250 ","End":"02:12.905","Text":"which would have been if there was 4 at the end."},{"Start":"02:12.905 ","End":"02:14.480","Text":"I may have missed copied the problem."},{"Start":"02:14.480 ","End":"02:17.245","Text":"Let\u0027s just change this 1 into 3,"},{"Start":"02:17.245 ","End":"02:24.975","Text":"then this will have to become 3 and this will have to become 3,"},{"Start":"02:24.975 ","End":"02:28.620","Text":"so I\u0027ll write here 3 and 3."},{"Start":"02:28.620 ","End":"02:31.875","Text":"Now, this becomes 4,"},{"Start":"02:31.875 ","End":"02:34.650","Text":"so erase the 2,"},{"Start":"02:34.650 ","End":"02:36.690","Text":"and write 4."},{"Start":"02:36.690 ","End":"02:38.310","Text":"Now I\u0027m happier."},{"Start":"02:38.310 ","End":"02:40.290","Text":"Even though it\u0027s fudging a bit,"},{"Start":"02:40.290 ","End":"02:44.445","Text":"I just presume I must have miscopied. Let\u0027s continue."},{"Start":"02:44.445 ","End":"02:50.810","Text":"What we need now is the integral from 0-4."},{"Start":"02:50.810 ","End":"02:54.060","Text":"Put the 0 and 4 back in view."},{"Start":"02:54.350 ","End":"02:57.210","Text":"Now, this thing I said is a perfect square,"},{"Start":"02:57.210 ","End":"02:59.780","Text":"it is in fact t plus 2 all squared,"},{"Start":"02:59.780 ","End":"03:02.555","Text":"as you can check by multiplying it out."},{"Start":"03:02.555 ","End":"03:10.780","Text":"The square root of x prime squared plus y prime squared is the square root of this,"},{"Start":"03:10.780 ","End":"03:13.465","Text":"which is just t plus 2."},{"Start":"03:13.465 ","End":"03:16.975","Text":"This is a straightforward integral."},{"Start":"03:16.975 ","End":"03:19.715","Text":"This is equal to,"},{"Start":"03:19.715 ","End":"03:22.995","Text":"integral of t is 1/2,"},{"Start":"03:22.995 ","End":"03:26.955","Text":"t squared, or t squared/2, plus 2t."},{"Start":"03:26.955 ","End":"03:31.575","Text":"All this taken between 0 and 4."},{"Start":"03:31.575 ","End":"03:34.560","Text":"If I put in 4,"},{"Start":"03:34.560 ","End":"03:42.270","Text":"I get 4 squared/2 is 16/2 is 8."},{"Start":"03:42.270 ","End":"03:46.320","Text":"2 times 4 is 8."},{"Start":"03:46.320 ","End":"03:50.295","Text":"When I put in 0, I get 1/2, t squared,"},{"Start":"03:50.295 ","End":"03:52.125","Text":"1/2 0 squared is 0,"},{"Start":"03:52.125 ","End":"03:54.435","Text":"and twice 0 is 0."},{"Start":"03:54.435 ","End":"04:00.615","Text":"According to this, what I get is 16."},{"Start":"04:00.615 ","End":"04:02.795","Text":"We are done."},{"Start":"04:02.795 ","End":"04:06.570","Text":"Forgive me for the miscopying of the problem."}],"Thumbnail":null,"ID":4570}],"ID":3986},{"Name":"Work","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"3m 22s","ChapterTopicVideoID":8139,"CourseChapterTopicPlaylistID":6161,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this clip, I\u0027ll be talking about another application of"},{"Start":"00:02.640 ","End":"00:06.345","Text":"integrals to the concept of work in physics."},{"Start":"00:06.345 ","End":"00:10.650","Text":"The main formula in physics relating to this is that"},{"Start":"00:10.650 ","End":"00:14.640","Text":"work is equal to force times distance."},{"Start":"00:14.640 ","End":"00:16.605","Text":"But I need to explain,"},{"Start":"00:16.605 ","End":"00:21.255","Text":"first of all, that the force has to be constant."},{"Start":"00:21.255 ","End":"00:25.695","Text":"It applies to some object or body,"},{"Start":"00:25.695 ","End":"00:31.155","Text":"and it moves in the direction of the force and a straight line,"},{"Start":"00:31.155 ","End":"00:35.760","Text":"a distance of d. If I draw a little diagram,"},{"Start":"00:35.760 ","End":"00:42.180","Text":"and typically, we let the body move along the x-axis in the positive direction."},{"Start":"00:42.180 ","End":"00:48.120","Text":"Suppose the body started here and ended up here,"},{"Start":"00:48.120 ","End":"00:53.930","Text":"so it moves and there\u0027s a force in this direction of F. Let\u0027s say it"},{"Start":"00:53.930 ","End":"01:00.610","Text":"starts at position a along the x-axis and ends at b along the x-axis."},{"Start":"01:00.610 ","End":"01:10.535","Text":"Then if we let this be d and d is b minus a,"},{"Start":"01:10.535 ","End":"01:16.335","Text":"then the work is F times d. Now,"},{"Start":"01:16.335 ","End":"01:19.710","Text":"this formula works when F is constant, but in general,"},{"Start":"01:19.710 ","End":"01:21.410","Text":"in real life problems,"},{"Start":"01:21.410 ","End":"01:25.700","Text":"F will not be a constant and we have to handle it more generally."},{"Start":"01:25.700 ","End":"01:30.140","Text":"The extension of this formula will be that the work,"},{"Start":"01:30.140 ","End":"01:32.375","Text":"instead of being F times d,"},{"Start":"01:32.375 ","End":"01:37.055","Text":"is the integral of Fdx,"},{"Start":"01:37.055 ","End":"01:41.675","Text":"where x goes from a to b."},{"Start":"01:41.675 ","End":"01:45.910","Text":"Actually, I should emphasize that it\u0027s not f but F of x."},{"Start":"01:45.910 ","End":"01:49.200","Text":"Let me indicate by saying F of x."},{"Start":"01:49.200 ","End":"01:54.215","Text":"I just like to show you that this actually is a special case of this."},{"Start":"01:54.215 ","End":"02:02.075","Text":"Because if I say that F of x equals some constant F and I use this formula,"},{"Start":"02:02.075 ","End":"02:12.155","Text":"then we get that w is equal to the integral from a to b of just Fdx."},{"Start":"02:12.155 ","End":"02:15.384","Text":"We can bring the F in front of the integral sign,"},{"Start":"02:15.384 ","End":"02:19.070","Text":"so F integral from a to b of just dx,"},{"Start":"02:19.070 ","End":"02:20.700","Text":"or write it as 1dx."},{"Start":"02:20.700 ","End":"02:23.540","Text":"The integral of 1 is just x,"},{"Start":"02:23.540 ","End":"02:27.190","Text":"so it\u0027s F times x from a to b,"},{"Start":"02:27.190 ","End":"02:28.580","Text":"and we plug in x equals b,"},{"Start":"02:28.580 ","End":"02:29.710","Text":"x equals a and subtract,"},{"Start":"02:29.710 ","End":"02:32.570","Text":"we get F times b minus a,"},{"Start":"02:32.570 ","End":"02:34.385","Text":"but b minus a is d,"},{"Start":"02:34.385 ","End":"02:42.690","Text":"so it\u0027s F times d. This is just a special case of this where F of x is a constant."},{"Start":"02:45.000 ","End":"02:48.795","Text":"The next thing is to just solve some problems and there\u0027s"},{"Start":"02:48.795 ","End":"02:52.790","Text":"a whole variety of problems in physics based on this."},{"Start":"02:52.790 ","End":"02:55.685","Text":"The first one will involve"},{"Start":"02:55.685 ","End":"03:00.635","Text":"the concept of a spring and I suggest maybe you review Hooke\u0027s law,"},{"Start":"03:00.635 ","End":"03:03.170","Text":"but if not, I\u0027ll bring it in the example."},{"Start":"03:03.170 ","End":"03:08.360","Text":"I also want to note that I will not be using the metric system,"},{"Start":"03:08.360 ","End":"03:12.710","Text":"I\u0027ll be using the system used in America with the pounds,"},{"Start":"03:12.710 ","End":"03:16.145","Text":"feet and inches, and so on."},{"Start":"03:16.145 ","End":"03:22.590","Text":"I\u0027ll keep this formula and we\u0027ll start the first problem."}],"Thumbnail":null,"ID":8293},{"Watched":false,"Name":"Example1","Duration":"4m 22s","ChapterTopicVideoID":8140,"CourseChapterTopicPlaylistID":6161,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:03.975","Text":"The example, which as I said,"},{"Start":"00:03.975 ","End":"00:06.674","Text":"will involve springs is as follows."},{"Start":"00:06.674 ","End":"00:09.550","Text":"I\u0027ll just write it shorthand."},{"Start":"00:10.790 ","End":"00:15.990","Text":"We are given a spring which has a natural length of 20 inches."},{"Start":"00:15.990 ","End":"00:19.080","Text":"Natural length I mean when it\u0027s not stretched."},{"Start":"00:19.080 ","End":"00:23.460","Text":"We\u0027re told that it takes a 4 pound force to stretch this same"},{"Start":"00:23.460 ","End":"00:28.425","Text":"spring from 20 inches to 30 inches."},{"Start":"00:28.425 ","End":"00:32.990","Text":"We want to compute how much work will it require to stretch"},{"Start":"00:32.990 ","End":"00:39.110","Text":"the same spring from 35 to 38 inches."},{"Start":"00:39.110 ","End":"00:45.870","Text":"To do this, we need to know a law about springs from physics,"},{"Start":"00:45.870 ","End":"00:48.070","Text":"it\u0027s called Hooke\u0027s Law."},{"Start":"00:48.070 ","End":"00:55.025","Text":"Basically what it says is that the force, the stretching force,"},{"Start":"00:55.025 ","End":"01:01.655","Text":"is equal to some constant k times the increase in length,"},{"Start":"01:01.655 ","End":"01:05.735","Text":"let\u0027s call it Delta x."},{"Start":"01:05.735 ","End":"01:09.305","Text":"This k is called the spring\u0027s constant,"},{"Start":"01:09.305 ","End":"01:11.075","Text":"the constant of the spring."},{"Start":"01:11.075 ","End":"01:18.445","Text":"We can actually compute this k because when we stretch from 20 to 30,"},{"Start":"01:18.445 ","End":"01:23.010","Text":"the Delta, let\u0027s call the extension,"},{"Start":"01:23.010 ","End":"01:26.055","Text":"the increase in x, x was 20 and x is 30,"},{"Start":"01:26.055 ","End":"01:28.125","Text":"Delta x is 10."},{"Start":"01:28.125 ","End":"01:33.000","Text":"We know that 4 pounds,"},{"Start":"01:33.000 ","End":"01:40.865","Text":"so 4 is equal to k times 10."},{"Start":"01:40.865 ","End":"01:45.540","Text":"As I said, the 10 is the 30 minus the 20."},{"Start":"01:45.540 ","End":"01:48.765","Text":"That\u0027s where the 10 comes from because it\u0027s the increase,"},{"Start":"01:48.765 ","End":"01:51.770","Text":"not the length itself."},{"Start":"01:51.770 ","End":"02:01.300","Text":"In that case, that gives us that k is equal to 4/10, which is 0.4."},{"Start":"02:01.300 ","End":"02:04.670","Text":"I don\u0027t really like using the letter Delta."},{"Start":"02:04.670 ","End":"02:08.080","Text":"Let\u0027s just call x the extension."},{"Start":"02:08.080 ","End":"02:14.510","Text":"I\u0027ll just call it x. I\u0027ll write that x is the extension, not the length."},{"Start":"02:14.510 ","End":"02:16.745","Text":"It\u0027s how much I stretched it to."},{"Start":"02:16.745 ","End":"02:19.850","Text":"In this case, the extension would be 10."},{"Start":"02:19.850 ","End":"02:25.850","Text":"What we have is that we want to know what the work"},{"Start":"02:25.850 ","End":"02:31.910","Text":"is when the extension here is 15,"},{"Start":"02:31.910 ","End":"02:33.785","Text":"just like here it was 10,"},{"Start":"02:33.785 ","End":"02:36.785","Text":"the extension here will be 18."},{"Start":"02:36.785 ","End":"02:39.530","Text":"We\u0027re going from 15 to 18."},{"Start":"02:39.530 ","End":"02:44.540","Text":"We have to subtract the 20 in order to work with extensions not length."},{"Start":"02:44.540 ","End":"02:48.620","Text":"It\u0027s the difference with from the original length."},{"Start":"02:48.620 ","End":"02:55.620","Text":"What we want is we know that f of x is kx."},{"Start":"02:55.620 ","End":"02:59.895","Text":"It\u0027s 0.4 times x,"},{"Start":"02:59.895 ","End":"03:02.815","Text":"and since we want to go from 15 to 18,"},{"Start":"03:02.815 ","End":"03:12.470","Text":"we\u0027ve got that the work is the integral from 15 to 18 of f of x dx,"},{"Start":"03:12.470 ","End":"03:17.565","Text":"which is 0.4x dx."},{"Start":"03:17.565 ","End":"03:20.730","Text":"Let\u0027s see what this comes out to be."},{"Start":"03:20.730 ","End":"03:25.184","Text":"I can bring the 0.4 out front."},{"Start":"03:25.184 ","End":"03:32.355","Text":"So 0.4 the integral of x is x squared over 2."},{"Start":"03:32.355 ","End":"03:37.875","Text":"I have to substitute 15 and 18 here."},{"Start":"03:37.875 ","End":"03:42.195","Text":"Let\u0027s see. What do we get?"},{"Start":"03:42.195 ","End":"03:46.635","Text":"Well, the 1/2 with the 0.4 can give me 0.2."},{"Start":"03:46.635 ","End":"03:48.435","Text":"Then x squared will be,"},{"Start":"03:48.435 ","End":"03:54.575","Text":"18 squared I happen to know by heart is 324,"},{"Start":"03:54.575 ","End":"03:58.970","Text":"15 squared is 225."},{"Start":"03:58.970 ","End":"04:00.830","Text":"The difference is, let\u0027s see,"},{"Start":"04:00.830 ","End":"04:06.065","Text":"this is 99 times 0.2."},{"Start":"04:06.065 ","End":"04:13.980","Text":"That would be 19.8 inch pounds."},{"Start":"04:14.330 ","End":"04:18.795","Text":"That\u0027s the answer for this example."},{"Start":"04:18.795 ","End":"04:22.930","Text":"Let\u0027s go on to another example."}],"Thumbnail":null,"ID":8294},{"Watched":false,"Name":"Example2","Duration":"4m 24s","ChapterTopicVideoID":8141,"CourseChapterTopicPlaylistID":6161,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"In this next example,"},{"Start":"00:01.920 ","End":"00:05.040","Text":"it\u0027s going to involve a real-life problem of"},{"Start":"00:05.040 ","End":"00:11.770","Text":"a coal mine and lifting a bucket of coal from a mine shaft."},{"Start":"00:12.050 ","End":"00:15.450","Text":"We have a chain and it\u0027s attached to a bucket of"},{"Start":"00:15.450 ","End":"00:18.945","Text":"coal and that bucket of coal is at the bottom of a mine shaft."},{"Start":"00:18.945 ","End":"00:22.710","Text":"Now let\u0027s have some data."},{"Start":"00:22.710 ","End":"00:28.665","Text":"Let\u0027s say the mine shaft is 500 feet deep."},{"Start":"00:28.665 ","End":"00:36.640","Text":"The bucket of coal weighs 800 pounds."},{"Start":"00:37.400 ","End":"00:42.224","Text":"The chain has a weight per unit length,"},{"Start":"00:42.224 ","End":"00:49.150","Text":"it\u0027s 2 pounds per foot length of chain."},{"Start":"00:49.150 ","End":"00:52.745","Text":"We need to determine the amount of work needed"},{"Start":"00:52.745 ","End":"00:56.735","Text":"to get the bucket all the way up the shaft."},{"Start":"00:56.735 ","End":"00:59.825","Text":"Then I just wrote that in telegraphic style."},{"Start":"00:59.825 ","End":"01:04.370","Text":"Just notice that when we\u0027re lifting it up,"},{"Start":"01:04.370 ","End":"01:07.940","Text":"the chain is getting successively shorter."},{"Start":"01:07.940 ","End":"01:13.055","Text":"The force is actually decreasing as we start raising it."},{"Start":"01:13.055 ","End":"01:15.650","Text":"That\u0027s why it\u0027s a variable force."},{"Start":"01:15.650 ","End":"01:22.400","Text":"Let\u0027s see if we can compute the function force as a function of x."},{"Start":"01:22.400 ","End":"01:25.070","Text":"In fact, before that, we have to say what x is."},{"Start":"01:25.070 ","End":"01:29.425","Text":"Let\u0027s think what would be advisable to set x to be?"},{"Start":"01:29.425 ","End":"01:38.180","Text":"What I suggest we let x be is how high up from the bottom the bucket is."},{"Start":"01:38.180 ","End":"01:42.820","Text":"Or if you like, how much chain we\u0027ve pulled up."},{"Start":"01:42.820 ","End":"01:46.895","Text":"On the 1 hand it\u0027s the height of the bucket from the bottom and on the other hand,"},{"Start":"01:46.895 ","End":"01:49.865","Text":"it will be the amount of chain pulled up."},{"Start":"01:49.865 ","End":"01:54.305","Text":"Clearly, x will go from 0-500."},{"Start":"01:54.305 ","End":"01:56.635","Text":"That\u0027s our range at the bottom,"},{"Start":"01:56.635 ","End":"02:02.400","Text":"it\u0027s 0, and then it goes up to the top, x becomes 500."},{"Start":"02:02.400 ","End":"02:06.705","Text":"Now at this point, f of x is made of 2 parts,"},{"Start":"02:06.705 ","End":"02:09.340","Text":"the bucket, and the chain."},{"Start":"02:09.340 ","End":"02:14.500","Text":"Now the weight of the bucket that will be constant."},{"Start":"02:14.500 ","End":"02:19.045","Text":"We also have the weight of the cable,"},{"Start":"02:19.045 ","End":"02:20.800","Text":"but not the whole cable,"},{"Start":"02:20.800 ","End":"02:25.810","Text":"the weight remaining because part of it is being pulled up already."},{"Start":"02:25.810 ","End":"02:28.285","Text":"Let\u0027s see if we can give an expression to each of these."},{"Start":"02:28.285 ","End":"02:33.220","Text":"The bucket is a constant and it\u0027s 800 pounds."},{"Start":"02:33.220 ","End":"02:35.210","Text":"Let\u0027s work in pounds."},{"Start":"02:35.210 ","End":"02:38.380","Text":"I\u0027m just saying that I\u0027m working in pounds."},{"Start":"02:38.380 ","End":"02:41.220","Text":"It\u0027s 800 plus."},{"Start":"02:41.220 ","End":"02:44.190","Text":"What is the weight of the cable?"},{"Start":"02:44.190 ","End":"02:49.725","Text":"It\u0027s the length of cable in feet times 2 pounds per feet."},{"Start":"02:49.725 ","End":"02:52.130","Text":"It\u0027s 2 times."},{"Start":"02:52.130 ","End":"02:53.690","Text":"What is the length?"},{"Start":"02:53.690 ","End":"02:59.550","Text":"It started off as 500."},{"Start":"03:00.490 ","End":"03:04.895","Text":"But it got shortened by x because"},{"Start":"03:04.895 ","End":"03:10.460","Text":"the bucket rose by x and the cable shrunk in length by x."},{"Start":"03:10.460 ","End":"03:15.785","Text":"This is the expression and this is equal to, let\u0027s see,"},{"Start":"03:15.785 ","End":"03:22.515","Text":"800 plus 2500 is 1800 minus 2x."},{"Start":"03:22.515 ","End":"03:25.305","Text":"This is our function,"},{"Start":"03:25.305 ","End":"03:32.030","Text":"and so what we want for the work is the integral looking at the formula here,"},{"Start":"03:32.030 ","End":"03:38.220","Text":"from 0-500 of f of x dx,"},{"Start":"03:38.220 ","End":"03:43.590","Text":"f of x is 1800 minus 2x dx."},{"Start":"03:43.590 ","End":"03:46.065","Text":"This is equal to,"},{"Start":"03:46.065 ","End":"03:54.335","Text":"we have 1800 x and then here the integral of 2x is x squared."},{"Start":"03:54.335 ","End":"03:59.965","Text":"We have to evaluate this from 0-500."},{"Start":"03:59.965 ","End":"04:04.910","Text":"At 0, we\u0027ll get nothing if we plug-in 500,"},{"Start":"04:04.910 ","End":"04:06.890","Text":"and this comes out to be,"},{"Start":"04:06.890 ","End":"04:08.450","Text":"I\u0027ll do the computation for you,"},{"Start":"04:08.450 ","End":"04:17.600","Text":"650,000 and the units, foot-pounds."},{"Start":"04:17.600 ","End":"04:21.300","Text":"That\u0027s the answer to this example,"},{"Start":"04:21.300 ","End":"04:24.640","Text":"and I\u0027ll do 1 more example."}],"Thumbnail":null,"ID":8295},{"Watched":false,"Name":"Example3","Duration":"10m 25s","ChapterTopicVideoID":8142,"CourseChapterTopicPlaylistID":6161,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.460","Text":"Our next example concerns a cylindrical tank filled with oil."},{"Start":"00:05.460 ","End":"00:15.225","Text":"Well, not full but only half-full of oil."},{"Start":"00:15.225 ","End":"00:17.790","Text":"Now, I need some data."},{"Start":"00:17.790 ","End":"00:22.605","Text":"For a cylinder I need a radius and a height."},{"Start":"00:22.605 ","End":"00:29.400","Text":"Let\u0027s say that the radius is equal to, I don\u0027t know,"},{"Start":"00:29.400 ","End":"00:34.330","Text":"4 feet and the height,"},{"Start":"00:35.540 ","End":"00:40.755","Text":"say that is equal to 9 feet."},{"Start":"00:40.755 ","End":"00:45.179","Text":"I\u0027m going to give you the density of the oil."},{"Start":"00:45.380 ","End":"00:49.215","Text":"The density Rho, doesn\u0027t matter,"},{"Start":"00:49.215 ","End":"00:59.180","Text":"is equal to 60 pounds per unit volume,"},{"Start":"00:59.180 ","End":"01:01.880","Text":"which I\u0027m going to use in cubic feet,"},{"Start":"01:01.880 ","End":"01:04.895","Text":"pounds per foot cubed."},{"Start":"01:04.895 ","End":"01:10.885","Text":"We\u0027re going to be pumping oil out but through the top of the tank."},{"Start":"01:10.885 ","End":"01:17.945","Text":"The question is, what is the work done in pumping out the oil via to the top of the tank?"},{"Start":"01:17.945 ","End":"01:22.880","Text":"Here\u0027s a diagram of the cylindrical tank."},{"Start":"01:22.880 ","End":"01:25.370","Text":"It\u0027s half-filled with oil."},{"Start":"01:25.370 ","End":"01:32.880","Text":"I guess that would make this 4.5 feet and 4.5 feet."},{"Start":"01:33.860 ","End":"01:42.175","Text":"The difficulty in this problem is to find a function f of x."},{"Start":"01:42.175 ","End":"01:44.940","Text":"It just doesn\u0027t work that way,"},{"Start":"01:44.940 ","End":"01:47.990","Text":"so we\u0027re going to have to use a different technique."},{"Start":"01:47.990 ","End":"01:50.915","Text":"I will take x as,"},{"Start":"01:50.915 ","End":"01:54.935","Text":"let\u0027s say, starting from 0 and going downward."},{"Start":"01:54.935 ","End":"02:01.195","Text":"Let\u0027s say this is 0 and I\u0027ll go this way."},{"Start":"02:01.195 ","End":"02:06.465","Text":"This would be 0 and here we\u0027d have what?"},{"Start":"02:06.465 ","End":"02:11.590","Text":"4 and a half, and here we\u0027d have 9."},{"Start":"02:11.590 ","End":"02:16.565","Text":"The idea is to use a Riemann sum."},{"Start":"02:16.565 ","End":"02:20.000","Text":"Here\u0027s what the Riemann integral looks like."},{"Start":"02:20.000 ","End":"02:26.525","Text":"I just went into the lesson on the definite integral."},{"Start":"02:26.525 ","End":"02:28.610","Text":"Just to jog your memory."},{"Start":"02:28.610 ","End":"02:35.320","Text":"Essentially the idea is to approximate the area by the sum of areas of"},{"Start":"02:35.320 ","End":"02:45.455","Text":"rectangles and we divide the interval into n pieces where n goes to infinity."},{"Start":"02:45.455 ","End":"02:48.889","Text":"Anyway, back to the present."},{"Start":"02:48.889 ","End":"02:52.385","Text":"Using that idea of the Riemann sum,"},{"Start":"02:52.385 ","End":"03:01.745","Text":"we divide this part with the oil to n equal layers,"},{"Start":"03:01.745 ","End":"03:05.375","Text":"where n will eventually go to infinity."},{"Start":"03:05.375 ","End":"03:08.645","Text":"Let\u0027s just call the ith one x_i."},{"Start":"03:08.645 ","End":"03:14.880","Text":"I will take the lower one of the two."},{"Start":"03:14.880 ","End":"03:16.965","Text":"This is like the ith,"},{"Start":"03:16.965 ","End":"03:18.915","Text":"what it would be a disk."},{"Start":"03:18.915 ","End":"03:22.665","Text":"On this last one would be x_n."},{"Start":"03:22.665 ","End":"03:26.005","Text":"We can actually compute the formula for that,"},{"Start":"03:26.005 ","End":"03:34.960","Text":"x_i would be 4.5 plus i times."},{"Start":"03:34.960 ","End":"03:41.190","Text":"We take this 4.5, that\u0027s the distance from here to here,"},{"Start":"03:41.260 ","End":"03:44.450","Text":"and divide it by n,"},{"Start":"03:44.450 ","End":"03:47.075","Text":"and that will give us x_i."},{"Start":"03:47.075 ","End":"03:55.820","Text":"What we do is we compute the work for just this disk shaped bit of oil."},{"Start":"03:55.820 ","End":"03:58.790","Text":"Since x doesn\u0027t vary much over the disk,"},{"Start":"03:58.790 ","End":"04:01.010","Text":"doesn\u0027t matter the top or the bottom."},{"Start":"04:01.010 ","End":"04:04.280","Text":"Everywhere the x is just x_i."},{"Start":"04:04.280 ","End":"04:06.905","Text":"We can compute the work."},{"Start":"04:06.905 ","End":"04:13.485","Text":"I first simplify that and say this is 4.5 times 1 plus i over"},{"Start":"04:13.485 ","End":"04:20.170","Text":"n. The disk is going to be using Pi r squared."},{"Start":"04:20.170 ","End":"04:23.810","Text":"The work is equal to,"},{"Start":"04:23.810 ","End":"04:29.870","Text":"we\u0027ll take the volume of this disk Pi times"},{"Start":"04:29.870 ","End":"04:36.320","Text":"the radius squared which is 4 squared times the height."},{"Start":"04:36.320 ","End":"04:46.660","Text":"Now, each of these strips has height 4.5 over n. I\u0027ll take this here,"},{"Start":"04:46.660 ","End":"04:55.055","Text":"times 4.5 over n. I\u0027ll just shade them both and you see is the same thing."},{"Start":"04:55.055 ","End":"05:00.250","Text":"Up to here it\u0027s the volume of this disk."},{"Start":"05:00.250 ","End":"05:07.060","Text":"Now, we need to get its weight by multiplying by the density which is 60."},{"Start":"05:07.060 ","End":"05:11.290","Text":"Volume times density is the weight."},{"Start":"05:11.290 ","End":"05:13.830","Text":"It\u0027s like force."},{"Start":"05:14.560 ","End":"05:19.670","Text":"We need force times distance and the distance would be x_i."},{"Start":"05:19.670 ","End":"05:25.000","Text":"Then let me call this W_i because it\u0027s specifically for this disk number"},{"Start":"05:25.000 ","End":"05:30.815","Text":"i. I need to multiply force times the distance to get the work."},{"Start":"05:30.815 ","End":"05:38.455","Text":"The distance here would be just x_i as written here, so x_i."},{"Start":"05:38.455 ","End":"05:47.010","Text":"Then I want to sum all these from i equals 1 to n. What I want is the total work which is"},{"Start":"05:47.010 ","End":"05:56.624","Text":"the sum of all these W_i\u0027s is i goes from 1 to n of the above,"},{"Start":"05:56.624 ","End":"06:00.900","Text":"Pi times 4 squared times"},{"Start":"06:00.900 ","End":"06:08.095","Text":"4 and a half over n times 60 times x_i."},{"Start":"06:08.095 ","End":"06:14.070","Text":"But not quite. I also want to take the limit as n goes to infinity."},{"Start":"06:17.480 ","End":"06:21.270","Text":"Now, this 4 and a half over n is the"},{"Start":"06:21.270 ","End":"06:26.299","Text":"thickness of this and we could call this thickness Delta"},{"Start":"06:26.299 ","End":"06:35.265","Text":"x. I could put this at the end and call it Delta x."},{"Start":"06:35.265 ","End":"06:43.340","Text":"Then I can write this Riemann sum limit as the integral."},{"Start":"06:43.340 ","End":"06:46.540","Text":"The x_i go from 4 and a half to 9,"},{"Start":"06:46.540 ","End":"06:51.765","Text":"so here 4 and a half and here 9."},{"Start":"06:51.765 ","End":"07:03.660","Text":"Then Pi times 4 squared times 60, and here x."},{"Start":"07:03.660 ","End":"07:07.020","Text":"Then dx instead of the Delta x."},{"Start":"07:07.020 ","End":"07:08.340","Text":"That\u0027s just how it works."},{"Start":"07:08.340 ","End":"07:09.600","Text":"The x_i becomes x,"},{"Start":"07:09.600 ","End":"07:12.304","Text":"the Delta x becomes dx."},{"Start":"07:12.304 ","End":"07:16.020","Text":"When we take the limit, we get the integral."},{"Start":"07:16.220 ","End":"07:19.465","Text":"Now, we can simplify it."},{"Start":"07:19.465 ","End":"07:23.360","Text":"Because constants come out in front,"},{"Start":"07:23.360 ","End":"07:27.035","Text":"and this bit here is, what is it?"},{"Start":"07:27.035 ","End":"07:33.270","Text":"Sixteen times 60 Pi,"},{"Start":"07:33.270 ","End":"07:40.960","Text":"which is 16 times 6 is 96, so that\u0027s 960Pi."},{"Start":"07:41.810 ","End":"07:45.480","Text":"I can pull that in front of the integral."},{"Start":"07:45.480 ","End":"07:50.430","Text":"We have 960Pi times the"},{"Start":"07:50.430 ","End":"07:57.375","Text":"integral from 4 and a half to 9 of x dx."},{"Start":"07:57.375 ","End":"08:01.380","Text":"This integral will also do it at the side. Let\u0027s see."},{"Start":"08:01.380 ","End":"08:04.445","Text":"This comes out to be x squared over"},{"Start":"08:04.445 ","End":"08:12.170","Text":"2 taken from 4 and a half to 9."},{"Start":"08:12.170 ","End":"08:14.255","Text":"The 1/2 also comes out."},{"Start":"08:14.255 ","End":"08:16.490","Text":"That\u0027s making it as a 1/2."},{"Start":"08:16.490 ","End":"08:18.845","Text":"Then all I need is x squared."},{"Start":"08:18.845 ","End":"08:23.790","Text":"When x is 9, that\u0027s 81 and when x is"},{"Start":"08:23.790 ","End":"08:31.455","Text":"4 and a half that comes out to be 20 and 1/4."},{"Start":"08:31.455 ","End":"08:33.935","Text":"Putting this back here,"},{"Start":"08:33.935 ","End":"08:41.010","Text":"what we get is the 1/2 with this makes it 480Pi."},{"Start":"08:42.740 ","End":"08:45.905","Text":"Then I have to multiply this"},{"Start":"08:45.905 ","End":"08:55.250","Text":"by 60 and 3/4."},{"Start":"08:55.250 ","End":"08:58.280","Text":"Again, I\u0027ll do a computation at the side."},{"Start":"08:58.280 ","End":"09:01.510","Text":"If I divide this by 4 and multiply this by 4,"},{"Start":"09:01.510 ","End":"09:05.530","Text":"then it\u0027s like a 120 times,"},{"Start":"09:05.530 ","End":"09:12.845","Text":"and this by 4 would give me 243."},{"Start":"09:12.845 ","End":"09:15.725","Text":"This would be Pi."},{"Start":"09:15.725 ","End":"09:17.380","Text":"Can\u0027t find my calculator,"},{"Start":"09:17.380 ","End":"09:20.230","Text":"never mind. 243 times 12."},{"Start":"09:20.230 ","End":"09:26.165","Text":"Let\u0027s do, 243 times 2 is 486."},{"Start":"09:26.165 ","End":"09:30.170","Text":"Here I have 243."},{"Start":"09:30.170 ","End":"09:32.210","Text":"That will give me 6,"},{"Start":"09:32.210 ","End":"09:33.440","Text":"8 and 3 is 11,"},{"Start":"09:33.440 ","End":"09:35.770","Text":"1 carry 1, 9."},{"Start":"09:35.770 ","End":"09:40.515","Text":"2,916, but I need the extra 0."},{"Start":"09:40.515 ","End":"09:46.250","Text":"What we get is 29,160."},{"Start":"09:46.250 ","End":"09:50.480","Text":"That\u0027s 29,160 Pi."},{"Start":"09:50.480 ","End":"09:58.940","Text":"As I said, I don\u0027t have my calculator with me but it\u0027s roughly 30,000 times 3."},{"Start":"09:58.940 ","End":"10:06.620","Text":"That\u0027s a very approximately something like 90,000."},{"Start":"10:06.890 ","End":"10:12.580","Text":"The units would be foot-pounds."},{"Start":"10:14.780 ","End":"10:21.170","Text":"Yeah, you do need a calculator or you could just leave it like this and also say that,"},{"Start":"10:21.170 ","End":"10:23.120","Text":"that is in foot-pounds."},{"Start":"10:23.120 ","End":"10:26.040","Text":"I guess we\u0027re done."}],"Thumbnail":null,"ID":8296}],"ID":6161},{"Name":"Center of Mass","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"4m 39s","ChapterTopicVideoID":8143,"CourseChapterTopicPlaylistID":6162,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this clip, we\u0027ll be talking about one of the applications of integrals,"},{"Start":"00:04.500 ","End":"00:06.840","Text":"something called center of mass."},{"Start":"00:06.840 ","End":"00:09.000","Text":"It also has another name."},{"Start":"00:09.000 ","End":"00:12.690","Text":"It\u0027s also called a centroid,"},{"Start":"00:12.690 ","End":"00:15.075","Text":"and actually also has another name."},{"Start":"00:15.075 ","End":"00:16.530","Text":"In terms of mass,"},{"Start":"00:16.530 ","End":"00:20.850","Text":"we could say center of gravity."},{"Start":"00:20.850 ","End":"00:23.625","Text":"Well, I\u0027ll just say center of mass."},{"Start":"00:23.625 ","End":"00:26.550","Text":"You might ask center of mass of what?"},{"Start":"00:26.550 ","End":"00:29.865","Text":"Well, we\u0027re going to have a thin plate."},{"Start":"00:29.865 ","End":"00:33.420","Text":"This plate is like a 2-dimensional object,"},{"Start":"00:33.420 ","End":"00:35.790","Text":"but it\u0027s 3-dimensional or very thin,"},{"Start":"00:35.790 ","End":"00:40.330","Text":"and it has a certain density."},{"Start":"00:40.330 ","End":"00:45.410","Text":"In fact, we\u0027re going to assume that this has a uniform density."},{"Start":"00:45.410 ","End":"00:52.475","Text":"The density here will be the mass per unit of area."},{"Start":"00:52.475 ","End":"00:58.430","Text":"The density, normally Rho is a function of x and y,"},{"Start":"00:58.430 ","End":"00:59.735","Text":"but here it\u0027s uniform,"},{"Start":"00:59.735 ","End":"01:02.405","Text":"so it\u0027s a constant Rho."},{"Start":"01:02.405 ","End":"01:07.950","Text":"We\u0027re going to assume that our plate is defined by 2 functions,"},{"Start":"01:07.950 ","End":"01:10.585","Text":"y of x in 1 case,"},{"Start":"01:10.585 ","End":"01:15.095","Text":"f of x for the upper and g of x for the lower."},{"Start":"01:15.095 ","End":"01:22.410","Text":"Let\u0027s also assume that the area of this plate is A."},{"Start":"01:22.410 ","End":"01:26.665","Text":"Well, we know what A is in terms of integrals."},{"Start":"01:26.665 ","End":"01:33.530","Text":"The area is the integral from a to b of the difference of these 2."},{"Start":"01:33.530 ","End":"01:37.080","Text":"I think it\u0027s probably best to put brackets around."},{"Start":"01:37.550 ","End":"01:42.045","Text":"We also need the mass of this plate."},{"Start":"01:42.045 ","End":"01:45.710","Text":"The mass is given simply by the density"},{"Start":"01:45.710 ","End":"01:49.150","Text":"Rho times the area."},{"Start":"01:49.150 ","End":"01:52.444","Text":"The center of mass at some point,"},{"Start":"01:52.444 ","End":"01:55.640","Text":"let\u0027s say in this case that it\u0027s here,"},{"Start":"01:55.640 ","End":"01:57.695","Text":"and I don\u0027t know where it is exactly,"},{"Start":"01:57.695 ","End":"02:01.294","Text":"but it\u0027s the point where if you suspend"},{"Start":"02:01.294 ","End":"02:04.760","Text":"the plate from this point, it will be perfectly balanced."},{"Start":"02:04.760 ","End":"02:09.320","Text":"Let\u0027s say that this point has coordinates x,"},{"Start":"02:09.320 ","End":"02:13.595","Text":"y, but let\u0027s call it x bar and y bar."},{"Start":"02:13.595 ","End":"02:18.320","Text":"I\u0027m going to give a formula for x bar and for y bar."},{"Start":"02:18.320 ","End":"02:22.460","Text":"But before that, we need to compute some other quantities."},{"Start":"02:22.460 ","End":"02:25.085","Text":"These are called moments."},{"Start":"02:25.085 ","End":"02:29.929","Text":"There\u0027s a moment about the x-axis and about the y-axis,"},{"Start":"02:29.929 ","End":"02:31.970","Text":"which is hard to explain,"},{"Start":"02:31.970 ","End":"02:34.865","Text":"but it\u0027s a tendency to rotate."},{"Start":"02:34.865 ","End":"02:37.120","Text":"In any event, we have 2 of them."},{"Start":"02:37.120 ","End":"02:43.945","Text":"These are the formulas for the moment about the x-axis and the moment about the y-axis."},{"Start":"02:43.945 ","End":"02:49.055","Text":"We use these to compute x bar and y bar."},{"Start":"02:49.055 ","End":"02:54.685","Text":"We\u0027re now ready for the formulas for x bar and y bar."},{"Start":"02:54.685 ","End":"03:01.220","Text":"Here they are. But we can simplify these."},{"Start":"03:01.220 ","End":"03:04.220","Text":"In fact, we are going to get rid of most of these formulas."},{"Start":"03:04.220 ","End":"03:09.200","Text":"Because if you look at the denominator here,"},{"Start":"03:09.200 ","End":"03:12.650","Text":"and I think it should have brackets,"},{"Start":"03:12.650 ","End":"03:21.754","Text":"this integral and this integral are both equal to the area."},{"Start":"03:21.754 ","End":"03:28.735","Text":"If I replace the 2 denominators by the letter A for area,"},{"Start":"03:28.735 ","End":"03:33.795","Text":"then we\u0027ll get these 2 formulas."},{"Start":"03:33.795 ","End":"03:35.540","Text":"This one, I\u0027ll go back a moment."},{"Start":"03:35.540 ","End":"03:38.420","Text":"You might say, where has Rho disappeared to."},{"Start":"03:38.420 ","End":"03:42.045","Text":"Well, the Rho from here,"},{"Start":"03:42.045 ","End":"03:45.300","Text":"from My was originally here,"},{"Start":"03:45.300 ","End":"03:48.675","Text":"but M was also Rho, a was here."},{"Start":"03:48.675 ","End":"03:52.545","Text":"Similarly, here, we had Rho and here we had Rho."},{"Start":"03:52.545 ","End":"03:55.640","Text":"In each case, they just canceled out this with this here,"},{"Start":"03:55.640 ","End":"03:57.170","Text":"this 2 canceled out."},{"Start":"03:57.170 ","End":"04:03.205","Text":"That\u0027s why we\u0027re left with this over this is y bar."},{"Start":"04:03.205 ","End":"04:05.715","Text":"The denominator, as I said is a,"},{"Start":"04:05.715 ","End":"04:07.245","Text":"and so it\u0027s much simpler."},{"Start":"04:07.245 ","End":"04:12.540","Text":"What I\u0027m going to do is erase all the unnecessary formulas."},{"Start":"04:12.760 ","End":"04:15.260","Text":"We\u0027ll just stick with"},{"Start":"04:15.260 ","End":"04:22.230","Text":"these 3 formulas because we also want the formula for the area to use here and here."},{"Start":"04:23.680 ","End":"04:27.530","Text":"I could have started with these formulas and just thrown them at you,"},{"Start":"04:27.530 ","End":"04:29.630","Text":"but I wanted to show you where they came from."},{"Start":"04:29.630 ","End":"04:32.960","Text":"Anyway, the point is that we don\u0027t actually need Rho anymore."},{"Start":"04:32.960 ","End":"04:35.530","Text":"I wrote it there but it\u0027s canceled out."},{"Start":"04:35.530 ","End":"04:40.110","Text":"Now, let\u0027s do a couple of exercises."}],"Thumbnail":null,"ID":8299},{"Watched":false,"Name":"Example 1","Duration":"6m 47s","ChapterTopicVideoID":8177,"CourseChapterTopicPlaylistID":6162,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:03.615","Text":"Here\u0027s the first example."},{"Start":"00:03.615 ","End":"00:10.290","Text":"This is the plate and we\u0027re given that this curve is f of x is 4 minus x squared,"},{"Start":"00:10.290 ","End":"00:15.240","Text":"and we want the plate to be bounded by the axis."},{"Start":"00:15.240 ","End":"00:19.370","Text":"Quite clearly, well, maybe not immediately clearly,"},{"Start":"00:19.370 ","End":"00:20.450","Text":"this is the point 2."},{"Start":"00:20.450 ","End":"00:24.230","Text":"I\u0027ve drawn it in, but we could have computed it because if 4 minus x"},{"Start":"00:24.230 ","End":"00:28.625","Text":"squared is 0 and x squared is 4 so is plus or minus 2."},{"Start":"00:28.625 ","End":"00:33.090","Text":"But because of the x bigger or equal to 0, x equals 2."},{"Start":"00:33.260 ","End":"00:36.405","Text":"This is from 0-2."},{"Start":"00:36.405 ","End":"00:41.970","Text":"The lower curve, the g of x is just 0."},{"Start":"00:41.970 ","End":"00:45.600","Text":"It\u0027s the x-axis where y is 0."},{"Start":"00:45.600 ","End":"00:50.175","Text":"First thing we want to do is compute the area."},{"Start":"00:50.175 ","End":"00:56.760","Text":"We say that a is equal to the integral from 0-2."},{"Start":"00:56.770 ","End":"01:02.330","Text":"F of x minus g of x is just f of x because g of x here is 0."},{"Start":"01:02.330 ","End":"01:08.800","Text":"It\u0027s just 4 minus x squared dx."},{"Start":"01:08.800 ","End":"01:10.710","Text":"This is straightforward."},{"Start":"01:10.710 ","End":"01:17.855","Text":"We get 4x minus x cubed over 3 for the integral from 0-2."},{"Start":"01:17.855 ","End":"01:19.580","Text":"When I put in 0 obviously it\u0027s 0,"},{"Start":"01:19.580 ","End":"01:21.185","Text":"so I just need to plug in 2."},{"Start":"01:21.185 ","End":"01:23.185","Text":"4 times 2 is 8,"},{"Start":"01:23.185 ","End":"01:27.495","Text":"minus 2 cubed is 8 over 3."},{"Start":"01:27.495 ","End":"01:31.650","Text":"This comes out 5-and-1/3 or 16 over 3,"},{"Start":"01:31.650 ","End":"01:34.365","Text":"I\u0027ll write it as. That\u0027s a."},{"Start":"01:34.365 ","End":"01:42.000","Text":"Next we need the x coordinate of the center of gravity, x-bar."},{"Start":"01:42.000 ","End":"01:45.215","Text":"That is equal to by the formula."},{"Start":"01:45.215 ","End":"01:49.330","Text":"1 over a is going to be 3 over 16,"},{"Start":"01:49.330 ","End":"01:52.060","Text":"because the inverse of a fraction,"},{"Start":"01:52.060 ","End":"01:53.990","Text":"just turn it upside down,"},{"Start":"01:53.990 ","End":"01:58.470","Text":"times the integral also from 0-2."},{"Start":"01:58.470 ","End":"02:03.520","Text":"But this time, I have the same thing that was before except multiplied by x."},{"Start":"02:03.520 ","End":"02:10.020","Text":"Instead of this, I have x times 4 minus x squared dx."},{"Start":"02:10.020 ","End":"02:12.224","Text":"Also not difficult."},{"Start":"02:12.224 ","End":"02:14.800","Text":"Let\u0027s do it 3 over 16."},{"Start":"02:14.800 ","End":"02:16.465","Text":"First I\u0027ll expand."},{"Start":"02:16.465 ","End":"02:21.605","Text":"It\u0027s 4x minus x cubed dx."},{"Start":"02:21.605 ","End":"02:25.145","Text":"Then the integral 4x gives me 2x squared."},{"Start":"02:25.145 ","End":"02:29.800","Text":"X cubed gives me 1/4 x^4."},{"Start":"02:29.800 ","End":"02:32.190","Text":"I want this from 0-2."},{"Start":"02:32.190 ","End":"02:33.390","Text":"0 gives me nothing,"},{"Start":"02:33.390 ","End":"02:34.875","Text":"so I just substitute 2."},{"Start":"02:34.875 ","End":"02:38.055","Text":"2 times 2 squared is 2 times 4 is 8."},{"Start":"02:38.055 ","End":"02:42.315","Text":"2^4 is 16 over 4 is 4."},{"Start":"02:42.315 ","End":"02:45.975","Text":"This is equal to 4."},{"Start":"02:45.975 ","End":"02:51.640","Text":"Oops, I forgot the 3/16."},{"Start":"02:51.860 ","End":"02:58.200","Text":"3/16 times 3/16. The answer,"},{"Start":"02:58.200 ","End":"03:04.095","Text":"I guess, is 4 into 16 goes 4 times 3/4."},{"Start":"03:04.095 ","End":"03:08.915","Text":"I think I can squeeze this into here, 3/4."},{"Start":"03:08.915 ","End":"03:11.285","Text":"It wouldn\u0027t hurt to highlight that."},{"Start":"03:11.285 ","End":"03:13.460","Text":"That\u0027s x-bar."},{"Start":"03:13.460 ","End":"03:16.720","Text":"Now, we also need y-bar."},{"Start":"03:16.720 ","End":"03:19.355","Text":"That\u0027s a different formula."},{"Start":"03:19.355 ","End":"03:21.290","Text":"But it starts off with 1 over a,"},{"Start":"03:21.290 ","End":"03:24.095","Text":"so it still starts off with 3/16,"},{"Start":"03:24.095 ","End":"03:27.635","Text":"and the integral from 0-2."},{"Start":"03:27.635 ","End":"03:35.745","Text":"But something else, we have 1.5 times,"},{"Start":"03:35.745 ","End":"03:38.340","Text":"now g of x is still 0,"},{"Start":"03:38.340 ","End":"03:40.635","Text":"so I just need f of x squared."},{"Start":"03:40.635 ","End":"03:43.500","Text":"F of x is 4 minus x squared."},{"Start":"03:43.500 ","End":"03:49.980","Text":"It\u0027s 4 minus x squared, squared dx."},{"Start":"03:49.980 ","End":"03:51.720","Text":"Let\u0027s compute this."},{"Start":"03:51.720 ","End":"03:53.390","Text":"The 1/2 I can bring it outside,"},{"Start":"03:53.390 ","End":"03:57.485","Text":"so I get 3 over 32 integral."},{"Start":"03:57.485 ","End":"04:02.330","Text":"Now, 4 minus x squared squared using the special binomial product,"},{"Start":"04:02.330 ","End":"04:06.920","Text":"4 squared is 16 minus twice this times this is"},{"Start":"04:06.920 ","End":"04:13.065","Text":"8x squared plus this squared is x^4 dx,"},{"Start":"04:13.065 ","End":"04:16.095","Text":"and then 3 over 32."},{"Start":"04:16.095 ","End":"04:19.785","Text":"The integral is 16x."},{"Start":"04:19.785 ","End":"04:25.380","Text":"For this, I get 8 over 3x cubed."},{"Start":"04:25.380 ","End":"04:30.420","Text":"Here 1/5, x^5."},{"Start":"04:30.420 ","End":"04:34.990","Text":"I have to substitute 0 and 2."},{"Start":"04:35.030 ","End":"04:38.595","Text":"0 just gives me 0."},{"Start":"04:38.595 ","End":"04:44.070","Text":"I just have to plug in the 2. What do I get?"},{"Start":"04:44.070 ","End":"04:53.500","Text":"Continuing here it\u0027s equal to 3 over 32."},{"Start":"04:55.070 ","End":"05:02.220","Text":"16 times 2 is 32 minus 8 over 3,"},{"Start":"05:02.220 ","End":"05:07.600","Text":"2 cubed is 8, plus 1/5."},{"Start":"05:07.600 ","End":"05:13.830","Text":"2^5 is 32. Tell you what I\u0027m going to do,"},{"Start":"05:13.830 ","End":"05:16.935","Text":"I see a 32 here and I see a 32 here."},{"Start":"05:16.935 ","End":"05:20.090","Text":"Here I have 8 times 8, which is 64."},{"Start":"05:20.090 ","End":"05:21.885","Text":"This with this 64."},{"Start":"05:21.885 ","End":"05:24.195","Text":"If I pull the 32 out,"},{"Start":"05:24.195 ","End":"05:28.125","Text":"I\u0027ll get 3 over 32 times 32."},{"Start":"05:28.125 ","End":"05:32.205","Text":"What am I left with? Here 1, here minus."},{"Start":"05:32.205 ","End":"05:33.570","Text":"I pull 32 out,"},{"Start":"05:33.570 ","End":"05:36.165","Text":"it\u0027s 2 over 3."},{"Start":"05:36.165 ","End":"05:38.025","Text":"Here plus 1/5."},{"Start":"05:38.025 ","End":"05:46.280","Text":"Well, 32 over 32 cancel, so it\u0027s 3."},{"Start":"05:46.280 ","End":"05:51.835","Text":"The fractional computation if I put it all over 15, let\u0027s see."},{"Start":"05:51.835 ","End":"06:00.165","Text":"If I put it over 15, I\u0027ve got 15 minus 10 over 15 plus 3 over 15."},{"Start":"06:00.165 ","End":"06:04.125","Text":"15 minus 10 is 5, plus 3 is 8."},{"Start":"06:04.125 ","End":"06:07.590","Text":"8 times 3 is 24,"},{"Start":"06:07.590 ","End":"06:11.854","Text":"over 15, I can divide by 3,"},{"Start":"06:11.854 ","End":"06:16.135","Text":"so I make it 8 over 5."},{"Start":"06:16.135 ","End":"06:22.455","Text":"That gives me y-bar is 8 over 5."},{"Start":"06:22.455 ","End":"06:26.750","Text":"I guess you could say the final answer that the center of"},{"Start":"06:26.750 ","End":"06:33.800","Text":"gravity is 3/4, 8/5."},{"Start":"06:33.800 ","End":"06:37.340","Text":"That would be this point here."},{"Start":"06:37.340 ","End":"06:39.950","Text":"Looks about right, 3/4 here,"},{"Start":"06:39.950 ","End":"06:43.765","Text":"could be 8/5 is 1.6, yeah."},{"Start":"06:43.765 ","End":"06:47.620","Text":"That\u0027s the first example. We\u0027ll do 1 more."}],"Thumbnail":null,"ID":8331},{"Watched":false,"Name":"Example 2","Duration":"5m 37s","ChapterTopicVideoID":8144,"CourseChapterTopicPlaylistID":6162,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.615","Text":"Now let\u0027s do another example of center of mass."},{"Start":"00:06.615 ","End":"00:10.470","Text":"This time we describe the plate just by formulas."},{"Start":"00:10.470 ","End":"00:18.420","Text":"It\u0027s the area between y equals the square root of x and y equals"},{"Start":"00:18.420 ","End":"00:21.900","Text":"x. I think we need"},{"Start":"00:21.900 ","End":"00:28.970","Text":"a sketch and here\u0027s a picture and it\u0027s pretty easy to find out where they intersect."},{"Start":"00:28.970 ","End":"00:31.190","Text":"If I let this equals this."},{"Start":"00:31.190 ","End":"00:32.960","Text":"Sorry. Did I say x?"},{"Start":"00:32.960 ","End":"00:34.980","Text":"I meant x squared."},{"Start":"00:35.210 ","End":"00:38.880","Text":"This is the square root of x,"},{"Start":"00:38.880 ","End":"00:41.505","Text":"and this is x squared."},{"Start":"00:41.505 ","End":"00:44.330","Text":"They intersect when this equals this,"},{"Start":"00:44.330 ","End":"00:46.880","Text":"and we\u0027ll get x equals 0 or 1."},{"Start":"00:46.880 ","End":"00:49.040","Text":"In fact, this will be 0,"},{"Start":"00:49.040 ","End":"00:50.555","Text":"0 and 1, 1."},{"Start":"00:50.555 ","End":"00:53.105","Text":"But what we need is 0 to 1."},{"Start":"00:53.105 ","End":"00:55.445","Text":"Let\u0027s say this is the center of mass,"},{"Start":"00:55.445 ","End":"01:01.950","Text":"we want to find out what its coordinates are, x-bar and y-bar."},{"Start":"01:01.950 ","End":"01:04.700","Text":"This function, well, we\u0027ve written them,"},{"Start":"01:04.700 ","End":"01:05.930","Text":"but this is the upper 1."},{"Start":"01:05.930 ","End":"01:12.320","Text":"This is maybe the f of x in the formula which is the upper 1."},{"Start":"01:12.320 ","End":"01:14.570","Text":"This will be the g of x,"},{"Start":"01:14.570 ","End":"01:19.160","Text":"which is the lower 1, and substitute them 1 at a time."},{"Start":"01:19.160 ","End":"01:22.135","Text":"First of all, we need the area A."},{"Start":"01:22.135 ","End":"01:30.560","Text":"We get that A is equal to the integral from 0 to 1 of the upper minus the lower."},{"Start":"01:30.560 ","End":"01:34.500","Text":"I\u0027ll do it in fractional powers,"},{"Start":"01:34.500 ","End":"01:41.310","Text":"x^1/2 is square root of x minus x squared dx."},{"Start":"01:41.310 ","End":"01:43.740","Text":"From here I get, let\u0027s see,"},{"Start":"01:43.740 ","End":"01:45.540","Text":"I add 1 and its 3/2."},{"Start":"01:45.540 ","End":"01:53.925","Text":"I get 2/3, x^3/2 minus here 1/3x cubed."},{"Start":"01:53.925 ","End":"01:58.470","Text":"I need to take this from 0 to 1."},{"Start":"01:58.470 ","End":"02:00.855","Text":"0 I get 0,"},{"Start":"02:00.855 ","End":"02:06.870","Text":"1 I get just 2/3 minus 1/3 this is equal to 1/3."},{"Start":"02:06.870 ","End":"02:11.200","Text":"Now let\u0027s do x-bar."},{"Start":"02:11.200 ","End":"02:18.905","Text":"X bar is equal to 1 over A is 1 over 1/3 is 3."},{"Start":"02:18.905 ","End":"02:29.775","Text":"I need 3 times the integral from 0 to 1 and it\u0027s going to be x times this."},{"Start":"02:29.775 ","End":"02:36.810","Text":"Let\u0027s take x and then copy this x^1/2 minus x squared dx."},{"Start":"02:36.810 ","End":"02:39.225","Text":"Best thing to do is multiply out."},{"Start":"02:39.225 ","End":"02:42.930","Text":"I\u0027ve got x^1.5, which is 3/2,"},{"Start":"02:42.930 ","End":"02:47.885","Text":"and x times x squared is x cubed, also dx."},{"Start":"02:47.885 ","End":"02:53.450","Text":"This integral comes out to be,"},{"Start":"02:53.450 ","End":"02:56.300","Text":"add 1, it\u0027s 5/2,"},{"Start":"02:56.300 ","End":"03:00.275","Text":"so it\u0027s 2/5, x^5/2."},{"Start":"03:00.275 ","End":"03:03.515","Text":"Here raise to 1 is 4, so 1/4,"},{"Start":"03:03.515 ","End":"03:08.190","Text":"x^1/4, I still need from 0 to 1."},{"Start":"03:10.060 ","End":"03:13.190","Text":"Then I get what?"},{"Start":"03:13.190 ","End":"03:15.559","Text":"0 doesn\u0027t give me anything,"},{"Start":"03:15.559 ","End":"03:16.775","Text":"just need the 1."},{"Start":"03:16.775 ","End":"03:22.980","Text":"It\u0027s 3 times 2/5 minus 1/4."},{"Start":"03:24.050 ","End":"03:27.500","Text":"See if I put it all over 20,"},{"Start":"03:27.500 ","End":"03:32.660","Text":"it\u0027s going to be 3 times something over 20."},{"Start":"03:32.660 ","End":"03:36.380","Text":"Now, 2/5 is 8 over 20,"},{"Start":"03:36.380 ","End":"03:40.200","Text":"1/4 is 5 over 20,"},{"Start":"03:40.520 ","End":"03:47.160","Text":"3 times 3 is 9,"},{"Start":"03:47.160 ","End":"03:51.930","Text":"so it\u0027s just 9 over 20."},{"Start":"03:51.930 ","End":"03:57.980","Text":"We found x-bar, the x of the center of mass."},{"Start":"03:57.980 ","End":"04:02.300","Text":"Now we need the y, different formula,"},{"Start":"04:02.300 ","End":"04:05.615","Text":"but starts out the same 1 over A so that\u0027s the 3,"},{"Start":"04:05.615 ","End":"04:08.910","Text":"the integral from 0 to 1."},{"Start":"04:08.910 ","End":"04:10.380","Text":"But different here."},{"Start":"04:10.380 ","End":"04:15.600","Text":"It\u0027s 1.5 f of x squared."},{"Start":"04:15.600 ","End":"04:18.570","Text":"F of x is square root of x."},{"Start":"04:18.570 ","End":"04:20.420","Text":"Square root of x squared,"},{"Start":"04:20.420 ","End":"04:22.970","Text":"I\u0027ll write it straight away as x."},{"Start":"04:22.970 ","End":"04:26.630","Text":"If f of x is square root of x, when you square it\u0027s just x,"},{"Start":"04:26.630 ","End":"04:35.320","Text":"g of x is x squared and x squared squared is x^4 dx."},{"Start":"04:35.320 ","End":"04:36.870","Text":"This is equal to,"},{"Start":"04:36.870 ","End":"04:40.770","Text":"I guess I can pull the 1/2 out, 3/2."},{"Start":"04:40.770 ","End":"04:44.910","Text":"Then the integral of x is x squared over 2,"},{"Start":"04:44.910 ","End":"04:51.530","Text":"here x^5 over 5 between 0 and 1."},{"Start":"04:51.530 ","End":"04:57.130","Text":"What do we get? 1/2 minus a 1/5."},{"Start":"04:57.130 ","End":"04:59.415","Text":"If I do it in tenths,"},{"Start":"04:59.415 ","End":"05:05.115","Text":"this is 5 minus 2 over 10 is 3/10."},{"Start":"05:05.115 ","End":"05:08.205","Text":"3 over 2 times 3 over 10,"},{"Start":"05:08.205 ","End":"05:13.740","Text":"I make it 9 over 20. Same answer."},{"Start":"05:13.740 ","End":"05:17.730","Text":"Notice the center of gravity, the x-bar,"},{"Start":"05:17.730 ","End":"05:23.475","Text":"y-bar will be 9 over 20, 9 over 20."},{"Start":"05:23.475 ","End":"05:26.490","Text":"That would be this point here."},{"Start":"05:26.490 ","End":"05:29.725","Text":"Makes sense. Let\u0027s see where is the a 1/2?"},{"Start":"05:29.725 ","End":"05:31.235","Text":"It would be about here."},{"Start":"05:31.235 ","End":"05:34.220","Text":"Just a bit less than a 1/2."},{"Start":"05:34.220 ","End":"05:38.490","Text":"Could be. Anyway, we are done."}],"Thumbnail":null,"ID":8300}],"ID":6162},{"Name":"Hydrostatic Pressure and Force","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"3m 31s","ChapterTopicVideoID":8145,"CourseChapterTopicPlaylistID":6163,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.920","Text":"In this clip we\u0027ll learn another application of integrals of the definite integral,"},{"Start":"00:04.920 ","End":"00:08.295","Text":"to something called hydrostatic pressure,"},{"Start":"00:08.295 ","End":"00:13.150","Text":"which is in physics, in hydrostatics."},{"Start":"00:14.000 ","End":"00:19.120","Text":"A typical situation will be where we have a flat object,"},{"Start":"00:19.120 ","End":"00:20.925","Text":"we usually call that a plate."},{"Start":"00:20.925 ","End":"00:22.500","Text":"It might be circular,"},{"Start":"00:22.500 ","End":"00:23.520","Text":"it might be square,"},{"Start":"00:23.520 ","End":"00:28.380","Text":"it might just be any shape."},{"Start":"00:28.380 ","End":"00:30.225","Text":"This is the plate,"},{"Start":"00:30.225 ","End":"00:31.560","Text":"and maybe I\u0027ll shade it."},{"Start":"00:31.560 ","End":"00:33.900","Text":"Actually, it doesn\u0027t have to be a plate,"},{"Start":"00:33.900 ","End":"00:39.330","Text":"it could be the sides of a tank or an aquarium which is filled with liquid,"},{"Start":"00:39.330 ","End":"00:43.800","Text":"but it should be planar, a flat surface."},{"Start":"00:43.800 ","End":"00:47.690","Text":"There\u0027s pressure on it from the liquid it\u0027s immersed in."},{"Start":"00:47.690 ","End":"00:50.040","Text":"Hydro doesn\u0027t have to be water,"},{"Start":"00:50.040 ","End":"00:52.570","Text":"it could be any liquid."},{"Start":"00:54.680 ","End":"00:57.075","Text":"Suppose that this, well,"},{"Start":"00:57.075 ","End":"00:59.760","Text":"maybe it\u0027s not a plate, it\u0027s,"},{"Start":"00:59.760 ","End":"01:02.730","Text":"say, the inside of a swimming pool,"},{"Start":"01:02.730 ","End":"01:05.595","Text":"or aquarium, or whatever,"},{"Start":"01:05.595 ","End":"01:08.380","Text":"and it has an area equals to A,"},{"Start":"01:08.380 ","End":"01:11.180","Text":"and then it\u0027s immersed in liquid,"},{"Start":"01:11.180 ","End":"01:12.650","Text":"so there\u0027s pressure on it."},{"Start":"01:12.650 ","End":"01:18.035","Text":"Let\u0027s say this is the bottom side of the pool or aquarium."},{"Start":"01:18.035 ","End":"01:22.530","Text":"Then there\u0027s going to be a constant pressure at every point."},{"Start":"01:24.170 ","End":"01:27.850","Text":"Now if that pressure is constant,"},{"Start":"01:27.850 ","End":"01:33.385","Text":"then there\u0027s a force acting on this surface."},{"Start":"01:33.385 ","End":"01:39.909","Text":"The rule is that the force is equal to the pressure times the area."},{"Start":"01:39.909 ","End":"01:44.410","Text":"But as I said, this only works if the pressure is constant."},{"Start":"01:44.410 ","End":"01:48.055","Text":"It would be, if this was at the bottom of the pool,"},{"Start":"01:48.055 ","End":"01:51.640","Text":"but if it\u0027s the side of the pool, or the aquarium,"},{"Start":"01:51.640 ","End":"01:56.260","Text":"or the tank, then the pressure is going to be greater, the deeper you go."},{"Start":"01:56.260 ","End":"01:58.445","Text":"In fact, there\u0027s a formula for that."},{"Start":"01:58.445 ","End":"02:01.660","Text":"The pressure does indeed depend on the depth."},{"Start":"02:01.660 ","End":"02:12.380","Text":"In fact, the pressure at a certain point on this surface which is submerged, is Rho, g,"},{"Start":"02:12.380 ","End":"02:17.480","Text":"d. Rho is the density"},{"Start":"02:17.480 ","End":"02:22.530","Text":"of the liquid in grams per centimeter cubed,"},{"Start":"02:22.530 ","End":"02:25.100","Text":"or maybe it\u0027s kilograms per meter cubed,"},{"Start":"02:25.100 ","End":"02:27.650","Text":"or whatever appropriate units there are,"},{"Start":"02:27.650 ","End":"02:29.270","Text":"pound per cubic foot,"},{"Start":"02:29.270 ","End":"02:31.480","Text":"that would be the density,"},{"Start":"02:31.480 ","End":"02:37.240","Text":"g is the acceleration due to gravity on the surface of the earth,"},{"Start":"02:37.240 ","End":"02:39.770","Text":"and d is the depth."},{"Start":"02:39.770 ","End":"02:42.810","Text":"The main thing is that,"},{"Start":"02:42.810 ","End":"02:49.380","Text":"the pressure depends on the depth, in fact,"},{"Start":"02:49.380 ","End":"02:50.940","Text":"it\u0027s proportional to it,"},{"Start":"02:50.940 ","End":"02:53.625","Text":"because Rho and g are constants,"},{"Start":"02:53.625 ","End":"02:57.595","Text":"for any given liquid and in any given place on earth."},{"Start":"02:57.595 ","End":"03:00.075","Text":"Maybe I\u0027ll draw the liquid."},{"Start":"03:00.075 ","End":"03:02.335","Text":"This thing is in some liquid,"},{"Start":"03:02.335 ","End":"03:07.770","Text":"like under the water in some pool,"},{"Start":"03:07.770 ","End":"03:10.030","Text":"or aquarium, or tank."},{"Start":"03:10.070 ","End":"03:14.580","Text":"Let me do an example problem,"},{"Start":"03:14.580 ","End":"03:17.100","Text":"rather than get too much into theory."},{"Start":"03:17.100 ","End":"03:19.355","Text":"Each example is a bit different."},{"Start":"03:19.355 ","End":"03:21.890","Text":"But basically these are the 2 formulas we\u0027re going to use;"},{"Start":"03:21.890 ","End":"03:26.010","Text":"the pressure in terms of the depth at a given point in a liquid,"},{"Start":"03:26.010 ","End":"03:31.930","Text":"and the force is pressure times area if the pressure is constant."}],"Thumbnail":null,"ID":8297},{"Watched":false,"Name":"Example","Duration":"14m 28s","ChapterTopicVideoID":8146,"CourseChapterTopicPlaylistID":6163,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"Here\u0027s an example of a tank filled with water."},{"Start":"00:04.890 ","End":"00:11.940","Text":"In our example, we\u0027ll take a tank of water and the tank has a funny shape,"},{"Start":"00:11.940 ","End":"00:18.810","Text":"not exactly funny, but it\u0027s like a trough that you would feed the horses or cows."},{"Start":"00:18.810 ","End":"00:27.510","Text":"Semicircular of the sides is a half disk and it has a certain length."},{"Start":"00:27.510 ","End":"00:35.805","Text":"The only thing that we need to know is that this is a semicircle with radius,"},{"Start":"00:35.805 ","End":"00:39.310","Text":"that\u0027s from here to here is the radius,"},{"Start":"00:40.460 ","End":"00:43.520","Text":"and this radius is 10."},{"Start":"00:43.520 ","End":"00:52.580","Text":"Let\u0027s work in meters this time instead of feet for a change and it\u0027s filled with water."},{"Start":"00:52.580 ","End":"00:59.180","Text":"What we have to compute is the hydrostatic force on 1 of these 2 ends, the side,"},{"Start":"00:59.180 ","End":"01:01.550","Text":"let\u0027s say this 1 from the inside,"},{"Start":"01:01.550 ","End":"01:07.175","Text":"it\u0027s got pressure and let\u0027s see how we approach this."},{"Start":"01:07.175 ","End":"01:11.390","Text":"Well, I\u0027ll draw a cross section just this bit."},{"Start":"01:11.620 ","End":"01:14.510","Text":"Here, I drew a side view,"},{"Start":"01:14.510 ","End":"01:17.270","Text":"this surface is this surface,"},{"Start":"01:17.270 ","End":"01:21.870","Text":"may be I\u0027ll shade it and look more like it."},{"Start":"01:22.550 ","End":"01:27.210","Text":"We know that from here to here,"},{"Start":"01:27.210 ","End":"01:31.600","Text":"it is 10 meters."},{"Start":"01:31.730 ","End":"01:34.980","Text":"Let\u0027s make the depth x,"},{"Start":"01:34.980 ","End":"01:42.295","Text":"we\u0027ll take the surface of the water of the tank to be 0 depth and work our way downwards."},{"Start":"01:42.295 ","End":"01:50.805","Text":"This is our x starting at 0 because it\u0027s a semicircle it also goes up to 10."},{"Start":"01:50.805 ","End":"01:54.795","Text":"X is like the d in the formula, it\u0027s the depth."},{"Start":"01:54.795 ","End":"02:01.250","Text":"Depth goes from 0-10 and what we\u0027ll do to figure out the force on it,"},{"Start":"02:01.250 ","End":"02:03.890","Text":"is to break it up into horizontal strips,"},{"Start":"02:03.890 ","End":"02:08.589","Text":"very narrow strips, which will have virtually the same pressure."},{"Start":"02:08.589 ","End":"02:12.500","Text":"What we\u0027re going to do is divide this surface,"},{"Start":"02:12.500 ","End":"02:19.640","Text":"the side of the tank into n horizontal strips."},{"Start":"02:19.640 ","End":"02:25.490","Text":"This might be strip i out of n so if divided it up into"},{"Start":"02:25.490 ","End":"02:33.015","Text":"n horizontal strips and"},{"Start":"02:33.015 ","End":"02:37.590","Text":"each of them would have a thickness,"},{"Start":"02:37.590 ","End":"02:41.315","Text":"when I say thickness, I mean the difference between the top and the bottom part."},{"Start":"02:41.315 ","End":"02:46.889","Text":"That would be, call it Delta x but it would equal,"},{"Start":"02:46.889 ","End":"02:50.510","Text":"we take 10 meters and divide it into n,"},{"Start":"02:50.510 ","End":"02:56.705","Text":"so each of them is 10 over n meters wide and we have n of them."},{"Start":"02:56.705 ","End":"03:01.260","Text":"Now the x of this strip,"},{"Start":"03:02.180 ","End":"03:04.730","Text":"it almost doesn\u0027t make any difference,"},{"Start":"03:04.730 ","End":"03:05.930","Text":"the x has the changes,"},{"Start":"03:05.930 ","End":"03:07.850","Text":"but let\u0027s take the breath of the bottom 1."},{"Start":"03:07.850 ","End":"03:10.835","Text":"This point will be x_i,"},{"Start":"03:10.835 ","End":"03:14.880","Text":"and then x_i will equal,"},{"Start":"03:14.880 ","End":"03:17.685","Text":"I just have to take i strips of these,"},{"Start":"03:17.685 ","End":"03:20.850","Text":"it\u0027ll will be 10_i over"},{"Start":"03:20.850 ","End":"03:29.320","Text":"n. If you plug in i equals n is just 10."},{"Start":"03:33.620 ","End":"03:36.660","Text":"This would be the last and nth strip,"},{"Start":"03:36.660 ","End":"03:38.820","Text":"this would be the first strip, the ith strip,"},{"Start":"03:38.820 ","End":"03:42.935","Text":"i goes from 1 to n. What we\u0027re going to do is compute"},{"Start":"03:42.935 ","End":"03:47.850","Text":"the pressure on each typical strip i,"},{"Start":"03:47.850 ","End":"03:54.710","Text":"and then take the sum from 1 to n and then we\u0027ll let n go to infinity and take the limit."},{"Start":"03:54.710 ","End":"03:58.430","Text":"This is sometimes called a Riemann sum."},{"Start":"03:58.430 ","End":"04:04.125","Text":"In fact, let me just do a flashback to the lesson."},{"Start":"04:04.125 ","End":"04:08.330","Text":"Here is just to give you a general idea that we approximate an"},{"Start":"04:08.330 ","End":"04:13.145","Text":"integral by breaking the area up into strips,"},{"Start":"04:13.145 ","End":"04:15.740","Text":"we break the range,"},{"Start":"04:15.740 ","End":"04:17.945","Text":"in this case it\u0027s 0, 1."},{"Start":"04:17.945 ","End":"04:22.490","Text":"It\u0027s actually a domain or a range from 0-1 break it up into n pieces."},{"Start":"04:22.490 ","End":"04:26.285","Text":"Take rectangles as an approximation and"},{"Start":"04:26.285 ","End":"04:30.275","Text":"add all the areas up together and then in the end take n to infinity,"},{"Start":"04:30.275 ","End":"04:32.300","Text":"and that\u0027s called a Riemann sum."},{"Start":"04:32.300 ","End":"04:34.490","Text":"You should go back and look at the lesson,"},{"Start":"04:34.490 ","End":"04:40.395","Text":"and let\u0027s get back to the present. Here we are."},{"Start":"04:40.395 ","End":"04:46.265","Text":"Now let\u0027s compute the total force on the side of the tank."},{"Start":"04:46.265 ","End":"04:52.835","Text":"What we have is that the force is going to be the sum of the forces on all n strips."},{"Start":"04:52.835 ","End":"05:03.495","Text":"It\u0027s going to be the sum from 1 to n. Sorry,"},{"Start":"05:03.495 ","End":"05:07.099","Text":"here the variable is i, and for each i,"},{"Start":"05:07.099 ","End":"05:11.150","Text":"I\u0027ll take the pressure on here."},{"Start":"05:11.150 ","End":"05:18.980","Text":"What we get is that the pressure will be,"},{"Start":"05:18.980 ","End":"05:24.479","Text":"using this formula, Rho times g,"},{"Start":"05:24.479 ","End":"05:30.325","Text":"now d would be x_i that\u0027s the pressure."},{"Start":"05:30.325 ","End":"05:33.830","Text":"Now, we have to multiply that by the area."},{"Start":"05:33.830 ","End":"05:41.005","Text":"Let me just say this part is the pressure and I need to multiply it by the area."},{"Start":"05:41.005 ","End":"05:46.270","Text":"Now the area is the length of this,"},{"Start":"05:46.270 ","End":"05:50.215","Text":"and I\u0027m going to have to do a computation on that side."},{"Start":"05:50.215 ","End":"05:52.015","Text":"But it\u0027s going to be something,"},{"Start":"05:52.015 ","End":"05:58.664","Text":"this length of this line times the Delta x."},{"Start":"05:58.664 ","End":"06:07.289","Text":"Let me put Delta x and after just compute the length of this from here to here,"},{"Start":"06:08.000 ","End":"06:10.460","Text":"what I\u0027m missing here is the length,"},{"Start":"06:10.460 ","End":"06:12.005","Text":"I put it in red, so you see it."},{"Start":"06:12.005 ","End":"06:13.445","Text":"I need the length,"},{"Start":"06:13.445 ","End":"06:17.255","Text":"l is the length of the red line here."},{"Start":"06:17.255 ","End":"06:20.825","Text":"Now I\u0027m going to do this using Pythagoras."},{"Start":"06:20.825 ","End":"06:24.800","Text":"If I join this point to this point, this line here,"},{"Start":"06:24.800 ","End":"06:26.960","Text":"it\u0027s length is also 10,"},{"Start":"06:26.960 ","End":"06:29.150","Text":"because 10 is the radius."},{"Start":"06:29.150 ","End":"06:33.094","Text":"Now I have a right-angled triangle, this,"},{"Start":"06:33.094 ","End":"06:38.060","Text":"this, and this and so what I get if I,"},{"Start":"06:38.060 ","End":"06:40.730","Text":"just to see where do I have room,"},{"Start":"06:40.730 ","End":"06:46.160","Text":"over here, I have this right-angle triangle,"},{"Start":"06:46.160 ","End":"06:50.419","Text":"where this bit is 10,"},{"Start":"06:50.419 ","End":"06:56.235","Text":"this here is x_i and"},{"Start":"06:56.235 ","End":"07:02.820","Text":"this is l over 2 because l is the whole length."},{"Start":"07:02.820 ","End":"07:07.230","Text":"By Pythagoras, I have that l over"},{"Start":"07:07.230 ","End":"07:15.545","Text":"2 is equal to the square root of this squared minus this squared."},{"Start":"07:15.545 ","End":"07:21.165","Text":"It\u0027s a 100 minus x_i squared."},{"Start":"07:21.165 ","End":"07:23.800","Text":"Now, I can put what l is,"},{"Start":"07:23.800 ","End":"07:26.510","Text":"because l is twice this."},{"Start":"07:26.520 ","End":"07:31.255","Text":"I will just put the 2 on the other side, like so,"},{"Start":"07:31.255 ","End":"07:33.445","Text":"and now when I put l in here,"},{"Start":"07:33.445 ","End":"07:38.215","Text":"I can put the 2 in front this bit."},{"Start":"07:38.215 ","End":"07:41.860","Text":"Then I can just instead of l,"},{"Start":"07:41.860 ","End":"07:45.835","Text":"the 2 and then the square root of"},{"Start":"07:45.835 ","End":"07:53.560","Text":"100 minus x i squared and what we want to do,"},{"Start":"07:53.560 ","End":"07:55.645","Text":"since this is a Riemann sum,"},{"Start":"07:55.645 ","End":"07:59.110","Text":"is to let n go to infinity,"},{"Start":"07:59.110 ","End":"08:08.275","Text":"then this sum becomes an integral and what we get is that the total force is twice."},{"Start":"08:08.275 ","End":"08:09.790","Text":"When you see a sigma,"},{"Start":"08:09.790 ","End":"08:11.785","Text":"you put an integral sign,"},{"Start":"08:11.785 ","End":"08:15.460","Text":"and you see where it goes from a to the x i,"},{"Start":"08:15.460 ","End":"08:20.035","Text":"or the x goes from 0 to 10."},{"Start":"08:20.035 ","End":"08:22.780","Text":"We have rho, we have g,"},{"Start":"08:22.780 ","End":"08:25.645","Text":"x i just becomes x,"},{"Start":"08:25.645 ","End":"08:31.840","Text":"square root of 100 minus x squared and delta x becomes dx."},{"Start":"08:31.840 ","End":"08:35.980","Text":"That\u0027s how you go from a Riemann sum in the limit to an integral."},{"Start":"08:35.980 ","End":"08:38.470","Text":"Now we just have to compute this."},{"Start":"08:38.470 ","End":"08:43.225","Text":"This looks like a case for integration by substitution."},{"Start":"08:43.225 ","End":"08:45.790","Text":"The several ways we could go about it,"},{"Start":"08:45.790 ","End":"08:48.700","Text":"we can substitute the whole square root of a 100 minus x squared."},{"Start":"08:48.700 ","End":"08:51.595","Text":"Oh, just a 100 minus x squared. They\u0027re both good."},{"Start":"08:51.595 ","End":"08:53.560","Text":"I\u0027ll just take say,"},{"Start":"08:53.560 ","End":"08:59.875","Text":"t is equal to 100 minus x squared in my substitution."},{"Start":"08:59.875 ","End":"09:09.850","Text":"Then dt will equal the derivative of this minus 2x dx."},{"Start":"09:09.850 ","End":"09:13.300","Text":"But we also have to substitute the limits."},{"Start":"09:13.300 ","End":"09:17.410","Text":"Notice that when x equals 0,"},{"Start":"09:17.410 ","End":"09:22.030","Text":"t is 100 minus 0 squared,"},{"Start":"09:22.030 ","End":"09:30.065","Text":"which is 100, and when x equals 10,"},{"Start":"09:30.065 ","End":"09:34.800","Text":"then t equals a 100 minus 10 squared,"},{"Start":"09:34.800 ","End":"09:38.205","Text":"then t equals 0,"},{"Start":"09:38.205 ","End":"09:44.950","Text":"and so what we get is,"},{"Start":"09:44.950 ","End":"09:46.750","Text":"now we have an integral,"},{"Start":"09:46.750 ","End":"09:48.730","Text":"but this time it won\u0027t be dx,"},{"Start":"09:48.730 ","End":"09:57.040","Text":"it will be dt from 100 to 0."},{"Start":"09:57.040 ","End":"10:00.260","Text":"Write the rho g here."},{"Start":"10:02.040 ","End":"10:07.150","Text":"What I say is that 2x dx."},{"Start":"10:07.150 ","End":"10:13.610","Text":"This with this, with this is just minus dt,"},{"Start":"10:13.920 ","End":"10:18.380","Text":"so I have minus dt,"},{"Start":"10:19.020 ","End":"10:22.675","Text":"and here a 100 minus x squared is t,"},{"Start":"10:22.675 ","End":"10:26.810","Text":"so I have the square root of t dt,"},{"Start":"10:26.880 ","End":"10:33.010","Text":"and what I can do is bring the constants outside."},{"Start":"10:33.010 ","End":"10:35.095","Text":"But there\u0027s something else I wanted to do."},{"Start":"10:35.095 ","End":"10:37.810","Text":"I don\u0027t like the minus here and I don\u0027t like the fact that"},{"Start":"10:37.810 ","End":"10:40.570","Text":"the upper limit of integration is less than the lower."},{"Start":"10:40.570 ","End":"10:43.540","Text":"If I switch these 2, I can get rid of the minus."},{"Start":"10:43.540 ","End":"10:46.585","Text":"It\u0027s the integral from 0 to a 100."},{"Start":"10:46.585 ","End":"10:48.295","Text":"Now without the minus,"},{"Start":"10:48.295 ","End":"10:55.370","Text":"I\u0027ll take the rho g in front and I\u0027ve got the square root of t dt."},{"Start":"10:55.980 ","End":"10:58.330","Text":"This is straightforward enough."},{"Start":"10:58.330 ","End":"11:01.570","Text":"This is square root of t is t^1.5."},{"Start":"11:01.570 ","End":"11:04.225","Text":"I\u0027ll just continue over here."},{"Start":"11:04.225 ","End":"11:10.720","Text":"So what I get is I raised the power by 1 and it\u0027s t to the 3 over 2,"},{"Start":"11:10.720 ","End":"11:12.430","Text":"but I have to divide by that."},{"Start":"11:12.430 ","End":"11:16.165","Text":"It\u0027s two-thirds and all this"},{"Start":"11:16.165 ","End":"11:24.370","Text":"between 0 and 100."},{"Start":"11:24.370 ","End":"11:28.270","Text":"I thought the right rho g. But the 2 thirds."},{"Start":"11:28.270 ","End":"11:33.385","Text":"You can also put outside the brackets and I\u0027ll also put here,"},{"Start":"11:33.385 ","End":"11:36.025","Text":"maybe I\u0027ll put it at the end,"},{"Start":"11:36.025 ","End":"11:43.700","Text":"times rho times g. Just have to compute this bit here."},{"Start":"11:44.130 ","End":"11:47.410","Text":"What we get is two-thirds."},{"Start":"11:47.410 ","End":"11:51.310","Text":"Now when we plug in t equals 100,"},{"Start":"11:51.310 ","End":"11:55.120","Text":"100^3 over 2 is a 1000."},{"Start":"11:55.120 ","End":"12:00.055","Text":"0^3 over 2 is 0, rho g,"},{"Start":"12:00.055 ","End":"12:06.805","Text":"and so the answer is just equal to 1000"},{"Start":"12:06.805 ","End":"12:13.840","Text":"times 2 over 3 is writing as 2000 over 3."},{"Start":"12:13.840 ","End":"12:20.655","Text":"Or if you want 666 and two-thirds times g. Now whether it rho get to, well,"},{"Start":"12:20.655 ","End":"12:24.105","Text":"rho is equal to 1 for water,"},{"Start":"12:24.105 ","End":"12:30.190","Text":"the density of water is 1."},{"Start":"12:31.430 ","End":"12:33.795","Text":"Take to on the last bit."},{"Start":"12:33.795 ","End":"12:42.670","Text":"We get 2000 over 3 rho g. Now I claim that we know what rho and g are."},{"Start":"12:42.990 ","End":"12:45.940","Text":"The medium is water,"},{"Start":"12:45.940 ","End":"12:55.060","Text":"and we know that rho is equal to 1000 for water in the metric system,"},{"Start":"12:55.060 ","End":"13:00.235","Text":"because a cubic meter of water weighs 1000,"},{"Start":"13:00.235 ","End":"13:09.130","Text":"which is a ton actually and we also know g is approximately 9.81,"},{"Start":"13:09.130 ","End":"13:13.135","Text":"the acceleration due to gravity in meters per second per second,"},{"Start":"13:13.135 ","End":"13:19.100","Text":"and so it\u0027s just a calculator exercise."},{"Start":"13:19.440 ","End":"13:22.270","Text":"I could write the expression,"},{"Start":"13:22.270 ","End":"13:28.360","Text":"could say that this is equal to one-third times 2000 times a"},{"Start":"13:28.360 ","End":"13:35.455","Text":"1000 is 2 million times"},{"Start":"13:35.455 ","End":"13:42.320","Text":"9.8 or 9.81 depending on how accurate you want to go."},{"Start":"13:42.930 ","End":"13:46.120","Text":"I\u0027m not actually going to compute it."},{"Start":"13:46.120 ","End":"13:48.670","Text":"I\u0027ll just leave it you to look it up."},{"Start":"13:48.670 ","End":"13:51.190","Text":"On the calculator. I can make a rough guess."},{"Start":"13:51.190 ","End":"13:53.410","Text":"A third of 9 is 3,"},{"Start":"13:53.410 ","End":"13:57.220","Text":"so you want to increase a bit 3 times 2 million is 6 million,"},{"Start":"13:57.220 ","End":"13:59.770","Text":"but it\u0027s a bit more so maybe 7 million."},{"Start":"13:59.770 ","End":"14:06.280","Text":"I would expect the answer to come about 7 million and the answer would be in newtons."},{"Start":"14:06.280 ","End":"14:14.920","Text":"The answer in Newtons would be the total force."},{"Start":"14:14.920 ","End":"14:16.960","Text":"Also, I think it\u0027s about 7 million,"},{"Start":"14:16.960 ","End":"14:20.120","Text":"but I don\u0027t have my calculator with me."},{"Start":"14:22.350 ","End":"14:25.735","Text":"This example will suffice,"},{"Start":"14:25.735 ","End":"14:28.400","Text":"so we are done here."}],"Thumbnail":null,"ID":8298}],"ID":6163},{"Name":"Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"5m 28s","ChapterTopicVideoID":8147,"CourseChapterTopicPlaylistID":6164,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this clip, we\u0027re going to consider another application"},{"Start":"00:02.970 ","End":"00:05.025","Text":"of integrals, and this time,"},{"Start":"00:05.025 ","End":"00:08.460","Text":"in the theory of probability and in particular,"},{"Start":"00:08.460 ","End":"00:10.785","Text":"probability density functions."},{"Start":"00:10.785 ","End":"00:13.350","Text":"We\u0027ll come to that. I want to, first of all,"},{"Start":"00:13.350 ","End":"00:17.350","Text":"talk about random variables."},{"Start":"00:18.380 ","End":"00:24.465","Text":"The random variable can be discrete or continuous,"},{"Start":"00:24.465 ","End":"00:26.505","Text":"so I\u0027ll just write that."},{"Start":"00:26.505 ","End":"00:30.600","Text":"But here, we\u0027re only going to be talking about"},{"Start":"00:30.600 ","End":"00:33.975","Text":"continuous random variables,"},{"Start":"00:33.975 ","End":"00:40.065","Text":"so I\u0027ll highlight the continuous and random variables."},{"Start":"00:40.065 ","End":"00:42.365","Text":"I\u0027ll say a couple of words about discrete,"},{"Start":"00:42.365 ","End":"00:44.600","Text":"but just as background."},{"Start":"00:44.600 ","End":"00:48.325","Text":"As an example of a discrete,"},{"Start":"00:48.325 ","End":"00:55.215","Text":"I\u0027ll take the case of a rolling a pair of dice."},{"Start":"00:55.215 ","End":"00:59.345","Text":"In both cases, we use a random variable,"},{"Start":"00:59.345 ","End":"01:02.510","Text":"usually a capital letter like X."},{"Start":"01:02.510 ","End":"01:07.340","Text":"In this case, I would say that X equals the sum"},{"Start":"01:07.340 ","End":"01:13.755","Text":"of 2 dice that are tossed randomly,"},{"Start":"01:13.755 ","End":"01:18.690","Text":"and then X would have particular values:"},{"Start":"01:18.690 ","End":"01:20.400","Text":"the lowest, it could be is 2,"},{"Start":"01:20.400 ","End":"01:23.070","Text":"and the highest, it can be is 12."},{"Start":"01:23.070 ","End":"01:27.660","Text":"Then we can also make a distribution."},{"Start":"01:27.660 ","End":"01:30.255","Text":"For each value, there\u0027s a probability."},{"Start":"01:30.255 ","End":"01:36.740","Text":"For example, the probability that X equals 6"},{"Start":"01:36.740 ","End":"01:40.800","Text":"happens to be 5 over 36,"},{"Start":"01:40.800 ","End":"01:43.610","Text":"and if you compute that in decimal, you\u0027ll get whatever."},{"Start":"01:43.610 ","End":"01:45.455","Text":"This is here for 6."},{"Start":"01:45.455 ","End":"01:49.010","Text":"Usually, there\u0027s a finite number, but not necessarily."},{"Start":"01:49.010 ","End":"01:52.295","Text":"It could be infinite, but countably infinite."},{"Start":"01:52.295 ","End":"01:57.020","Text":"For example, how many times you have"},{"Start":"01:57.020 ","End":"02:00.065","Text":"to toss a coin until you get heads?"},{"Start":"02:00.065 ","End":"02:01.790","Text":"It could be anything."},{"Start":"02:01.790 ","End":"02:05.300","Text":"You could go a thousand tosses without getting a head,"},{"Start":"02:05.300 ","End":"02:10.150","Text":"but it\u0027s still discrete, its individual numbers."},{"Start":"02:10.150 ","End":"02:12.695","Text":"Now, continuous is something else."},{"Start":"02:12.695 ","End":"02:16.700","Text":"In real life, continuous would be something like"},{"Start":"02:16.700 ","End":"02:20.990","Text":"a person\u0027s height or the number of hours"},{"Start":"02:20.990 ","End":"02:24.485","Text":"the light bulb stays lit before it burns out,"},{"Start":"02:24.485 ","End":"02:26.990","Text":"it could be the age of a person."},{"Start":"02:26.990 ","End":"02:30.830","Text":"But some of these could be discrete or continuous, like the age."},{"Start":"02:30.830 ","End":"02:32.750","Text":"If you just take it as a whole number,"},{"Start":"02:32.750 ","End":"02:35.090","Text":"age 20, age 35,"},{"Start":"02:35.090 ","End":"02:36.725","Text":"and so on, then it\u0027s discrete,"},{"Start":"02:36.725 ","End":"02:40.115","Text":"but if you allow time and days,"},{"Start":"02:40.115 ","End":"02:42.770","Text":"hours, minutes, seconds, fractions of a second,"},{"Start":"02:42.770 ","End":"02:44.655","Text":"it\u0027s a continuous."},{"Start":"02:44.655 ","End":"02:47.470","Text":"Usually, for continuous random variables,"},{"Start":"02:47.470 ","End":"02:51.520","Text":"it could take on any value from minus infinity to infinity."},{"Start":"02:51.520 ","End":"02:54.525","Text":"Even the age of a person,"},{"Start":"02:54.525 ","End":"02:56.060","Text":"we say it can be negative,"},{"Start":"02:56.060 ","End":"02:58.430","Text":"but probability is 0."},{"Start":"02:58.430 ","End":"03:02.749","Text":"Usually, continuous is from minus infinity to infinity,"},{"Start":"03:02.749 ","End":"03:06.470","Text":"and then sometimes it\u0027s on a smaller interval,"},{"Start":"03:06.470 ","End":"03:08.600","Text":"but in any event, it\u0027s on an interval."},{"Start":"03:08.600 ","End":"03:12.215","Text":"For a continuous variable X,"},{"Start":"03:12.215 ","End":"03:17.730","Text":"it could have a range of values continuous, so real number."},{"Start":"03:19.190 ","End":"03:24.260","Text":"Instead of individual probabilities for outcomes,"},{"Start":"03:24.260 ","End":"03:27.110","Text":"we have what is called a probability density"},{"Start":"03:27.110 ","End":"03:29.075","Text":"function or a density curve."},{"Start":"03:29.075 ","End":"03:35.040","Text":"Let\u0027s say that this 1 is f of x or f of big X,"},{"Start":"03:35.040 ","End":"03:37.370","Text":"and in that case,"},{"Start":"03:37.370 ","End":"03:42.739","Text":"the probability that X is in a certain range,"},{"Start":"03:42.739 ","End":"03:46.670","Text":"say from a to b, is given by the area under the curve,"},{"Start":"03:46.670 ","End":"03:48.410","Text":"but if we know integrals,"},{"Start":"03:48.410 ","End":"03:55.550","Text":"this is equal to the integral from a to b of f of x dx,"},{"Start":"03:55.550 ","End":"03:58.220","Text":"where f is the probability density function."},{"Start":"03:58.220 ","End":"04:00.305","Text":"There are certain similarities."},{"Start":"04:00.305 ","End":"04:03.860","Text":"Notice that all these values can be only positive."},{"Start":"04:03.860 ","End":"04:06.205","Text":"Probability can\u0027t be negative."},{"Start":"04:06.205 ","End":"04:09.430","Text":"It could be 0, it\u0027s non-negative."},{"Start":"04:09.430 ","End":"04:12.980","Text":"Similarly, the density curve or the density"},{"Start":"04:12.980 ","End":"04:15.680","Text":"function is always non-negative,"},{"Start":"04:15.680 ","End":"04:17.120","Text":"but more than that,"},{"Start":"04:17.120 ","End":"04:20.420","Text":"just like here, all the probabilities add up to 1."},{"Start":"04:20.420 ","End":"04:22.430","Text":"In the case of a continuous,"},{"Start":"04:22.430 ","End":"04:28.590","Text":"the total area under the curve is going to be 1"},{"Start":"04:28.590 ","End":"04:31.265","Text":"so that the basic properties for"},{"Start":"04:31.265 ","End":"04:36.770","Text":"a probability density function are that f of x is"},{"Start":"04:36.770 ","End":"04:42.885","Text":"bigger or equal to 0 for all x,"},{"Start":"04:42.885 ","End":"04:47.150","Text":"and also that the integral minus infinity to infinity"},{"Start":"04:47.150 ","End":"04:50.700","Text":"of f of x dx is equal to 1."},{"Start":"04:50.700 ","End":"04:52.995","Text":"1 is the maximum probability."},{"Start":"04:52.995 ","End":"04:55.820","Text":"That\u0027s the introduction. Now, let\u0027s get working"},{"Start":"04:55.820 ","End":"04:58.910","Text":"on some sample problems."},{"Start":"04:58.910 ","End":"05:04.250","Text":"Keep the diagram, and I\u0027ll keep the definition of a PDF,"},{"Start":"05:04.250 ","End":"05:06.810","Text":"probability density function."},{"Start":"05:06.850 ","End":"05:13.020","Text":"I\u0027d like to write the term probability density function."},{"Start":"05:13.510 ","End":"05:16.970","Text":"In fact, this is the definition that makes f"},{"Start":"05:16.970 ","End":"05:21.485","Text":"a probability density function is that it\u0027s non-negative,"},{"Start":"05:21.485 ","End":"05:29.270","Text":"and its integral from minus infinity to infinity is 1."}],"Thumbnail":null,"ID":8301},{"Watched":false,"Name":"Example1","Duration":"8m 18s","ChapterTopicVideoID":8148,"CourseChapterTopicPlaylistID":6164,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.530 ","End":"00:04.545","Text":"Now let me bring the example."},{"Start":"00:04.545 ","End":"00:06.660","Text":"I\u0027m going to give you"},{"Start":"00:06.660 ","End":"00:13.740","Text":"that f of x"},{"Start":"00:13.740 ","End":"00:20.220","Text":"is equal to 3x squared,"},{"Start":"00:20.220 ","End":"00:26.230","Text":"provided that x is between 0 and 1,"},{"Start":"00:27.020 ","End":"00:29.545","Text":"less than 0 or bigger than 1,"},{"Start":"00:29.545 ","End":"00:33.290","Text":"and we just write otherwise or elsewhere."},{"Start":"00:33.290 ","End":"00:43.310","Text":"Everywhere between 0 and 1 it\u0027s 0 and we have to show that f is a PDF."},{"Start":"00:43.310 ","End":"00:46.430","Text":"PDF, I mean probability density function."},{"Start":"00:46.430 ","End":"00:51.815","Text":"I know it\u0027s an acronym for other things too."},{"Start":"00:51.815 ","End":"00:53.585","Text":"There\u0027s the file format,"},{"Start":"00:53.585 ","End":"00:57.320","Text":"PDF I will sometimes say,"},{"Start":"00:57.320 ","End":"01:00.275","Text":"and I will mean probability density function."},{"Start":"01:00.275 ","End":"01:04.730","Text":"Let\u0027s show that first and then I\u0027ll ask some other questions about it."},{"Start":"01:04.730 ","End":"01:06.950","Text":"Let\u0027s renumber these properties,"},{"Start":"01:06.950 ","End":"01:08.815","Text":"let\u0027s say 1 and 2."},{"Start":"01:08.815 ","End":"01:11.010","Text":"1 is of course,"},{"Start":"01:11.010 ","End":"01:18.500","Text":"f of x is certainly bigger or equal to 0 because x squared"},{"Start":"01:18.500 ","End":"01:26.420","Text":"is bigger or equal to 0 or ways and hence 3x squared and 0 is also bigger or equal to 0."},{"Start":"01:26.420 ","End":"01:31.115","Text":"Both in this interval and outside it was bigger or equal to 0."},{"Start":"01:31.115 ","End":"01:35.390","Text":"The second property and each show that the integral from"},{"Start":"01:35.390 ","End":"01:40.820","Text":"minus infinity to infinity of f of x is 1."},{"Start":"01:40.820 ","End":"01:45.680","Text":"Now what we can do is break it up into 3 separate integrals."},{"Start":"01:45.680 ","End":"01:49.969","Text":"We can take the integral from minus infinity to 0."},{"Start":"01:49.969 ","End":"01:55.100","Text":"I won\u0027t write the contents just the idea that we couldn\u0027t break it up then from 0-1,"},{"Start":"01:55.100 ","End":"01:58.220","Text":"and then from 1 to infinity."},{"Start":"01:58.220 ","End":"02:03.365","Text":"We get from minus infinity to infinity in 3 bits."},{"Start":"02:03.365 ","End":"02:06.259","Text":"The thing is that in this range,"},{"Start":"02:06.259 ","End":"02:07.850","Text":"f is equal to 0,"},{"Start":"02:07.850 ","End":"02:10.930","Text":"and in this range, f of x is equal to 0."},{"Start":"02:10.930 ","End":"02:14.930","Text":"I\u0027ll just write f equals 0 here and f equals 0 here,"},{"Start":"02:14.930 ","End":"02:18.710","Text":"and here, f of x is 3x squared."},{"Start":"02:18.710 ","End":"02:29.010","Text":"Basically what I just get is the integral from 0-1 of 3x squared dx."},{"Start":"02:29.010 ","End":"02:31.310","Text":"It\u0027s from minus infinity to infinity,"},{"Start":"02:31.310 ","End":"02:34.505","Text":"but it\u0027s 0 everywhere outside so we just take it here."},{"Start":"02:34.505 ","End":"02:36.965","Text":"That\u0027s a common thing to do."},{"Start":"02:36.965 ","End":"02:41.180","Text":"This is equal to now 3x squared."},{"Start":"02:41.180 ","End":"02:43.130","Text":"The integral is x cubed,"},{"Start":"02:43.130 ","End":"02:46.940","Text":"and we have to evaluate that from 0-1."},{"Start":"02:46.940 ","End":"02:50.435","Text":"We get 1 cubed minus 0 cubed."},{"Start":"02:50.435 ","End":"02:53.480","Text":"This is indeed equal to 1."},{"Start":"02:53.480 ","End":"03:02.360","Text":"X is between 1/3 and 2/3."},{"Start":"03:02.360 ","End":"03:07.250","Text":"I\u0027m going to ask, what is the probability that x,"},{"Start":"03:07.250 ","End":"03:10.155","Text":"I should really be saying big X,"},{"Start":"03:10.155 ","End":"03:13.520","Text":"instead of little x, doesn\u0027t matter."},{"Start":"03:13.520 ","End":"03:22.520","Text":"What\u0027s the probability that x is bigger or equal to 1/2?"},{"Start":"03:24.620 ","End":"03:32.330","Text":"I also want the probability that x equals,"},{"Start":"03:32.330 ","End":"03:33.500","Text":"let\u0027s"},{"Start":"03:33.500 ","End":"03:35.280","Text":"say"},{"Start":"03:36.080 ","End":"03:44.760","Text":"3/4."},{"Start":"03:44.760 ","End":"03:48.240","Text":"Maybe I\u0027ll give them labels."},{"Start":"03:48.240 ","End":"03:53.390","Text":"Let\u0027s called this part A, part B,"},{"Start":"03:53.390 ","End":"03:59.480","Text":"and part C. Let\u0027s begin with solving part A."},{"Start":"03:59.480 ","End":"04:03.365","Text":"To evaluate this, if you look at what we had above,"},{"Start":"04:03.365 ","End":"04:08.540","Text":"the probability that the variable is between 2 values,"},{"Start":"04:08.540 ","End":"04:11.335","Text":"a and b is just the integral"},{"Start":"04:11.335 ","End":"04:19.620","Text":"from 1/3 to 2/3 of f of x dx."},{"Start":"04:19.620 ","End":"04:22.245","Text":"But between 1/3 and 2/3,"},{"Start":"04:22.245 ","End":"04:24.330","Text":"we\u0027re using this definition,"},{"Start":"04:24.330 ","End":"04:29.430","Text":"so it\u0027s just the integral from"},{"Start":"04:29.430 ","End":"04:37.180","Text":"1/3 to 2/3 of 3x squared dx."},{"Start":"04:39.040 ","End":"04:42.964","Text":"The indefinite integral we already did, it\u0027s x cubed,"},{"Start":"04:42.964 ","End":"04:50.900","Text":"so we need x cubed between 1/3 and 2/3 This is equal"},{"Start":"04:50.900 ","End":"05:00.395","Text":"to 2/3 cubed is 8 over 27 minus 1 over 27."},{"Start":"05:00.395 ","End":"05:04.070","Text":"This comes out to be 7 over 27."},{"Start":"05:04.070 ","End":"05:05.180","Text":"I\u0027ll leave it as a fraction."},{"Start":"05:05.180 ","End":"05:07.640","Text":"If you want to convert it to a decimal."},{"Start":"05:07.640 ","End":"05:10.190","Text":"It has to come out between 0 and 1."},{"Start":"05:10.190 ","End":"05:14.780","Text":"That\u0027s for sure any probability has to come out like that."},{"Start":"05:14.780 ","End":"05:18.395","Text":"Now, in part B,"},{"Start":"05:18.395 ","End":"05:21.905","Text":"we don\u0027t have that x is between something and something."},{"Start":"05:21.905 ","End":"05:23.510","Text":"But in a way we do,"},{"Start":"05:23.510 ","End":"05:33.250","Text":"because I could say that this is the same as x being between a half and infinity."},{"Start":"05:33.250 ","End":"05:44.630","Text":"I would say that to do the integral from 1/2 to infinity of f of x dx."},{"Start":"05:44.630 ","End":"05:47.555","Text":"Now, do the same trick as I did here."},{"Start":"05:47.555 ","End":"05:54.885","Text":"I could break it up from 1/2 to 1 plus from 1 to infinity."},{"Start":"05:54.885 ","End":"05:59.100","Text":"The part from 1 to infinity is 0."},{"Start":"05:59.100 ","End":"06:02.610","Text":"I just have to take it from 1/2 to"},{"Start":"06:02.610 ","End":"06:12.335","Text":"1 because of the 1/3 above 1, it\u0027s 0."},{"Start":"06:12.335 ","End":"06:19.360","Text":"In fact, we even know that between 1/2 and 1 that it\u0027s equal to 3x squared."},{"Start":"06:19.360 ","End":"06:22.725","Text":"I could have just written 3x squared right away."},{"Start":"06:22.725 ","End":"06:25.070","Text":"You know what? When you\u0027re reviewing this,"},{"Start":"06:25.070 ","End":"06:26.180","Text":"I might just say,"},{"Start":"06:26.180 ","End":"06:29.840","Text":"let\u0027s put also from 1 to infinity."},{"Start":"06:29.840 ","End":"06:32.960","Text":"But now the function is 0 dx."},{"Start":"06:32.960 ","End":"06:38.270","Text":"Just so you don\u0027t think later that I forgot. This part is 0."},{"Start":"06:38.270 ","End":"06:41.690","Text":"The integral of 0 over any range is 0,"},{"Start":"06:41.690 ","End":"06:43.740","Text":"even an improper integral."},{"Start":"06:43.740 ","End":"06:47.195","Text":"What I have here is x cubed,"},{"Start":"06:47.195 ","End":"06:48.845","Text":"same as we had here."},{"Start":"06:48.845 ","End":"06:51.270","Text":"Just the limits of integration of change,"},{"Start":"06:51.270 ","End":"06:55.170","Text":"so we have to do it from 1/2 to 1."},{"Start":"06:55.170 ","End":"06:57.210","Text":"At 1 It\u0027s equal to 1."},{"Start":"06:57.210 ","End":"06:59.955","Text":"At 1/2 it\u0027s equal to 1/8,"},{"Start":"06:59.955 ","End":"07:02.145","Text":"so it\u0027s 1 minus 1/8."},{"Start":"07:02.145 ","End":"07:04.510","Text":"So it\u0027s 7/8."},{"Start":"07:05.270 ","End":"07:09.030","Text":"Part C is a bit of a strange one."},{"Start":"07:09.030 ","End":"07:14.310","Text":"Part C, to say that x equals 3/4."},{"Start":"07:14.310 ","End":"07:23.235","Text":"I could say it\u0027s between 3/4 and 3/4 and use the formula."},{"Start":"07:23.235 ","End":"07:27.315","Text":"What I get is the integral from"},{"Start":"07:27.315 ","End":"07:33.930","Text":"3/4 to 3/4 of f of x dx."},{"Start":"07:33.930 ","End":"07:37.310","Text":"It doesn\u0027t matter what this function is,"},{"Start":"07:37.310 ","End":"07:40.985","Text":"it happens to be 3x squared and the integral is x cubed."},{"Start":"07:40.985 ","End":"07:45.210","Text":"The point is I\u0027m going to be substituting the same upper and lower limit,"},{"Start":"07:45.210 ","End":"07:46.880","Text":"so it\u0027s going to subtract itself,"},{"Start":"07:46.880 ","End":"07:49.340","Text":"it\u0027s going to have to be 0."},{"Start":"07:49.340 ","End":"07:59.165","Text":"The probability that x is exactly some number for a continuous random variable is 0,"},{"Start":"07:59.165 ","End":"08:02.780","Text":"which just, I find this of interest because it\u0027s"},{"Start":"08:02.780 ","End":"08:06.110","Text":"actually possible to choose a number at random."},{"Start":"08:06.110 ","End":"08:08.720","Text":"Let\u0027s say it could be 3/4,"},{"Start":"08:08.720 ","End":"08:11.255","Text":"but the probability of that is 0."},{"Start":"08:11.255 ","End":"08:14.210","Text":"Probability 0 does not mean impossible."},{"Start":"08:14.210 ","End":"08:19.050","Text":"That\u0027s just a bit of a philosophical observation."}],"Thumbnail":null,"ID":8302},{"Watched":false,"Name":"Example2","Duration":"3m 3s","ChapterTopicVideoID":8149,"CourseChapterTopicPlaylistID":6164,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.425","Text":"Now I want to introduce another concept for a continuous random variable,"},{"Start":"00:07.425 ","End":"00:09.285","Text":"capital X let\u0027s say,"},{"Start":"00:09.285 ","End":"00:14.355","Text":"that the mean of X,"},{"Start":"00:14.355 ","End":"00:18.360","Text":"where X has a probability density function little f,"},{"Start":"00:18.360 ","End":"00:25.830","Text":"the mean of X is given by the integral from minus infinity to infinity,"},{"Start":"00:25.830 ","End":"00:29.265","Text":"but not f of x like here,"},{"Start":"00:29.265 ","End":"00:34.575","Text":"x times f of x dx."},{"Start":"00:34.575 ","End":"00:37.590","Text":"That\u0027s the mean or average,"},{"Start":"00:37.590 ","End":"00:42.690","Text":"just like in statistics or in basic probability theory."},{"Start":"00:42.690 ","End":"00:46.620","Text":"Let\u0027s just do an example of this."},{"Start":"00:46.620 ","End":"00:51.950","Text":"I just copy pasted the previous example that we already know."},{"Start":"00:51.950 ","End":"00:55.350","Text":"This is the probability density function 3x squared"},{"Start":"00:55.350 ","End":"00:59.115","Text":"between 0 and 1 and 0 otherwise and we want to find the mean."},{"Start":"00:59.115 ","End":"01:01.005","Text":"I forgot to say the mean,"},{"Start":"01:01.005 ","End":"01:07.195","Text":"we use the Greek letter Mu for m, which is mean."},{"Start":"01:07.195 ","End":"01:10.790","Text":"I want to find Mu or the mean of"},{"Start":"01:10.790 ","End":"01:18.975","Text":"this random variable X with this distribution, probability density function."},{"Start":"01:18.975 ","End":"01:23.200","Text":"Yeah, that\u0027s the right word distribution you could also say."},{"Start":"01:23.570 ","End":"01:34.295","Text":"From the formula we get that Mu is the integral from minus infinity to infinity,"},{"Start":"01:34.295 ","End":"01:39.425","Text":"x times f of x dx."},{"Start":"01:39.425 ","End":"01:47.550","Text":"We already know how to handle the fact that it\u0027s piece-wise and its 0 outside this range."},{"Start":"01:47.550 ","End":"01:49.730","Text":"Basically break it up into 3 integrals,"},{"Start":"01:49.730 ","End":"01:56.900","Text":"but the lower 1 and the top 1 will be 0 because f is 0 outside of a certain range,"},{"Start":"01:56.900 ","End":"02:00.200","Text":"0 to 1, we can just take the integral here."},{"Start":"02:00.200 ","End":"02:02.900","Text":"So it\u0027s x times f of x."},{"Start":"02:02.900 ","End":"02:05.030","Text":"But between 0 and 1,"},{"Start":"02:05.030 ","End":"02:11.765","Text":"f of x is 3_x squared dx."},{"Start":"02:11.765 ","End":"02:18.720","Text":"What do we get? x times 3_x squared is 3_x cubed,"},{"Start":"02:19.790 ","End":"02:22.145","Text":"I\u0027ll just write this here."},{"Start":"02:22.145 ","End":"02:27.290","Text":"This here is 3_x cubed and we know how to do this integral."},{"Start":"02:27.290 ","End":"02:30.560","Text":"You raise the power by 1 is 4 and divide by it,"},{"Start":"02:30.560 ","End":"02:36.060","Text":"so it\u0027s 3/4_x to the 4th."},{"Start":"02:36.610 ","End":"02:42.080","Text":"Then we have to substitute the limits of integration 0 and"},{"Start":"02:42.080 ","End":"02:47.420","Text":"1 and this is equal to when x is 0 is just 0,"},{"Start":"02:47.420 ","End":"02:49.910","Text":"when x is 1, x to the 4th is 1."},{"Start":"02:49.910 ","End":"02:55.880","Text":"This is just equal to 3/4 and so the mean of"},{"Start":"02:55.880 ","End":"03:04.590","Text":"the probability density function or the random variable is 3/4 and that\u0027s it."}],"Thumbnail":null,"ID":8303},{"Watched":false,"Name":"Example3","Duration":"11m 17s","ChapterTopicVideoID":8150,"CourseChapterTopicPlaylistID":6164,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.655","Text":"I\u0027d like to do another example."},{"Start":"00:02.655 ","End":"00:07.860","Text":"Let me just erase what I don\u0027t need, just rearrange this."},{"Start":"00:07.860 ","End":"00:10.500","Text":"In this example, I won\u0027t use the letter X,"},{"Start":"00:10.500 ","End":"00:14.310","Text":"so I\u0027ll use the letter T because it\u0027s going to be a time."},{"Start":"00:14.310 ","End":"00:21.720","Text":"The variable will be the time to wait in line,"},{"Start":"00:21.720 ","End":"00:25.545","Text":"at a counter, at some store,"},{"Start":"00:25.545 ","End":"00:27.450","Text":"it\u0027s some waiting time,"},{"Start":"00:27.450 ","End":"00:29.565","Text":"doesn\u0027t really matter exactly."},{"Start":"00:29.565 ","End":"00:36.750","Text":"We\u0027re given that its probability density function is where f,"},{"Start":"00:36.750 ","End":"00:45.075","Text":"and I\u0027ll use the letter t instead of x to be equal to, well,"},{"Start":"00:45.075 ","End":"00:48.755","Text":"0 for t less than 0,"},{"Start":"00:48.755 ","End":"00:52.925","Text":"you don\u0027t expect anything for negative times,"},{"Start":"00:52.925 ","End":"01:03.240","Text":"and it\u0027s going to be 0.2e^t over 5,"},{"Start":"01:03.240 ","End":"01:06.285","Text":"make that minus t over 5,"},{"Start":"01:06.285 ","End":"01:11.445","Text":"and that\u0027s for t bigger or equal to 0."},{"Start":"01:11.445 ","End":"01:16.095","Text":"Now, there are 3 parts to the question."},{"Start":"01:16.095 ","End":"01:19.310","Text":"The 3 parts of the question are as follows;"},{"Start":"01:19.310 ","End":"01:24.250","Text":"a is just to verify that this really is"},{"Start":"01:24.250 ","End":"01:30.265","Text":"a probability density function so verify the PDF."},{"Start":"01:30.265 ","End":"01:36.130","Text":"Then part b will be to compute"},{"Start":"01:36.130 ","End":"01:44.760","Text":"the probability that we have to wait longer than 5 minutes."},{"Start":"01:44.760 ","End":"01:49.140","Text":"I should have said that t is in minutes,"},{"Start":"01:49.140 ","End":"01:56.460","Text":"we need a unit for time, that\u0027s part b."},{"Start":"01:56.460 ","End":"02:03.475","Text":"Then part c is to compute the mean for this,"},{"Start":"02:03.475 ","End":"02:08.360","Text":"in other words, the average waiting time."},{"Start":"02:08.600 ","End":"02:13.900","Text":"Let\u0027s start with part a."},{"Start":"02:15.140 ","End":"02:18.315","Text":"For a, we need 2 properties."},{"Start":"02:18.315 ","End":"02:24.700","Text":"We need that f of t has to be bigger or equal to 0 for"},{"Start":"02:24.700 ","End":"02:32.340","Text":"all t. It certainly is because 0 is bigger or equal to 0,"},{"Start":"02:32.340 ","End":"02:36.250","Text":"and e to the anything is also bigger or equal to 0,"},{"Start":"02:36.250 ","End":"02:39.580","Text":"times a positive number is still bigger or equal to 0,"},{"Start":"02:39.580 ","End":"02:41.530","Text":"so check on that."},{"Start":"02:41.530 ","End":"02:49.525","Text":"The second thing was the integral from minus infinity to infinity of f of t,"},{"Start":"02:49.525 ","End":"02:52.020","Text":"dt has to equal 1."},{"Start":"02:52.020 ","End":"02:58.280","Text":"Well, let\u0027s see, this integral at 0 when t is less than 0,"},{"Start":"02:58.280 ","End":"03:02.330","Text":"so we just need the integral from 0 to infinity,"},{"Start":"03:02.330 ","End":"03:09.870","Text":"in which case the function is 0.2e^ minus t over 5dt."},{"Start":"03:14.000 ","End":"03:17.525","Text":"Now to do the integral of an exponent,"},{"Start":"03:17.525 ","End":"03:24.650","Text":"I have to divide by the coefficient of t so what I"},{"Start":"03:24.650 ","End":"03:32.375","Text":"get is 0.2 divided by minus a 1/5,"},{"Start":"03:32.375 ","End":"03:38.185","Text":"and then e^ minus t over 5,"},{"Start":"03:38.185 ","End":"03:40.190","Text":"and this is a constant,"},{"Start":"03:40.190 ","End":"03:46.320","Text":"so we just need to evaluate this part from 0 to infinity."},{"Start":"03:46.670 ","End":"03:54.095","Text":"Now, 0.2 over minus a 1/5 is 0.2 times minus 5,"},{"Start":"03:54.095 ","End":"03:56.350","Text":"which is just minus 1."},{"Start":"03:56.350 ","End":"04:05.480","Text":"So it\u0027s minus e^ minus t over 5 from 0 to infinity."},{"Start":"04:05.480 ","End":"04:08.675","Text":"What I like to do is when there\u0027s a minus,"},{"Start":"04:08.675 ","End":"04:13.655","Text":"I can get rid of the minus by switching the order of the subtraction,"},{"Start":"04:13.655 ","End":"04:21.390","Text":"so let me write this as e^ minus t over 5 from infinity to 0,"},{"Start":"04:22.660 ","End":"04:25.995","Text":"and this is just equal to."},{"Start":"04:25.995 ","End":"04:28.195","Text":"If I plug in 0,"},{"Start":"04:28.195 ","End":"04:32.420","Text":"e^0 is 1, if I plug in infinity,"},{"Start":"04:32.420 ","End":"04:38.030","Text":"e^ minus infinity over 5 is still e^ minus infinity is 0,"},{"Start":"04:38.030 ","End":"04:40.525","Text":"so this is equal to 1,"},{"Start":"04:40.525 ","End":"04:43.710","Text":"and so that has been checked too."},{"Start":"04:43.710 ","End":"04:46.695","Text":"We have verified that it\u0027s a PDF,"},{"Start":"04:46.695 ","End":"04:49.020","Text":"that\u0027s part a, done."},{"Start":"04:49.020 ","End":"04:56.670","Text":"Now part b; bigger or equal to 5 means from 5 to infinity,"},{"Start":"04:56.670 ","End":"05:05.455","Text":"so we want the integral from 5 to infinity of f of t, dt."},{"Start":"05:05.455 ","End":"05:08.210","Text":"When we\u0027re bigger than 5,"},{"Start":"05:08.210 ","End":"05:14.960","Text":"we\u0027re also bigger than 0 so it\u0027s just equal to the integral"},{"Start":"05:14.960 ","End":"05:23.280","Text":"of 0.2e^ minus t over 5dt from 5 to infinity."},{"Start":"05:23.280 ","End":"05:26.710","Text":"Now, we\u0027ve already done the indefinite integral over here,"},{"Start":"05:26.710 ","End":"05:30.945","Text":"so we can just reuse this result,"},{"Start":"05:30.945 ","End":"05:35.370","Text":"e^ minus t over 5,"},{"Start":"05:35.370 ","End":"05:39.630","Text":"but instead of 0 and infinity,"},{"Start":"05:39.630 ","End":"05:44.975","Text":"we\u0027d have 5 and infinity and reversed also."},{"Start":"05:44.975 ","End":"05:47.510","Text":"This is equal to,"},{"Start":"05:47.510 ","End":"05:51.230","Text":"if I plug in t equals 5,"},{"Start":"05:51.230 ","End":"05:54.629","Text":"I\u0027ve got e^ minus 1,"},{"Start":"05:54.629 ","End":"06:00.525","Text":"1 over e. When I plug an infinity to the minus infinity is 0 like here,"},{"Start":"06:00.525 ","End":"06:06.875","Text":"the result for b is the probability is 1 over e,"},{"Start":"06:06.875 ","End":"06:12.215","Text":"it\u0027s about between 1/2 and a 1/3."},{"Start":"06:12.215 ","End":"06:16.575","Text":"Now we\u0027re just left with part c, the mean."},{"Start":"06:16.575 ","End":"06:22.945","Text":"For part c, we want the integral."},{"Start":"06:22.945 ","End":"06:27.110","Text":"Now we could say from minus infinity to infinity,"},{"Start":"06:27.110 ","End":"06:35.580","Text":"but just take the integral from 0 to infinity and it\u0027s xf of x dx,"},{"Start":"06:35.630 ","End":"06:39.290","Text":"perhaps I\u0027ll just indicate this was minus infinity,"},{"Start":"06:39.290 ","End":"06:43.550","Text":"but I ignored that because anyway,"},{"Start":"06:43.550 ","End":"06:50.900","Text":"I\u0027m using t. This is equal to the integral from 0 to infinity."},{"Start":"06:50.900 ","End":"06:55.520","Text":"Now we have f of t, from 0 to infinity it\u0027s this."},{"Start":"06:55.520 ","End":"06:59.360","Text":"What we have is 0.2,"},{"Start":"06:59.360 ","End":"07:05.340","Text":"but this time we have an extra te^ minus t over 5dt."},{"Start":"07:08.390 ","End":"07:13.640","Text":"I don\u0027t want to waste time solving integrals,"},{"Start":"07:13.640 ","End":"07:19.505","Text":"I\u0027ll just tell you that you can do it using the technique of integration by parts,"},{"Start":"07:19.505 ","End":"07:23.730","Text":"and I\u0027m just going to quote the result for you."},{"Start":"07:24.010 ","End":"07:35.960","Text":"That this is equal to minus t plus 5 e^ minus t over 5,"},{"Start":"07:35.960 ","End":"07:44.430","Text":"and this is what we have to take from 0 to infinity."},{"Start":"07:44.430 ","End":"07:46.880","Text":"You know me, I don\u0027t like to have"},{"Start":"07:46.880 ","End":"07:51.710","Text":"this minus so I usually just get rid of the minus and I switch"},{"Start":"07:51.710 ","End":"07:55.985","Text":"the upper and lower limit so I\u0027ll take it as t plus"},{"Start":"07:55.985 ","End":"08:01.935","Text":"5 e^ minus t over 5 without the minus,"},{"Start":"08:01.935 ","End":"08:07.415","Text":"and just do it from infinity to 0."},{"Start":"08:07.415 ","End":"08:09.320","Text":"Let\u0027s see what we get."},{"Start":"08:09.320 ","End":"08:14.705","Text":"If we plug in t equals 0,"},{"Start":"08:14.705 ","End":"08:18.185","Text":"e^ minus 0 is 1,"},{"Start":"08:18.185 ","End":"08:26.240","Text":"and 0 plus 5 is 5 so I have 5."},{"Start":"08:26.240 ","End":"08:29.350","Text":"When I plug in infinity,"},{"Start":"08:29.350 ","End":"08:35.660","Text":"it\u0027s a strange limit because e^ minus infinity is"},{"Start":"08:35.660 ","End":"08:44.720","Text":"0 and infinity plus 5 is infinity so we do get an infinity times 0 case."},{"Start":"08:44.720 ","End":"08:48.650","Text":"Now I\u0027m claiming that this is equal to 0 anyway,"},{"Start":"08:48.650 ","End":"08:50.210","Text":"and I\u0027ll show you that in a moment,"},{"Start":"08:50.210 ","End":"08:53.975","Text":"so it\u0027s 5 minus 0, so it\u0027s 5."},{"Start":"08:53.975 ","End":"08:59.810","Text":"So 5 minutes is the average waiting time."},{"Start":"08:59.810 ","End":"09:03.455","Text":"There is something which you might find peculiar, but really isn\u0027t."},{"Start":"09:03.455 ","End":"09:07.520","Text":"The average waiting time is 5 minutes, but in part b,"},{"Start":"09:07.520 ","End":"09:12.590","Text":"we computed the probability of waiting 5 minutes or more,"},{"Start":"09:12.590 ","End":"09:18.995","Text":"and that came out to be 1 over e. It wasn\u0027t a 50-50 or 1/2 probability."},{"Start":"09:18.995 ","End":"09:24.530","Text":"The probability of waiting longer than the average is actually less than a 1/2,"},{"Start":"09:24.530 ","End":"09:26.330","Text":"I was just pointing that out."},{"Start":"09:26.330 ","End":"09:30.410","Text":"I still owe you the limit here,"},{"Start":"09:30.410 ","End":"09:34.640","Text":"and basically what we can do is use"},{"Start":"09:34.640 ","End":"09:40.500","Text":"L\u0027Hopital\u0027s rule because what we have here is substitute infinity,"},{"Start":"09:40.500 ","End":"09:42.185","Text":"as I say it\u0027s really a limit."},{"Start":"09:42.185 ","End":"09:47.090","Text":"We need the limit as t goes to infinity of this,"},{"Start":"09:47.090 ","End":"09:52.325","Text":"but because it\u0027s a infinity times 0 case,"},{"Start":"09:52.325 ","End":"09:56.400","Text":"we convert it into either infinity over infinity or 0 over 0,"},{"Start":"09:56.400 ","End":"10:03.440","Text":"and the easiest way to do this is to put this in the denominator so we have t"},{"Start":"10:03.440 ","End":"10:11.120","Text":"plus 5 over e^t over 5 without the minus."},{"Start":"10:11.120 ","End":"10:15.290","Text":"Now, this time we have an infinity over infinity,"},{"Start":"10:15.290 ","End":"10:17.150","Text":"this is equal to,"},{"Start":"10:17.150 ","End":"10:23.130","Text":"by L\u0027Hopital\u0027s rule for the infinity over infinity case."},{"Start":"10:23.950 ","End":"10:30.330","Text":"The shorthand notation L for, remember L\u0027Hopital\u0027s rule?"},{"Start":"10:30.330 ","End":"10:36.515","Text":"It\u0027s a French mathematician who said that if you have infinity over infinity or 0 over 0,"},{"Start":"10:36.515 ","End":"10:38.840","Text":"you can differentiate the numerator and"},{"Start":"10:38.840 ","End":"10:41.750","Text":"denominator separately and that will give the same limit."},{"Start":"10:41.750 ","End":"10:46.020","Text":"We have the limit as t goes to infinity."},{"Start":"10:46.020 ","End":"10:48.580","Text":"The derivative of this is 1,"},{"Start":"10:48.580 ","End":"10:57.785","Text":"the derivative of this is 1/5e^t over 5."},{"Start":"10:57.785 ","End":"11:03.000","Text":"Now, it\u0027s clearly a case of 1 over infinity,"},{"Start":"11:03.000 ","End":"11:04.844","Text":"and this is equal to 0,"},{"Start":"11:04.844 ","End":"11:09.075","Text":"and that\u0027s the 0 here so we verified that."},{"Start":"11:09.075 ","End":"11:12.209","Text":"We\u0027ve answered all the parts of the question,"},{"Start":"11:12.209 ","End":"11:17.550","Text":"and this is now the end of the lesson."}],"Thumbnail":null,"ID":8304}],"ID":6164},{"Name":"Mean Value Theorem for Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Mean Value Theorem for Integrals","Duration":"6m 11s","ChapterTopicVideoID":8422,"CourseChapterTopicPlaylistID":6165,"HasSubtitles":false,"VideoComments":[],"Subtitles":[],"Thumbnail":null,"ID":8616},{"Watched":false,"Name":"Exercise MVT","Duration":"2m 40s","ChapterTopicVideoID":8423,"CourseChapterTopicPlaylistID":6165,"HasSubtitles":false,"VideoComments":[],"Subtitles":[],"Thumbnail":null,"ID":8617}],"ID":6165},{"Name":"Average Function Value","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Average Function Value","Duration":"7m 4s","ChapterTopicVideoID":8424,"CourseChapterTopicPlaylistID":6166,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"In this clip, starts a new topic."},{"Start":"00:02.790 ","End":"00:04.020","Text":"It\u0027s a small topic,"},{"Start":"00:04.020 ","End":"00:09.480","Text":"something you should know about though, average function value."},{"Start":"00:09.480 ","End":"00:11.310","Text":"The setup is this."},{"Start":"00:11.310 ","End":"00:16.650","Text":"We\u0027re given a function f of x on some interval from a to b."},{"Start":"00:16.650 ","End":"00:19.740","Text":"Say x is between a and b,"},{"Start":"00:19.740 ","End":"00:22.560","Text":"although we sometimes use interval notation to"},{"Start":"00:22.560 ","End":"00:26.445","Text":"say that f is defined on the interval a, b."},{"Start":"00:26.445 ","End":"00:31.920","Text":"Then the average, I\u0027ll give it a name,"},{"Start":"00:31.920 ","End":"00:34.620","Text":"f_avg, although it\u0027s not standard,"},{"Start":"00:34.620 ","End":"00:43.070","Text":"but we could just say here the average of the function f on the interval a,"},{"Start":"00:43.070 ","End":"00:48.929","Text":"b is 1 over b minus a,"},{"Start":"00:48.929 ","End":"00:51.470","Text":"because I\u0027m assuming that a is not equal to b,"},{"Start":"00:51.470 ","End":"00:53.900","Text":"that b is bigger than a,"},{"Start":"00:53.900 ","End":"01:03.195","Text":"times the integral from a to b of f of x dx."},{"Start":"01:03.195 ","End":"01:06.255","Text":"I think I\u0027ll highlight this."},{"Start":"01:06.255 ","End":"01:09.280","Text":"Now I\u0027ll give you some intuition of what this is about."},{"Start":"01:09.280 ","End":"01:12.445","Text":"Let me sketch some axis and say,"},{"Start":"01:12.445 ","End":"01:20.035","Text":"a y-axis and an x-axis,"},{"Start":"01:20.035 ","End":"01:24.090","Text":"and let\u0027s say we have 2 points,"},{"Start":"01:24.090 ","End":"01:29.470","Text":"a and b on the x-axis,"},{"Start":"01:29.540 ","End":"01:32.595","Text":"and we have a function."},{"Start":"01:32.595 ","End":"01:35.850","Text":"Let\u0027s say something like this."},{"Start":"01:35.850 ","End":"01:37.575","Text":"I\u0027m going to change my mind,"},{"Start":"01:37.575 ","End":"01:44.169","Text":"I\u0027ll take something more like this and it\u0027s defined from a to b may be further."},{"Start":"01:44.169 ","End":"01:49.320","Text":"I\u0027ll label it y equals f of x,"},{"Start":"01:49.320 ","End":"01:54.275","Text":"draw a vertical line from a and from"},{"Start":"01:54.275 ","End":"02:00.815","Text":"b and now I have a closed region and it has a certain area."},{"Start":"02:00.815 ","End":"02:04.340","Text":"Now 1 way you might look at the average is to say, yeah,"},{"Start":"02:04.340 ","End":"02:07.955","Text":"okay, the value of the function changes that\u0027s the red line."},{"Start":"02:07.955 ","End":"02:11.629","Text":"But what if I was to flatten out this area,"},{"Start":"02:11.629 ","End":"02:17.280","Text":"keep the area but make it rectangular and the height here is such"},{"Start":"02:17.280 ","End":"02:23.340","Text":"that the area of the rectangle,"},{"Start":"02:23.340 ","End":"02:31.440","Text":"I\u0027m doing it in orange will be the same as the shaded area in light blue,"},{"Start":"02:31.440 ","End":"02:35.430","Text":"whatever it is and this value here,"},{"Start":"02:35.430 ","End":"02:38.455","Text":"what we call f_avg,"},{"Start":"02:38.455 ","End":"02:42.620","Text":"the average value of f. Just write it in words."},{"Start":"02:42.620 ","End":"02:46.580","Text":"This is the average value of the function along here."},{"Start":"02:46.580 ","End":"02:51.170","Text":"I still have to tell you how this picture relates to this formula."},{"Start":"02:51.170 ","End":"02:54.200","Text":"I\u0027ll just quickly go through this."},{"Start":"02:54.200 ","End":"02:56.015","Text":"I mean, you know geometry,"},{"Start":"02:56.015 ","End":"03:00.860","Text":"area of a rectangle is base times height and so you could look at it the"},{"Start":"03:00.860 ","End":"03:06.185","Text":"other way that height is area over base."},{"Start":"03:06.185 ","End":"03:08.645","Text":"Now this part here is the area,"},{"Start":"03:08.645 ","End":"03:12.740","Text":"the integral, the base is b minus a."},{"Start":"03:12.740 ","End":"03:15.440","Text":"So area over base would be"},{"Start":"03:15.440 ","End":"03:20.875","Text":"average height and that\u0027s how we get from this picture to this."},{"Start":"03:20.875 ","End":"03:22.790","Text":"Okay. That\u0027s the intuition."},{"Start":"03:22.790 ","End":"03:26.060","Text":"You can forget about that if it doesn\u0027t help and just take"},{"Start":"03:26.060 ","End":"03:29.960","Text":"the formula definition of the average of a function."},{"Start":"03:29.960 ","End":"03:34.615","Text":"You integrate it and divide by the width of the interval."},{"Start":"03:34.615 ","End":"03:36.500","Text":"There\u0027s not really much to it,"},{"Start":"03:36.500 ","End":"03:42.170","Text":"but we\u0027ll do an example anyway just to see if we can follow this."},{"Start":"03:42.170 ","End":"03:52.985","Text":"I want to find the average of the function f of x"},{"Start":"03:52.985 ","End":"03:58.505","Text":"equals the square root of x on"},{"Start":"03:58.505 ","End":"04:05.320","Text":"the interval from 0 to 4."},{"Start":"04:05.320 ","End":"04:06.530","Text":"Or if you prefer,"},{"Start":"04:06.530 ","End":"04:11.160","Text":"you can write it x between 0 and 4."},{"Start":"04:11.160 ","End":"04:12.690","Text":"We have f of x."},{"Start":"04:12.690 ","End":"04:14.330","Text":"We also need b and a,"},{"Start":"04:14.330 ","End":"04:15.530","Text":"which is just this."},{"Start":"04:15.530 ","End":"04:17.405","Text":"This is a and this is b."},{"Start":"04:17.405 ","End":"04:22.310","Text":"The average of f is equal to 1 over,"},{"Start":"04:22.310 ","End":"04:29.285","Text":"now b minus a would be 4 minus 0 times the integral"},{"Start":"04:29.285 ","End":"04:38.675","Text":"from 0 to 4 of the square root of x dx."},{"Start":"04:38.675 ","End":"04:42.620","Text":"I\u0027ll do this indefinite integral, the side."},{"Start":"04:42.620 ","End":"04:45.320","Text":"Let\u0027s see, the integral of the square root of"},{"Start":"04:45.320 ","End":"04:52.170","Text":"x dx would be the integral of x to the power of 1/2."},{"Start":"04:52.190 ","End":"04:56.570","Text":"Now what do we do? We raise the power by 1,"},{"Start":"04:56.570 ","End":"04:59.855","Text":"that makes it x to the power of 1 and 1/2,"},{"Start":"04:59.855 ","End":"05:04.670","Text":"which is 3 over 2 and then we have to divide by the 3 over 2."},{"Start":"05:04.670 ","End":"05:10.535","Text":"But dividing by a fraction is like multiplying by the inverse fraction."},{"Start":"05:10.535 ","End":"05:13.880","Text":"Strictly speaking, we have to write a plus C here,"},{"Start":"05:13.880 ","End":"05:20.520","Text":"but we don\u0027t use it when we compute a definite integral, it cancels."},{"Start":"05:20.590 ","End":"05:27.275","Text":"We have to keep the 1 over 4 and then what we have"},{"Start":"05:27.275 ","End":"05:32.915","Text":"is 2/3x to the 3 over"},{"Start":"05:32.915 ","End":"05:39.475","Text":"2 evaluated from 0 to 4."},{"Start":"05:39.475 ","End":"05:45.200","Text":"At 0, we just get nothing because 0 to the 3 over 2,"},{"Start":"05:45.200 ","End":"05:47.825","Text":"0 to anything is 0."},{"Start":"05:47.825 ","End":"05:50.660","Text":"So we just have to plug in 4."},{"Start":"05:50.660 ","End":"05:59.785","Text":"We get 1/4 times 2/3 times 4 to the 3 over 2,"},{"Start":"05:59.785 ","End":"06:01.880","Text":"like I said, minus 0,"},{"Start":"06:01.880 ","End":"06:04.920","Text":"which is what we would get if we plug this in."},{"Start":"06:05.300 ","End":"06:14.540","Text":"Do a bit of canceling because 2 goes into 4 twice and another side exercise,"},{"Start":"06:14.540 ","End":"06:19.310","Text":"4 to the 3 over 2 is 4 to the 1 and 1/2,"},{"Start":"06:19.310 ","End":"06:23.070","Text":"which is 4 times 4 to the 1/2,"},{"Start":"06:24.140 ","End":"06:28.845","Text":"which is 4 root 4,"},{"Start":"06:28.845 ","End":"06:33.910","Text":"which is 4 times 2, which is 8."},{"Start":"06:37.630 ","End":"06:41.000","Text":"I didn\u0027t really need to cancel these."},{"Start":"06:41.000 ","End":"06:46.620","Text":"I\u0027ll just replace this by 8 and the other stuff I\u0027ll just copy."},{"Start":"06:47.950 ","End":"06:51.330","Text":"Now 2 times 8 is 16 over 4 is 4."},{"Start":"06:51.330 ","End":"06:56.330","Text":"So the answer is 4 over 3 is"},{"Start":"06:56.330 ","End":"07:02.300","Text":"the average value of the square root of x on the interval from 0 to 4."},{"Start":"07:02.300 ","End":"07:05.070","Text":"We\u0027re done for now."}],"Thumbnail":null,"ID":8618},{"Watched":false,"Name":"Exercise AFV","Duration":"3m 30s","ChapterTopicVideoID":8425,"CourseChapterTopicPlaylistID":6166,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"This exercise, it\u0027s a 2-in-1,"},{"Start":"00:03.285 ","End":"00:08.819","Text":"we have to find the average value of each of these functions over the given intervals."},{"Start":"00:08.819 ","End":"00:10.950","Text":"Just to make it a bit different,"},{"Start":"00:10.950 ","End":"00:13.350","Text":"instead of always having f of x,"},{"Start":"00:13.350 ","End":"00:16.320","Text":"the first one is the function g and the second one,"},{"Start":"00:16.320 ","End":"00:18.000","Text":"the variable I called it r."},{"Start":"00:18.000 ","End":"00:23.070","Text":"I\u0027m not going to repeat the definition of the average value."},{"Start":"00:23.070 ","End":"00:24.450","Text":"I\u0027m assuming you know it."},{"Start":"00:24.450 ","End":"00:27.160","Text":"Let\u0027s just get started."},{"Start":"00:27.160 ","End":"00:30.040","Text":"It\u0027s pretty straightforward."},{"Start":"00:30.040 ","End":"00:36.010","Text":"The average, let\u0027s call it g average, avg,"},{"Start":"00:36.010 ","End":"00:43.475","Text":"is equal to, we take 1 over the difference of the endpoints, 4 minus 1."},{"Start":"00:43.475 ","End":"00:52.910","Text":"Then the integral over this interval of the function, 1/x dx."},{"Start":"00:52.910 ","End":"00:56.255","Text":"Now, the rest of it is just computation."},{"Start":"00:56.255 ","End":"00:59.930","Text":"We have 1/3."},{"Start":"00:59.930 ","End":"01:06.905","Text":"Then 1/x, its integral is natural logarithm of x."},{"Start":"01:06.905 ","End":"01:11.495","Text":"We want to take this between 1 and 4."},{"Start":"01:11.495 ","End":"01:14.675","Text":"Now natural log of 1 is 0,"},{"Start":"01:14.675 ","End":"01:23.375","Text":"so the answer is just equal to 1/3 natural log of 4."},{"Start":"01:23.375 ","End":"01:27.769","Text":"That\u0027s all there is to it. Onto part b."},{"Start":"01:27.769 ","End":"01:34.010","Text":"In this case, we have the function f. We want its average."},{"Start":"01:34.010 ","End":"01:36.350","Text":"Same thing."},{"Start":"01:36.350 ","End":"01:39.340","Text":"We take 1 over,"},{"Start":"01:39.340 ","End":"01:43.485","Text":"this time it\u0027s 6 minus 1 from here."},{"Start":"01:43.485 ","End":"01:56.340","Text":"Then the integral from 1 to 6 of 3/1 plus r squared dr."},{"Start":"01:56.340 ","End":"02:00.930","Text":"Just moved over here more space."},{"Start":"02:00.930 ","End":"02:07.040","Text":"This is 1/5, but I can also take the 3 outside the integral."},{"Start":"02:07.040 ","End":"02:09.335","Text":"I\u0027ve got 3/5."},{"Start":"02:09.335 ","End":"02:24.150","Text":"Now, the integral of 1/1 plus r squared is minus 1/1 plus r."},{"Start":"02:24.150 ","End":"02:30.095","Text":"I have to take this from 1 to 6,"},{"Start":"02:30.095 ","End":"02:33.310","Text":"that will give me 3/5."},{"Start":"02:33.310 ","End":"02:36.480","Text":"Now, when I plug in 6,"},{"Start":"02:36.480 ","End":"02:43.275","Text":"I have minus 1/7."},{"Start":"02:43.275 ","End":"02:46.510","Text":"When I plug in 1,"},{"Start":"02:46.510 ","End":"02:54.820","Text":"I have minus 1/1 plus 1 is 2."},{"Start":"02:54.890 ","End":"03:01.480","Text":"Let\u0027s see. Here I have plus a 1/2 minus a 7th."},{"Start":"03:02.180 ","End":"03:07.535","Text":"This subtraction, I can do it in common denominator 14."},{"Start":"03:07.535 ","End":"03:14.925","Text":"I got 7/14 minus 2/14, which is 5/14."},{"Start":"03:14.925 ","End":"03:19.444","Text":"Then the 5 cancel so"},{"Start":"03:19.444 ","End":"03:27.120","Text":"the answer is just 3/14."},{"Start":"03:27.120 ","End":"03:28.550","Text":"The answer to part a,"},{"Start":"03:28.550 ","End":"03:31.770","Text":"the answer for part b, and we\u0027re done."}],"Thumbnail":null,"ID":8619}],"ID":6166},{"Name":"Numerical Integration","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Numerical Integration","Duration":"14m 11s","ChapterTopicVideoID":8378,"CourseChapterTopicPlaylistID":6167,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.430","Text":"In this clip, we\u0027re starting a new topic of numerical integration."},{"Start":"00:05.430 ","End":"00:12.059","Text":"It\u0027s a numerical method for approximating the value of definite integrals."},{"Start":"00:12.059 ","End":"00:15.000","Text":"Now why do we need the approximation?"},{"Start":"00:15.000 ","End":"00:16.725","Text":"Well, the thing is this,"},{"Start":"00:16.725 ","End":"00:25.065","Text":"suppose that I wanted to find the integral of e^x squared."},{"Start":"00:25.065 ","End":"00:27.615","Text":"Let\u0027s say the anti-derivative,"},{"Start":"00:27.615 ","End":"00:30.970","Text":"the primitive of e to the x squared."},{"Start":"00:32.630 ","End":"00:36.725","Text":"In fact there is no formula for such an integral."},{"Start":"00:36.725 ","End":"00:39.890","Text":"With derivatives, pretty much anything you can differentiate,"},{"Start":"00:39.890 ","End":"00:41.990","Text":"but integrals you often get stuck."},{"Start":"00:41.990 ","End":"00:47.600","Text":"If I had a definite integral like 0 to 2,"},{"Start":"00:47.600 ","End":"00:52.160","Text":"the usual method of taking a primitive of this, substituting 2,"},{"Start":"00:52.160 ","End":"00:54.035","Text":"substituting 0 and subtracting,"},{"Start":"00:54.035 ","End":"00:59.615","Text":"won\u0027t work because we just don\u0027t know how to find an indefinite integral of e^x squared,"},{"Start":"00:59.615 ","End":"01:02.630","Text":"what we do is the next best thing,"},{"Start":"01:02.630 ","End":"01:10.530","Text":"is to find an approximation to this numerical value and there are various techniques,"},{"Start":"01:10.530 ","End":"01:14.700","Text":"we\u0027re going to learn 3 of them and the first"},{"Start":"01:14.700 ","End":"01:20.659","Text":"1 will be something called the Midpoint rule,"},{"Start":"01:20.659 ","End":"01:23.810","Text":"3 will be midpoint rule,"},{"Start":"01:23.810 ","End":"01:28.235","Text":"trapezoidal rule and Simpson\u0027s rule anyway, 1 at a time."},{"Start":"01:28.235 ","End":"01:33.545","Text":"But I won\u0027t just be discussing this particular definite integral, in general,"},{"Start":"01:33.545 ","End":"01:41.170","Text":"we\u0027ll discuss the integral from a to b of f of x d x,"},{"Start":"01:41.170 ","End":"01:46.295","Text":"where presumably f of x is difficult to integrate,"},{"Start":"01:46.295 ","End":"01:50.070","Text":"although, it could be any f of x."},{"Start":"01:50.140 ","End":"01:52.505","Text":"The idea is this,"},{"Start":"01:52.505 ","End":"01:55.940","Text":"we take the interval ab,"},{"Start":"01:55.940 ","End":"01:59.755","Text":"the closed interval, written like this,"},{"Start":"01:59.755 ","End":"02:04.910","Text":"and we divide this interval into n equal pieces,"},{"Start":"02:04.910 ","End":"02:07.740","Text":"I\u0027ll talk more about that in a moment,"},{"Start":"02:09.370 ","End":"02:13.620","Text":"with each piece we\u0027ll have width delta x,"},{"Start":"02:13.620 ","End":"02:16.310","Text":"let\u0027s call it, which will be b minus a,"},{"Start":"02:16.310 ","End":"02:21.460","Text":"which is the total width divided by n. Then"},{"Start":"02:21.460 ","End":"02:28.980","Text":"this ab becomes n separate contiguous sub-intervals,"},{"Start":"02:28.980 ","End":"02:33.540","Text":"let\u0027s call them x0 to x1."},{"Start":"02:33.540 ","End":"02:38.725","Text":"Next interval from x1 to x2,"},{"Start":"02:38.725 ","End":"02:40.540","Text":"and so on and so on."},{"Start":"02:40.540 ","End":"02:47.915","Text":"The last an nth interval will be from x sub n minus 1 to x sub n,"},{"Start":"02:47.915 ","End":"02:55.390","Text":"where of course, this is a and this is b."},{"Start":"02:55.390 ","End":"02:57.800","Text":"Now each of these n intervals has"},{"Start":"02:57.800 ","End":"03:01.610","Text":"a midpoint and this is why it\u0027s called the midpoint rule,"},{"Start":"03:01.610 ","End":"03:03.650","Text":"and it\u0027s called the midpoint."},{"Start":"03:03.650 ","End":"03:07.645","Text":"For this 1 it will be x1 asterisk,"},{"Start":"03:07.645 ","End":"03:10.505","Text":"here we\u0027ll have x2 asterisk,"},{"Start":"03:10.505 ","End":"03:15.155","Text":"and so on up to x n with an asterisk."},{"Start":"03:15.155 ","End":"03:18.770","Text":"Of course, we compute each 1 by taking this plus this over 2,"},{"Start":"03:18.770 ","End":"03:19.910","Text":"it means the midpoint,"},{"Start":"03:19.910 ","End":"03:21.470","Text":"this plus this over 2,"},{"Start":"03:21.470 ","End":"03:24.055","Text":"I won\u0027t write that formula down."},{"Start":"03:24.055 ","End":"03:28.010","Text":"There are actually 2 lesser known rules where we"},{"Start":"03:28.010 ","End":"03:31.895","Text":"take the left-hand point and there\u0027s a rule for the right end point,"},{"Start":"03:31.895 ","End":"03:35.755","Text":"but we\u0027ll just use the rule for the midpoint."},{"Start":"03:35.755 ","End":"03:38.475","Text":"Once we\u0027ve done this,"},{"Start":"03:38.475 ","End":"03:40.370","Text":"yeah, brought in a picture,"},{"Start":"03:40.370 ","End":"03:45.365","Text":"it\u0027s going to really help. Here we are."},{"Start":"03:45.365 ","End":"03:51.475","Text":"This is the interval from a to b,"},{"Start":"03:51.475 ","End":"03:53.000","Text":"and we\u0027ve divided it up."},{"Start":"03:53.000 ","End":"03:56.645","Text":"In this case, we took n equals 6,"},{"Start":"03:56.645 ","End":"03:59.630","Text":"I still have to explain about the choice of n,"},{"Start":"03:59.630 ","End":"04:01.100","Text":"but in this case,"},{"Start":"04:01.100 ","End":"04:05.870","Text":"so we have 6 separate intervals,"},{"Start":"04:05.870 ","End":"04:08.000","Text":"and for each of these,"},{"Start":"04:08.000 ","End":"04:12.385","Text":"I draw a rectangle."},{"Start":"04:12.385 ","End":"04:14.360","Text":"Well, let me just take 1 as an example."},{"Start":"04:14.360 ","End":"04:18.230","Text":"Suppose I take the interval from x2 to x3."},{"Start":"04:18.230 ","End":"04:22.160","Text":"This here is x3 asterisk."},{"Start":"04:22.160 ","End":"04:27.650","Text":"Then I look at the value of the function, which is here,"},{"Start":"04:27.650 ","End":"04:35.100","Text":"I should mention that this is the graph y equals f of x as here."},{"Start":"04:35.120 ","End":"04:39.135","Text":"What we do is we take the y of this."},{"Start":"04:39.135 ","End":"04:46.875","Text":"In this case, y would be f of x3 asterisk."},{"Start":"04:46.875 ","End":"04:51.405","Text":"That\u0027s the y value over here, somewhere."},{"Start":"04:51.405 ","End":"04:57.050","Text":"What we\u0027re doing, is we\u0027re estimating the area under the curve,"},{"Start":"04:57.050 ","End":"05:00.905","Text":"which would be this area from here to here."},{"Start":"05:00.905 ","End":"05:04.815","Text":"Everything underneath this curve,"},{"Start":"05:04.815 ","End":"05:06.945","Text":"if I shaded it, it would look messy."},{"Start":"05:06.945 ","End":"05:14.365","Text":"But underneath this red curve between a and b and the x axis."},{"Start":"05:14.365 ","End":"05:18.755","Text":"We do this by using these rectangles,"},{"Start":"05:18.755 ","End":"05:24.589","Text":"the sum of the areas of the rectangles as an approximation to the area under the curve."},{"Start":"05:24.589 ","End":"05:31.805","Text":"Of course, the width of each rectangle is this delta x,"},{"Start":"05:31.805 ","End":"05:34.370","Text":"which is over here."},{"Start":"05:34.370 ","End":"05:36.755","Text":"If we add all these up,"},{"Start":"05:36.755 ","End":"05:38.960","Text":"we\u0027ll get that the area,"},{"Start":"05:38.960 ","End":"05:45.240","Text":"which is the integral from a to b of f of x d x,"},{"Start":"05:45.240 ","End":"05:49.790","Text":"the area under the curve is equal, no,"},{"Start":"05:49.790 ","End":"05:57.410","Text":"is approximately equal to the area of the first rectangle, width times height."},{"Start":"05:57.410 ","End":"06:07.795","Text":"The width is delta x and the height is f at the point x1 asterisk."},{"Start":"06:07.795 ","End":"06:11.645","Text":"Next 1 also width delta x,"},{"Start":"06:11.645 ","End":"06:14.960","Text":"but f of x2 asterisk,"},{"Start":"06:14.960 ","End":"06:18.380","Text":"and so on and so on up to the last 1,"},{"Start":"06:18.380 ","End":"06:25.734","Text":"which is also width delta x times f of the last,"},{"Start":"06:25.734 ","End":"06:27.840","Text":"well, here it shows with 6,"},{"Start":"06:27.840 ","End":"06:33.940","Text":"but this is in general n f of x n asterisk."},{"Start":"06:33.940 ","End":"06:37.370","Text":"Now, delta x is common to all of these terms."},{"Start":"06:37.370 ","End":"06:41.230","Text":"I can take it outside the brackets,"},{"Start":"06:41.230 ","End":"06:43.190","Text":"and this is what we get."},{"Start":"06:43.190 ","End":"06:49.865","Text":"Just take the delta x out and we have the sum of all the f of the x asterisks."},{"Start":"06:49.865 ","End":"06:53.000","Text":"That is the midpoint rule."},{"Start":"06:53.000 ","End":"06:56.075","Text":"For those who like sigma notation,"},{"Start":"06:56.075 ","End":"07:03.140","Text":"I could write this as the integral from a to b of f of"},{"Start":"07:03.140 ","End":"07:12.170","Text":"x d x is approximately equal to delta x times the sum."},{"Start":"07:12.170 ","End":"07:23.105","Text":"Let\u0027s say i goes from 1-n of f of x_i asterisk."},{"Start":"07:23.105 ","End":"07:28.025","Text":"It\u0027s an approximation and we\u0027ll see an example."},{"Start":"07:28.025 ","End":"07:30.335","Text":"But 1 thing I want to say,"},{"Start":"07:30.335 ","End":"07:33.750","Text":"I want to get back to this topic of n."},{"Start":"07:34.570 ","End":"07:42.575","Text":"The choice of n is such that the larger the n,"},{"Start":"07:42.575 ","End":"07:45.395","Text":"the more accurate the approximation is,"},{"Start":"07:45.395 ","End":"07:48.785","Text":"but also the more difficult the calculation becomes."},{"Start":"07:48.785 ","End":"07:51.875","Text":"It\u0027s a trade-off. You can get more accurate,"},{"Start":"07:51.875 ","End":"07:53.960","Text":"but you have to work harder."},{"Start":"07:53.960 ","End":"07:57.515","Text":"We\u0027ll continue to the next rule,"},{"Start":"07:57.515 ","End":"08:00.230","Text":"so I\u0027ll erase what I don\u0027t need."},{"Start":"08:00.230 ","End":"08:07.590","Text":"The next rule will be the trapezoid rule."},{"Start":"08:09.580 ","End":"08:20.370","Text":"As before, we divide the interval into n equal intervals."},{"Start":"08:20.650 ","End":"08:25.310","Text":"As before, the larger the value of n,"},{"Start":"08:25.310 ","End":"08:26.600","Text":"the more accurate it will be,"},{"Start":"08:26.600 ","End":"08:29.300","Text":"but also more computations."},{"Start":"08:29.300 ","End":"08:36.230","Text":"The picture here, and this is a bit different from the previous time."},{"Start":"08:36.230 ","End":"08:42.215","Text":"We still have n pieces from a to b."},{"Start":"08:42.215 ","End":"08:44.450","Text":"This is a and this is b."},{"Start":"08:44.450 ","End":"08:48.989","Text":"We demonstrated here with n equals 6,"},{"Start":"08:50.830 ","End":"08:55.460","Text":"the graph y equals f of x,"},{"Start":"08:55.460 ","End":"08:59.495","Text":"where the f is the f from here."},{"Start":"08:59.495 ","End":"09:08.690","Text":"We want the area under the graph between a and b bordered by the x axis."},{"Start":"09:08.690 ","End":"09:18.095","Text":"What we do this time is we join each of the values that the x_1, x_2, x_3,"},{"Start":"09:18.095 ","End":"09:19.835","Text":"the x_i in general,"},{"Start":"09:19.835 ","End":"09:26.150","Text":"and we connect them with straight lines so that each bar"},{"Start":"09:26.150 ","End":"09:32.885","Text":"is a trapezoid shape from here to here."},{"Start":"09:32.885 ","End":"09:36.110","Text":"We use the geometric formula for the trapezoid."},{"Start":"09:36.110 ","End":"09:38.720","Text":"This time, I\u0027m going to show you the whole development,"},{"Start":"09:38.720 ","End":"09:42.870","Text":"I\u0027ll just give you the end result."},{"Start":"09:43.180 ","End":"09:46.565","Text":"This time, this is what it looks like."},{"Start":"09:46.565 ","End":"09:48.395","Text":"We take Delta x."},{"Start":"09:48.395 ","End":"09:56.645","Text":"In the picture, Delta x is the width of each of these bars and it\u0027s given by this,"},{"Start":"09:56.645 ","End":"10:00.365","Text":"and then we divide it by 2."},{"Start":"10:00.365 ","End":"10:06.920","Text":"We add all the values of f and multiply them all by 2,"},{"Start":"10:06.920 ","End":"10:08.720","Text":"except for the first and last."},{"Start":"10:08.720 ","End":"10:11.329","Text":"Here I have a 1 and a 1 in the coefficient,"},{"Start":"10:11.329 ","End":"10:13.715","Text":"here I have a 2 everywhere."},{"Start":"10:13.715 ","End":"10:16.805","Text":"For those who wanted in Sigma notation,"},{"Start":"10:16.805 ","End":"10:19.700","Text":"if I combine the first and the last,"},{"Start":"10:19.700 ","End":"10:25.025","Text":"I\u0027ll get f of x naught plus"},{"Start":"10:25.025 ","End":"10:31.355","Text":"f of x_n over 2."},{"Start":"10:31.355 ","End":"10:34.680","Text":"I\u0027ll leave the Delta x to the end."},{"Start":"10:35.680 ","End":"10:37.970","Text":"For the rest of them,"},{"Start":"10:37.970 ","End":"10:39.380","Text":"I\u0027ll get the Sigma,"},{"Start":"10:39.380 ","End":"10:41.465","Text":"the 2 will cancel with the 1/2,"},{"Start":"10:41.465 ","End":"10:50.120","Text":"I\u0027ll get i goes from 1 up to n minus 1 of f of x_i."},{"Start":"10:50.120 ","End":"10:55.415","Text":"All this is multiplied by Delta x."},{"Start":"10:55.415 ","End":"11:00.364","Text":"But I think we just go with the dot dot dot version."},{"Start":"11:00.364 ","End":"11:05.735","Text":"Just be easier, but just for those who like it in Sigma without the dot dot dot,"},{"Start":"11:05.735 ","End":"11:09.810","Text":"ellipsis, it\u0027s called, there it is."},{"Start":"11:09.880 ","End":"11:16.625","Text":"We will see the numerical examples soon."},{"Start":"11:16.625 ","End":"11:21.320","Text":"I\u0027ll just get to the last of the 3 rules."},{"Start":"11:21.320 ","End":"11:25.055","Text":"Let me erase again what I don\u0027t need."},{"Start":"11:25.055 ","End":"11:32.760","Text":"The last of the 3 rules is called Simpson\u0027s rule."},{"Start":"11:33.940 ","End":"11:37.100","Text":"Here\u0027s the picture and as before,"},{"Start":"11:37.100 ","End":"11:43.580","Text":"this is a and this is b and the width of each of these is Delta x,"},{"Start":"11:43.580 ","End":"11:46.355","Text":"which is given by the same formula as above."},{"Start":"11:46.355 ","End":"11:48.860","Text":"There is 1 restriction on this rule,"},{"Start":"11:48.860 ","End":"11:53.670","Text":"Simpson\u0027s rule, is that n has to be even."},{"Start":"11:53.670 ","End":"12:02.775","Text":"What this is based on is taking it in pairs and each pair of bars is 3 points."},{"Start":"12:02.775 ","End":"12:08.390","Text":"I\u0027ll illustrate on the last 3 points on this color."},{"Start":"12:08.390 ","End":"12:15.700","Text":"Of course the red line is our y equals f of x and we want"},{"Start":"12:15.700 ","End":"12:19.810","Text":"the area under the graph between the graph on"},{"Start":"12:19.810 ","End":"12:25.010","Text":"the x-axis and bounded by x equals a and x equals b."},{"Start":"12:25.010 ","End":"12:32.885","Text":"What we do with these 3 points is to connect them with a parabola."},{"Start":"12:32.885 ","End":"12:35.209","Text":"That\u0027s for these."},{"Start":"12:35.209 ","End":"12:42.365","Text":"Similarly, this is a parabola even though it may look like a straight line."},{"Start":"12:42.365 ","End":"12:48.840","Text":"Also here we have a piece of parabola."},{"Start":"12:49.090 ","End":"12:53.675","Text":"I\u0027m not going to go into the details of how we get the final formula,"},{"Start":"12:53.675 ","End":"12:56.165","Text":"I\u0027ll just present it."},{"Start":"12:56.165 ","End":"13:01.295","Text":"This time, the rule comes out a little bit more complicated."},{"Start":"13:01.295 ","End":"13:06.530","Text":"What we have for an approximation for the definite integral,"},{"Start":"13:06.530 ","End":"13:08.645","Text":"the area under the red curve,"},{"Start":"13:08.645 ","End":"13:13.010","Text":"is Delta x over 3."},{"Start":"13:13.010 ","End":"13:18.830","Text":"We have here f of x naught up to f of x_n."},{"Start":"13:18.830 ","End":"13:21.559","Text":"But the coefficients are a bit peculiar,"},{"Start":"13:21.559 ","End":"13:23.584","Text":"the first and the last are 1,"},{"Start":"13:23.584 ","End":"13:25.580","Text":"and all the rest alternate: 4,"},{"Start":"13:25.580 ","End":"13:30.470","Text":"2 and so on and ending in 2, 4."},{"Start":"13:30.470 ","End":"13:33.500","Text":"Just alternates 4, 2, 4, 2."},{"Start":"13:33.500 ","End":"13:35.540","Text":"This is the formula."},{"Start":"13:35.540 ","End":"13:40.340","Text":"In general, the Simpson\u0027s rule gives the best results from"},{"Start":"13:40.340 ","End":"13:46.220","Text":"the 3 estimations we had if you remember: the midpoint rule,"},{"Start":"13:46.220 ","End":"13:48.890","Text":"the trapezoid rule, and Simpson\u0027s rule."},{"Start":"13:48.890 ","End":"13:52.230","Text":"This is the most recommended."},{"Start":"13:53.950 ","End":"13:59.660","Text":"The solved exercises will be in separate clips."},{"Start":"13:59.660 ","End":"14:07.760","Text":"Hopefully, I\u0027ll do this particular 1 using all 3 approximation methods."},{"Start":"14:07.760 ","End":"14:12.510","Text":"But meanwhile, I am done here."}],"Thumbnail":null,"ID":8501},{"Watched":false,"Name":"Numerical Integration - Error Bound","Duration":"12m ","ChapterTopicVideoID":8379,"CourseChapterTopicPlaylistID":6167,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In the main tutorial clip,"},{"Start":"00:02.310 ","End":"00:06.540","Text":"we discussed the concept of numerical integration for"},{"Start":"00:06.540 ","End":"00:11.440","Text":"a function which was difficult to integrate by regular methods,"},{"Start":"00:11.440 ","End":"00:16.740","Text":"and we had to use numerical methods on the example we gave was e^x squared."},{"Start":"00:16.740 ","End":"00:19.200","Text":"We don\u0027t know it\u0027s indefinite integral,"},{"Start":"00:19.200 ","End":"00:23.190","Text":"but there are numeric methods to compute it,"},{"Start":"00:23.190 ","End":"00:25.380","Text":"say between zero and 2."},{"Start":"00:25.380 ","End":"00:30.480","Text":"In exercise 1, we actually did it using"},{"Start":"00:30.480 ","End":"00:37.710","Text":"3 different methods involving a certain n which is cutting up the interval a,"},{"Start":"00:37.710 ","End":"00:42.280","Text":"b into n pieces of equal width."},{"Start":"00:42.800 ","End":"00:49.135","Text":"I\u0027d like to just show you the summary of results we got for this exercise."},{"Start":"00:49.135 ","End":"00:51.380","Text":"We used the 3 methods,"},{"Start":"00:51.380 ","End":"00:55.910","Text":"the midpoint rule, the trapezoid rule, and Simpson\u0027s rule."},{"Start":"00:55.910 ","End":"01:00.005","Text":"We\u0027ve got 3 different answers using each of them."},{"Start":"01:00.005 ","End":"01:01.850","Text":"Now, in this case,"},{"Start":"01:01.850 ","End":"01:04.520","Text":"I produced like a rabbit out of the hat,"},{"Start":"01:04.520 ","End":"01:07.205","Text":"the actual value, because there is"},{"Start":"01:07.205 ","End":"01:12.845","Text":"more advanced software that will do such things to say 8 decimal places or whatever."},{"Start":"01:12.845 ","End":"01:14.660","Text":"In this particular case,"},{"Start":"01:14.660 ","End":"01:16.505","Text":"when we know the actual value,"},{"Start":"01:16.505 ","End":"01:18.980","Text":"then we can say what the error is."},{"Start":"01:18.980 ","End":"01:22.640","Text":"I just called them error midpoint, error trapezoid,"},{"Start":"01:22.640 ","End":"01:28.580","Text":"error Simpson, and the smallest was the Simpson rule method."},{"Start":"01:28.580 ","End":"01:32.510","Text":"But in actual fact,"},{"Start":"01:32.510 ","End":"01:34.205","Text":"when we do such an exercise,"},{"Start":"01:34.205 ","End":"01:42.245","Text":"we don\u0027t have the real value and we don\u0027t know what the error is."},{"Start":"01:42.245 ","End":"01:44.990","Text":"But we\u0027d like to bound it,"},{"Start":"01:44.990 ","End":"01:49.295","Text":"given up a bound to say that the error\u0027s going to be at most so and so,"},{"Start":"01:49.295 ","End":"01:52.700","Text":"so we can say with some confidence what the actual value might"},{"Start":"01:52.700 ","End":"01:57.170","Text":"be and hopefully not too far from what we\u0027ve estimated it as,"},{"Start":"01:57.170 ","End":"02:00.590","Text":"and that\u0027s the topic of this clip."},{"Start":"02:00.590 ","End":"02:04.910","Text":"For cases where I don\u0027t have an actual value instead of the actual error,"},{"Start":"02:04.910 ","End":"02:10.615","Text":"at least I can bound the error and say the error is no more than such and such a value."},{"Start":"02:10.615 ","End":"02:13.160","Text":"I\u0027m going to give you some formulas."},{"Start":"02:13.160 ","End":"02:17.060","Text":"Now we\u0027re going to assume this is technical matters,"},{"Start":"02:17.060 ","End":"02:21.080","Text":"but we have to assume that f double prime of x is"},{"Start":"02:21.080 ","End":"02:28.605","Text":"continuous on our interval which is a,b in general."},{"Start":"02:28.605 ","End":"02:32.855","Text":"Then we also have to assume,"},{"Start":"02:32.855 ","End":"02:37.790","Text":"at least for the Simpson\u0027s rule,"},{"Start":"02:37.790 ","End":"02:41.240","Text":"that we also have the fourth derivative."},{"Start":"02:41.240 ","End":"02:47.270","Text":"I can write it in Roman numerals or f prime,"},{"Start":"02:47.270 ","End":"02:51.380","Text":"prime, prime, prime, or f with a 4."},{"Start":"02:51.380 ","End":"02:58.685","Text":"Let\u0027s go with the foreign brackets of x. I guess we\u0027d like this to be continuous."},{"Start":"02:58.685 ","End":"03:01.805","Text":"This one is used for the Simpson\u0027s rule."},{"Start":"03:01.805 ","End":"03:06.365","Text":"We only need the second derivatives for the midpoint and trapezoid."},{"Start":"03:06.365 ","End":"03:10.865","Text":"Now suppose that on the interval a, b,"},{"Start":"03:10.865 ","End":"03:14.060","Text":"that when x is in,"},{"Start":"03:14.060 ","End":"03:15.890","Text":"well, let me put it this way."},{"Start":"03:15.890 ","End":"03:19.520","Text":"I\u0027ll use regular notation not interval notation,"},{"Start":"03:19.520 ","End":"03:21.770","Text":"that for all x in a, b,"},{"Start":"03:21.770 ","End":"03:27.770","Text":"let\u0027s say we find a constant K,"},{"Start":"03:27.770 ","End":"03:33.980","Text":"such that the second derivative is always less than K on this interval."},{"Start":"03:33.980 ","End":"03:37.865","Text":"Also if we\u0027re going to use estimate for Simpson\u0027 rule,"},{"Start":"03:37.865 ","End":"03:44.210","Text":"we\u0027ll need to bound the fourth derivative of x on"},{"Start":"03:44.210 ","End":"03:51.860","Text":"the interval by some M. The moment we\u0027ll get into how do we find K and M,"},{"Start":"03:51.860 ","End":"03:57.755","Text":"the quick answer is that you could take these to be the maximum values"},{"Start":"03:57.755 ","End":"04:04.130","Text":"of the double derivative or the fourth derivative on the interval."},{"Start":"04:04.130 ","End":"04:08.060","Text":"I\u0027m using maximum-minimum methods."},{"Start":"04:08.060 ","End":"04:10.910","Text":"If we find these K and these M,"},{"Start":"04:10.910 ","End":"04:13.130","Text":"then we can estimate,"},{"Start":"04:13.130 ","End":"04:17.275","Text":"we\u0027ll call them E_M, E_T, and E_S."},{"Start":"04:17.275 ","End":"04:21.740","Text":"Here are the 3 formulas for the 3 errors."},{"Start":"04:21.740 ","End":"04:27.170","Text":"Notice that the midpoint and trapezoid ones involve K and the"},{"Start":"04:27.170 ","End":"04:33.780","Text":"Simpson one involves the constant M that we found."},{"Start":"04:33.860 ","End":"04:42.049","Text":"The absolute value is here because sometimes we give a sign to the error plus or minus,"},{"Start":"04:42.049 ","End":"04:45.710","Text":"take the estimated value minus the actual value."},{"Start":"04:45.710 ","End":"04:47.875","Text":"In this case, for example,"},{"Start":"04:47.875 ","End":"04:53.030","Text":"I\u0027ll put a minus in front because this minus, this is negative."},{"Start":"04:53.030 ","End":"04:55.055","Text":"If you do the error that way,"},{"Start":"04:55.055 ","End":"04:56.600","Text":"then it could be plus or minus,"},{"Start":"04:56.600 ","End":"04:57.620","Text":"so that would be safer,"},{"Start":"04:57.620 ","End":"04:59.855","Text":"just put it in bars."},{"Start":"04:59.855 ","End":"05:01.970","Text":"This will give us some estimates."},{"Start":"05:01.970 ","End":"05:08.000","Text":"Now what I want to do is actually compute this for our case,"},{"Start":"05:08.000 ","End":"05:13.020","Text":"where as above, a would be zero,"},{"Start":"05:13.020 ","End":"05:14.670","Text":"b would equal 2,"},{"Start":"05:14.670 ","End":"05:16.080","Text":"I\u0027m reading off here."},{"Start":"05:16.080 ","End":"05:18.755","Text":"The function of x,"},{"Start":"05:18.755 ","End":"05:19.790","Text":"although isn\u0027t written here,"},{"Start":"05:19.790 ","End":"05:25.620","Text":"but let\u0027s just say the function of x is e^x squared."},{"Start":"05:26.080 ","End":"05:31.040","Text":"We also know that n is equal to 4."},{"Start":"05:31.040 ","End":"05:34.640","Text":"I want to put the arrow bounds and I\u0027ll"},{"Start":"05:34.640 ","End":"05:38.990","Text":"compute all 3 of them for the 3 cases and put them in the table,"},{"Start":"05:38.990 ","End":"05:47.790","Text":"and let\u0027s just see if our arrows really do fall within the bounds, and yeah."},{"Start":"05:48.250 ","End":"05:52.380","Text":"Let\u0027s start looking for K and M. First we\u0027ll need"},{"Start":"05:52.380 ","End":"05:59.010","Text":"the second and fourth derivatives of f of x."},{"Start":"05:59.010 ","End":"06:05.630","Text":"Now, I\u0027ll leave it to you as an exercise since differentiation is not that difficult,"},{"Start":"06:05.630 ","End":"06:06.680","Text":"you know how do this."},{"Start":"06:06.680 ","End":"06:16.370","Text":"That the second derivative of x is equal to 2e_x squared,"},{"Start":"06:16.370 ","End":"06:24.920","Text":"1 plus 2x squared, and f,"},{"Start":"06:24.920 ","End":"06:30.785","Text":"fourth derivative of x"},{"Start":"06:30.785 ","End":"06:36.980","Text":"is equal to 4e^x squared,"},{"Start":"06:36.980 ","End":"06:45.210","Text":"3 plus 12x squared plus 4x^4."},{"Start":"06:46.330 ","End":"06:56.765","Text":"Now what we want to find K and M are the global maximum values on the interval from 0-2."},{"Start":"06:56.765 ","End":"06:58.265","Text":"Now if you look at them,"},{"Start":"06:58.265 ","End":"07:02.180","Text":"these are both increasing functions."},{"Start":"07:02.180 ","End":"07:05.075","Text":"So there\u0027s no need to do all kinds of derivative stuff."},{"Start":"07:05.075 ","End":"07:08.585","Text":"The maximum occurs at the right endpoint."},{"Start":"07:08.585 ","End":"07:18.920","Text":"Let me just write something that they are both increasing functions,"},{"Start":"07:18.920 ","End":"07:24.330","Text":"and so global max"},{"Start":"07:24.550 ","End":"07:32.225","Text":"occurs at the right endpoint at x equals 2."},{"Start":"07:32.225 ","End":"07:37.465","Text":"What I want to do is to figure out"},{"Start":"07:37.465 ","End":"07:43.919","Text":"f double-prime of 2 and see what this equals."},{"Start":"07:43.919 ","End":"07:51.945","Text":"Also, f quadruple prime fourth derivative"},{"Start":"07:51.945 ","End":"07:57.555","Text":"when x is 2 and just plugging in 2."},{"Start":"07:57.555 ","End":"08:00.600","Text":"Here we get 982,"},{"Start":"08:00.600 ","End":"08:02.525","Text":"of course we use a calculator."},{"Start":"08:02.525 ","End":"08:08.555","Text":"0.7666 looks like it just keeps on with sixes,"},{"Start":"08:08.555 ","End":"08:12.590","Text":"and the fourth derivative comes out considerably"},{"Start":"08:12.590 ","End":"08:20.245","Text":"larger to"},{"Start":"08:20.245 ","End":"08:25.250","Text":"25,115.14901 something."},{"Start":"08:25.250 ","End":"08:31.340","Text":"We can be generous and just round up"},{"Start":"08:31.340 ","End":"08:39.200","Text":"and take K to equal 983."},{"Start":"08:39.200 ","End":"08:42.875","Text":"That will certainly do if it\u0027s less than or equal to this."},{"Start":"08:42.875 ","End":"08:48.275","Text":"There\u0027s not much difference and not wasting much in my error estimation."},{"Start":"08:48.275 ","End":"08:57.600","Text":"M we shall take as 25,116."},{"Start":"08:57.600 ","End":"09:00.350","Text":"Now we can get our 3 bounds."},{"Start":"09:00.350 ","End":"09:07.370","Text":"We can say that the error using the midpoint rule is less than or equal to."},{"Start":"09:07.370 ","End":"09:11.630","Text":"I\u0027m looking here now, 983."},{"Start":"09:11.630 ","End":"09:15.535","Text":"Now b minus a is 2."},{"Start":"09:15.535 ","End":"09:25.875","Text":"It\u0027s 2 minus zero times 2 cubed over 24,"},{"Start":"09:25.875 ","End":"09:31.330","Text":"and n we had was 4 squared."},{"Start":"09:31.550 ","End":"09:39.460","Text":"This comes out to 20.479 something."},{"Start":"09:40.790 ","End":"09:48.305","Text":"The error using the trapezoid rule it\u0027ll be less than or equal to,"},{"Start":"09:48.305 ","End":"09:52.035","Text":"it\u0027s very similar, it\u0027s just that there\u0027s 12 here."},{"Start":"09:52.035 ","End":"09:58.525","Text":"I\u0027ll just erase the 24 and write the 12."},{"Start":"09:58.525 ","End":"10:02.620","Text":"This comes out roughly"},{"Start":"10:02.620 ","End":"10:11.055","Text":"twice this, 40.958 something."},{"Start":"10:11.055 ","End":"10:17.780","Text":"The last one, the Simpson estimation,"},{"Start":"10:17.780 ","End":"10:22.835","Text":"the error bound for using Simpson\u0027s rule"},{"Start":"10:22.835 ","End":"10:33.410","Text":"would be 25,116 times."},{"Start":"10:33.410 ","End":"10:40.625","Text":"This times 2^5 over 180,"},{"Start":"10:40.625 ","End":"10:44.280","Text":"and n is 4^4,"},{"Start":"10:44.280 ","End":"10:53.110","Text":"and this comes out roughly 17.441 something."},{"Start":"10:53.110 ","End":"11:03.340","Text":"Now if I take these values and I\u0027m going to scroll back up and place them in the table."},{"Start":"11:03.340 ","End":"11:06.940","Text":"Here they are, I put those 3 values we computed in the table,"},{"Start":"11:06.940 ","End":"11:09.370","Text":"and you can see that"},{"Start":"11:09.370 ","End":"11:17.815","Text":"the actual errors are much smaller than the error bounds."},{"Start":"11:17.815 ","End":"11:19.420","Text":"Normally we wouldn\u0027t have these."},{"Start":"11:19.420 ","End":"11:21.220","Text":"As I said, you don\u0027t have the actual value."},{"Start":"11:21.220 ","End":"11:25.845","Text":"We have these, but these are like real worst-case scenario,"},{"Start":"11:25.845 ","End":"11:28.690","Text":"and in practice we do much better."},{"Start":"11:28.690 ","End":"11:31.210","Text":"But if you want to be sure,"},{"Start":"11:31.210 ","End":"11:33.370","Text":"you use these error bound."},{"Start":"11:33.370 ","End":"11:37.230","Text":"If you don\u0027t have anything else that\u0027s maybe better than nothing."},{"Start":"11:37.230 ","End":"11:39.785","Text":"Especially when n gets large,"},{"Start":"11:39.785 ","End":"11:41.360","Text":"these things get much smaller,"},{"Start":"11:41.360 ","End":"11:43.580","Text":"as you can see from these denominators,"},{"Start":"11:43.580 ","End":"11:47.900","Text":"when n gets larger, especially the Simpson, gets very small."},{"Start":"11:47.900 ","End":"11:51.600","Text":"The error estimate when n gets large."},{"Start":"11:52.330 ","End":"12:01.890","Text":"That\u0027s it on bounding the error, and we\u0027re done."}],"Thumbnail":null,"ID":8502},{"Watched":false,"Name":"Exercise1","Duration":"9m 53s","ChapterTopicVideoID":8380,"CourseChapterTopicPlaylistID":6167,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.295","Text":"In this exercise, I\u0027ve used the very same integral we used in the tutorial."},{"Start":"00:05.295 ","End":"00:09.375","Text":"The integral from 0 to 2 of e^x squared dx."},{"Start":"00:09.375 ","End":"00:11.265","Text":"I\u0027m going to estimate it,"},{"Start":"00:11.265 ","End":"00:13.695","Text":"using each of the 3 rules we learned,"},{"Start":"00:13.695 ","End":"00:17.280","Text":"the midpoint rule, the trapezoid rule, and Simpson\u0027s rule."},{"Start":"00:17.280 ","End":"00:22.790","Text":"But we also have to know how many sub-intervals to divide into,"},{"Start":"00:22.790 ","End":"00:26.040","Text":"and we\u0027re given that this is equal to 4."},{"Start":"00:26.040 ","End":"00:28.470","Text":"It has to be even for Simpson\u0027s rule."},{"Start":"00:28.470 ","End":"00:30.105","Text":"For the rest, it could be odd or even,"},{"Start":"00:30.105 ","End":"00:32.340","Text":"but 4 is fine."},{"Start":"00:32.340 ","End":"00:36.440","Text":"It\u0027s pretty small for an accurate estimation,"},{"Start":"00:36.440 ","End":"00:40.430","Text":"but we want it to be easy to compute and just to demonstrate the method,"},{"Start":"00:40.430 ","End":"00:43.075","Text":"so we\u0027re not expecting great accuracy."},{"Start":"00:43.075 ","End":"00:46.640","Text":"Now, there are various software tools that compute much"},{"Start":"00:46.640 ","End":"00:49.990","Text":"more accurately such kind of integrals."},{"Start":"00:49.990 ","End":"00:53.535","Text":"Here\u0027s a spoiler, that to 8 decimal places,"},{"Start":"00:53.535 ","End":"00:55.455","Text":"this is what the answer should be."},{"Start":"00:55.455 ","End":"00:58.790","Text":"But we\u0027ll just use it for comparison purposes when we do it"},{"Start":"00:58.790 ","End":"01:02.870","Text":"using our methods of these or them."},{"Start":"01:02.870 ","End":"01:06.465","Text":"Let\u0027s start with the midpoint rule."},{"Start":"01:06.465 ","End":"01:08.540","Text":"So for all 4 methods,"},{"Start":"01:08.540 ","End":"01:13.995","Text":"we have to divide the interval from 0 to 2."},{"Start":"01:13.995 ","End":"01:18.370","Text":"We have to divide it into 4 sub-intervals,"},{"Start":"01:18.370 ","End":"01:20.860","Text":"and in this case,"},{"Start":"01:20.860 ","End":"01:28.390","Text":"and in all of the cases we\u0027ll get, let\u0027s say, 0-1/2."},{"Start":"01:28.390 ","End":"01:34.885","Text":"Perhaps I should add that we also had the formula that Delta x is b minus a,"},{"Start":"01:34.885 ","End":"01:41.750","Text":"so it\u0027s 2 minus 0/4, which is 1.2."},{"Start":"01:41.750 ","End":"01:44.145","Text":"So each interval is 1/2 wide."},{"Start":"01:44.145 ","End":"01:47.580","Text":"The next 1 is from 1/2 to 1."},{"Start":"01:47.580 ","End":"01:51.795","Text":"The next interval from 1 to 1 1/2,"},{"Start":"01:51.795 ","End":"01:58.305","Text":"and the fourth interval from 1 1/2 to 2."},{"Start":"01:58.305 ","End":"02:02.405","Text":"This will serve us for all 3 cases."},{"Start":"02:02.405 ","End":"02:03.980","Text":"But in the midpoint rule,"},{"Start":"02:03.980 ","End":"02:07.690","Text":"we also have to find the midpoint of each of these intervals."},{"Start":"02:07.690 ","End":"02:11.375","Text":"Now, I brought in the formula for the midpoint rule,"},{"Start":"02:11.375 ","End":"02:17.990","Text":"and it requires these x_i asterisk for the various x_i,"},{"Start":"02:17.990 ","End":"02:19.879","Text":"and those are the midpoints,"},{"Start":"02:19.879 ","End":"02:23.840","Text":"so here the midpoint will be 1/4."},{"Start":"02:23.840 ","End":"02:27.635","Text":"Here, it\u0027s halfway between this and this, it\u0027s 3.4."},{"Start":"02:27.635 ","End":"02:30.850","Text":"You can work in fraction or decimal, it doesn\u0027t really matter."},{"Start":"02:30.850 ","End":"02:33.860","Text":"Here, we have 1 1.4,"},{"Start":"02:33.860 ","End":"02:36.930","Text":"and here we have 1 3/4."},{"Start":"02:36.930 ","End":"02:38.625","Text":"That\u0027s x_1, x_2, x_3,"},{"Start":"02:38.625 ","End":"02:41.584","Text":"x_4, in our case, with an asterisk."},{"Start":"02:41.584 ","End":"02:43.280","Text":"Actually, when I think about it,"},{"Start":"02:43.280 ","End":"02:44.880","Text":"it\u0027s better to go with decimal,"},{"Start":"02:44.880 ","End":"02:46.310","Text":"so I don\u0027t need to erase these."},{"Start":"02:46.310 ","End":"02:52.680","Text":"I\u0027ll just add the decimal value to each of these, 1.25 and 1.75."},{"Start":"02:52.680 ","End":"02:54.470","Text":"Now, using this formula,"},{"Start":"02:54.470 ","End":"02:58.800","Text":"we get that the integral from 0 to 2,"},{"Start":"02:58.800 ","End":"03:05.195","Text":"f of x is e^x squared dx is going to be approximately equal to,"},{"Start":"03:05.195 ","End":"03:08.530","Text":"now Delta x is 1/2,"},{"Start":"03:08.530 ","End":"03:10.605","Text":"and then we get this sum,"},{"Start":"03:10.605 ","End":"03:13.275","Text":"and this is going to be 4 terms because n is 4."},{"Start":"03:13.275 ","End":"03:18.975","Text":"We\u0027ll have e^0.25 squared."},{"Start":"03:18.975 ","End":"03:20.850","Text":"It\u0027s just e^x squared,"},{"Start":"03:20.850 ","End":"03:29.340","Text":"where x takes each of these values plus e^0.75 squared"},{"Start":"03:29.340 ","End":"03:34.185","Text":"plus e^1.25"},{"Start":"03:34.185 ","End":"03:40.995","Text":"squared plus e^1.75 squared."},{"Start":"03:40.995 ","End":"03:43.020","Text":"If we do this computation,"},{"Start":"03:43.020 ","End":"03:46.565","Text":"I\u0027m not going to go into that level of detail,"},{"Start":"03:46.565 ","End":"03:51.700","Text":"and this comes out to be 14.48561253."},{"Start":"03:59.810 ","End":"04:03.990","Text":"That\u0027s the first method, the midpoint."},{"Start":"04:03.990 ","End":"04:08.070","Text":"Next, we\u0027ll go on to the trapezoid."},{"Start":"04:08.070 ","End":"04:10.960","Text":"I\u0027ll just highlight them."},{"Start":"04:11.260 ","End":"04:14.510","Text":"For the trapezoid method,"},{"Start":"04:14.510 ","End":"04:17.760","Text":"we use a different formula,"},{"Start":"04:18.460 ","End":"04:21.845","Text":"and here\u0027s the formula."},{"Start":"04:21.845 ","End":"04:24.730","Text":"We get that the integral,"},{"Start":"04:24.730 ","End":"04:32.690","Text":"in our case from 0 to 2 e^x squared dx is approximately equal to."},{"Start":"04:32.690 ","End":"04:36.380","Text":"Now we have Delta x/2."},{"Start":"04:36.380 ","End":"04:39.770","Text":"Delta x is 1/2,"},{"Start":"04:39.770 ","End":"04:45.210","Text":"so this is going to be 1/2,"},{"Start":"04:45.210 ","End":"04:46.800","Text":"1/2 over 2 is 1/4."},{"Start":"04:46.800 ","End":"04:50.745","Text":"Then we\u0027re going to have e to the."},{"Start":"04:50.745 ","End":"04:57.635","Text":"Now, these x_0 through x_n are just these numbers from the intervals."},{"Start":"04:57.635 ","End":"05:01.285","Text":"They are, like this is 0,"},{"Start":"05:01.285 ","End":"05:07.110","Text":"1/2, 1, and so on."},{"Start":"05:07.270 ","End":"05:17.100","Text":"I mean, we\u0027ve almost got them all here, 1 1/2, 2."},{"Start":"05:17.100 ","End":"05:20.695","Text":"This was almost fitted to n equals 4."},{"Start":"05:20.695 ","End":"05:26.150","Text":"In any event, you put all the endpoints, the first, the last,"},{"Start":"05:26.150 ","End":"05:28.970","Text":"and the middle subdivisions,"},{"Start":"05:28.970 ","End":"05:32.465","Text":"and we take e^x squared,"},{"Start":"05:32.465 ","End":"05:34.175","Text":"the function of x for each of them."},{"Start":"05:34.175 ","End":"05:40.075","Text":"So we have e^0 squared plus,"},{"Start":"05:40.075 ","End":"05:44.765","Text":"only thing to notice is that there are 2s in the middle except for the first and last,"},{"Start":"05:44.765 ","End":"05:47.250","Text":"so we need 2e^1/2,"},{"Start":"05:49.820 ","End":"05:52.620","Text":"and we\u0027ll work in decimal here,"},{"Start":"05:52.620 ","End":"06:01.070","Text":"0.5 squared plus twice e^1 squared plus twice"},{"Start":"06:01.070 ","End":"06:11.290","Text":"e^1.5 squared plus once, only e^2 squared."},{"Start":"06:11.290 ","End":"06:17.580","Text":"If we do this computation, we get 20.64455905."},{"Start":"06:25.070 ","End":"06:28.710","Text":"I\u0027ll highlight that\u0027s 2 out of 3."},{"Start":"06:28.710 ","End":"06:31.805","Text":"Now, after we\u0027ve done the midpoint and the trapezoid,"},{"Start":"06:31.805 ","End":"06:36.800","Text":"we\u0027ll go and do the Simpson\u0027s rule."},{"Start":"06:36.800 ","End":"06:43.830","Text":"I\u0027ll write that down using Simpson\u0027s rule,"},{"Start":"06:43.830 ","End":"06:47.700","Text":"and I\u0027ll bring in the formula for that."},{"Start":"06:47.700 ","End":"06:51.245","Text":"Here\u0027s this formula, it\u0027s a bit lengthier."},{"Start":"06:51.245 ","End":"06:55.350","Text":"It\u0027s similar to this one, but not quite."},{"Start":"06:55.350 ","End":"06:56.765","Text":"Well, we\u0027ll see the differences."},{"Start":"06:56.765 ","End":"07:02.810","Text":"What we get is the integral from 0 to 2 of e^x squared dx,"},{"Start":"07:02.810 ","End":"07:04.745","Text":"where this is our f of x,"},{"Start":"07:04.745 ","End":"07:08.770","Text":"is going to equal approximately."},{"Start":"07:08.770 ","End":"07:13.155","Text":"Now Delta x/3 will be 1/6."},{"Start":"07:13.155 ","End":"07:14.580","Text":"It\u0027s just off screen,"},{"Start":"07:14.580 ","End":"07:19.210","Text":"but Delta x was 1/2 and 1/2 over 1/3 is 1/6."},{"Start":"07:19.210 ","End":"07:22.790","Text":"Now we get a similar thing to this,"},{"Start":"07:22.790 ","End":"07:26.100","Text":"except that we have different numbers."},{"Start":"07:26.100 ","End":"07:28.560","Text":"The first and last are also 1,"},{"Start":"07:28.560 ","End":"07:31.830","Text":"but other than that, it alternates between 4 and 2."},{"Start":"07:31.830 ","End":"07:42.630","Text":"So we get e^0 squared plus 4,"},{"Start":"07:42.630 ","End":"07:48.045","Text":"this time e^0.5 squared,"},{"Start":"07:48.045 ","End":"07:49.830","Text":"and then remember 1,"},{"Start":"07:49.830 ","End":"07:51.510","Text":"4, 2, 4, 2, and so on."},{"Start":"07:51.510 ","End":"07:56.415","Text":"Then 2e^1 squared,"},{"Start":"07:56.415 ","End":"08:03.075","Text":"and then again a 4e^1.5 squared."},{"Start":"08:03.075 ","End":"08:08.850","Text":"The last one is always a 1, it\u0027s e^2 squared."},{"Start":"08:08.850 ","End":"08:14.300","Text":"Now, of course, if you\u0027re writing your calculations and saving the intermediate values,"},{"Start":"08:14.300 ","End":"08:19.190","Text":"you could reuse these numbers here without the coefficients."},{"Start":"08:19.190 ","End":"08:20.550","Text":"You could reuse them here."},{"Start":"08:20.550 ","End":"08:25.050","Text":"Anyway, this computation comes out."},{"Start":"08:25.050 ","End":"08:27.765","Text":"Here it is, I just wrote it out."},{"Start":"08:27.765 ","End":"08:31.755","Text":"Now we have, for 3 different methods,"},{"Start":"08:31.755 ","End":"08:34.155","Text":"we have 3 different values."},{"Start":"08:34.155 ","End":"08:41.535","Text":"Let me bring back the actual value from that software."},{"Start":"08:41.535 ","End":"08:43.710","Text":"Here it is again,"},{"Start":"08:43.710 ","End":"08:47.250","Text":"I\u0027ll call it the actual value."},{"Start":"08:47.250 ","End":"08:49.440","Text":"We\u0027re done with the exercise,"},{"Start":"08:49.440 ","End":"08:52.110","Text":"but I just want to say a few more words just to look at them."},{"Start":"08:52.110 ","End":"08:54.080","Text":"They\u0027re all not very good."},{"Start":"08:54.080 ","End":"08:56.600","Text":"We see that the Simpson\u0027s rule is the closest."},{"Start":"08:56.600 ","End":"09:02.179","Text":"One of the main reasons they\u0027re not very good is that n was too small."},{"Start":"09:02.179 ","End":"09:05.195","Text":"But in general, even for small n,"},{"Start":"09:05.195 ","End":"09:09.850","Text":"Simpson\u0027s turns out to be better than the other 2."},{"Start":"09:09.850 ","End":"09:12.165","Text":"Just for interest\u0027s sake,"},{"Start":"09:12.165 ","End":"09:15.710","Text":"besides the actual I brought in in red here,"},{"Start":"09:15.710 ","End":"09:19.730","Text":"the differences between the estimate and the actual."},{"Start":"09:19.730 ","End":"09:22.865","Text":"The error for the midpoint method,"},{"Start":"09:22.865 ","End":"09:25.040","Text":"the error for the trapezoid method,"},{"Start":"09:25.040 ","End":"09:27.860","Text":"and the error for the Simpson method."},{"Start":"09:27.860 ","End":"09:30.980","Text":"Clearly this is the closest,"},{"Start":"09:30.980 ","End":"09:37.220","Text":"but it\u0027s still almost 1 away from the actual."},{"Start":"09:37.220 ","End":"09:42.829","Text":"There is a theory of estimating the errors,"},{"Start":"09:42.829 ","End":"09:44.330","Text":"putting a bound on them,"},{"Start":"09:44.330 ","End":"09:45.830","Text":"but that doesn\u0027t belong here."},{"Start":"09:45.830 ","End":"09:47.845","Text":"Some of you will learn it, some of you not."},{"Start":"09:47.845 ","End":"09:53.400","Text":"Any event, for this exercise I\u0027ve done more than enough, and we\u0027re done."}],"Thumbnail":null,"ID":8503},{"Watched":false,"Name":"Exercise2","Duration":"14m 26s","ChapterTopicVideoID":8381,"CourseChapterTopicPlaylistID":6167,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"This exercise is similar to the previous exercise,"},{"Start":"00:03.360 ","End":"00:06.104","Text":"just a different function, different interval,"},{"Start":"00:06.104 ","End":"00:14.460","Text":"and we have to compute or estimate the integral using the midpoint,"},{"Start":"00:14.460 ","End":"00:17.670","Text":"the trapezoid, and Simpson\u0027s rule."},{"Start":"00:17.670 ","End":"00:23.355","Text":"Now in this case, we actually could compute the integral and get the exact answer,"},{"Start":"00:23.355 ","End":"00:28.120","Text":"but we need to practice in these techniques."},{"Start":"00:29.330 ","End":"00:33.209","Text":"In part a, the midpoint method,"},{"Start":"00:33.209 ","End":"00:37.230","Text":"the interval we have is the interval from 2-4,"},{"Start":"00:37.230 ","End":"00:43.110","Text":"and we divide it into n equals 4 intervals."},{"Start":"00:43.110 ","End":"00:45.065","Text":"Instead of writing the intervals,"},{"Start":"00:45.065 ","End":"00:48.125","Text":"why don\u0027t I just write the end points?"},{"Start":"00:48.125 ","End":"00:57.780","Text":"We have 2, and then if we take Delta x, in all cases,"},{"Start":"00:57.780 ","End":"01:03.845","Text":"will be 4 minus 2 over the number of intervals,"},{"Start":"01:03.845 ","End":"01:06.500","Text":"which is equal to 2/4,"},{"Start":"01:06.500 ","End":"01:12.140","Text":"which is 1/2, so we\u0027ll get 2, 2.5,"},{"Start":"01:12.140 ","End":"01:20.310","Text":"3, 3.5, and 4."},{"Start":"01:20.310 ","End":"01:22.610","Text":"Like the interval, 1 interval is from here to here,"},{"Start":"01:22.610 ","End":"01:25.350","Text":"the other interval is here,"},{"Start":"01:26.290 ","End":"01:28.970","Text":"and the 4th 1 is here."},{"Start":"01:28.970 ","End":"01:31.610","Text":"Now with the midpoint rule,"},{"Start":"01:31.610 ","End":"01:38.670","Text":"we need to find the asterisks, the midpoint values,"},{"Start":"01:38.670 ","End":"01:42.120","Text":"so between here and here,"},{"Start":"01:42.120 ","End":"01:46.830","Text":"I\u0027ve got 2.25, between here and here,"},{"Start":"01:46.830 ","End":"01:54.885","Text":"I\u0027ve got 2.75, between here and here,"},{"Start":"01:54.885 ","End":"02:04.710","Text":"I\u0027ve got 3.25, and the last 1 between this and this, 3.75."},{"Start":"02:04.710 ","End":"02:07.350","Text":"These are my x_1 asterisk,"},{"Start":"02:07.350 ","End":"02:09.865","Text":"x_2 asterisk and so on."},{"Start":"02:09.865 ","End":"02:15.560","Text":"I\u0027ll do it in a bit more detail here than in the previous exercise."},{"Start":"02:15.560 ","End":"02:18.845","Text":"I want to make a table with values,"},{"Start":"02:18.845 ","End":"02:27.335","Text":"and the values are going to be x_1 asterisk,"},{"Start":"02:27.335 ","End":"02:34.235","Text":"x_2 asterisk, x_3 asterisk, x_4 asterisk."},{"Start":"02:34.235 ","End":"02:36.995","Text":"I\u0027m going to put the values themselves,"},{"Start":"02:36.995 ","End":"02:42.360","Text":"and then the value of the function."},{"Start":"02:42.820 ","End":"02:49.680","Text":"This will be 1/2 of x, x/x minus 1,"},{"Start":"02:49.680 ","End":"02:50.940","Text":"so here I\u0027ll have x,"},{"Start":"02:50.940 ","End":"02:52.800","Text":"and here I\u0027ll have f of x,"},{"Start":"02:52.800 ","End":"02:56.070","Text":"so these are just copy from here."},{"Start":"02:56.070 ","End":"03:05.415","Text":"We had 2.25, 2.75, 3.25, 3.75."},{"Start":"03:05.415 ","End":"03:07.710","Text":"Now the function of x,"},{"Start":"03:07.710 ","End":"03:16.120","Text":"which is x/x minus 1 is not that hard to compute."},{"Start":"03:16.520 ","End":"03:24.330","Text":"Well, I\u0027ll write first and then I\u0027ll simplify."},{"Start":"03:24.330 ","End":"03:29.340","Text":"2.25 is x, minus 1 is 1.25."},{"Start":"03:29.340 ","End":"03:37.185","Text":"Here, I have 2.75/1.75, here,"},{"Start":"03:37.185 ","End":"03:46.245","Text":"3.25/2.25, and here 3.75/2.75."},{"Start":"03:46.245 ","End":"03:51.590","Text":"I could actually get these infractions if I multiply top and bottom by 4 in each case,"},{"Start":"03:51.590 ","End":"03:53.060","Text":"and so this comes out to"},{"Start":"03:53.060 ","End":"04:02.045","Text":"be 9/5,"},{"Start":"04:02.045 ","End":"04:08.600","Text":"this comes out to be 11/7."},{"Start":"04:08.600 ","End":"04:11.130","Text":"Multiply this by 4 is 13,"},{"Start":"04:11.130 ","End":"04:17.795","Text":"and then so over 9, and here if I multiply this by 4,"},{"Start":"04:17.795 ","End":"04:20.460","Text":"it\u0027s going to be"},{"Start":"04:20.630 ","End":"04:31.390","Text":"15/11."},{"Start":"04:31.390 ","End":"04:34.735","Text":"That brought in the rule for the midpoint rule,"},{"Start":"04:34.735 ","End":"04:42.230","Text":"and so what we get is that the integral from 2-4 of"},{"Start":"04:42.230 ","End":"04:50.660","Text":"x/x minus 1dx is approximately equal to,"},{"Start":"04:50.660 ","End":"04:52.865","Text":"we can also write it this way,"},{"Start":"04:52.865 ","End":"04:57.990","Text":"Delta x which is 1/2,"},{"Start":"04:57.990 ","End":"05:01.455","Text":"and then we have these which are just here,"},{"Start":"05:01.455 ","End":"05:11.710","Text":"so it\u0027s 9/5 plus 11/7 plus 13/9, plus 15/11."},{"Start":"05:13.030 ","End":"05:16.070","Text":"This part I did on the calculator for you,"},{"Start":"05:16.070 ","End":"05:18.605","Text":"though it\u0027s an easy exercise in fractions,"},{"Start":"05:18.605 ","End":"05:20.660","Text":"and to 8 decimal places,"},{"Start":"05:20.660 ","End":"05:22.280","Text":"this is what it comes out."},{"Start":"05:22.280 ","End":"05:25.610","Text":"Now on to part b."},{"Start":"05:25.610 ","End":"05:31.280","Text":"In part b, I\u0027m also going to do more detail like in part a,"},{"Start":"05:31.280 ","End":"05:32.660","Text":"part of the details."},{"Start":"05:32.660 ","End":"05:37.085","Text":"We\u0027ll also make a table with things that we need."},{"Start":"05:37.085 ","End":"05:39.765","Text":"This time we don\u0027t need the midpoints,"},{"Start":"05:39.765 ","End":"05:44.430","Text":"we need the values themselves,"},{"Start":"05:44.430 ","End":"05:45.900","Text":"is going to be 5 of them."},{"Start":"05:45.900 ","End":"05:50.240","Text":"We\u0027ll need x_0, x_1,"},{"Start":"05:50.240 ","End":"05:56.970","Text":"x_2, x_3, and x_4."},{"Start":"05:56.970 ","End":"05:59.925","Text":"This is the x and this will be the f of x,"},{"Start":"05:59.925 ","End":"06:04.800","Text":"which as we said is x/x minus 1."},{"Start":"06:04.800 ","End":"06:07.575","Text":"I\u0027m just copying the 2,"},{"Start":"06:07.575 ","End":"06:15.760","Text":"2.5, 3, 3.5, and 4."},{"Start":"06:16.040 ","End":"06:19.710","Text":"X/x minus 1, 2/1 is 2,"},{"Start":"06:19.710 ","End":"06:21.855","Text":"2.5/1.5 is 5/3, 3/2 is just 3/2,"},{"Start":"06:21.855 ","End":"06:34.490","Text":"3.5/2.5 is 7/5,"},{"Start":"06:34.490 ","End":"06:40.350","Text":"and 4/3 is just what it is."},{"Start":"06:40.350 ","End":"06:43.905","Text":"Now I\u0027ll bring in the formula."},{"Start":"06:43.905 ","End":"06:46.830","Text":"Here it is. It\u0027s squashed,"},{"Start":"06:46.830 ","End":"06:48.465","Text":"we can read it."},{"Start":"06:48.465 ","End":"06:51.660","Text":"We get that the integral in our case,"},{"Start":"06:51.660 ","End":"06:55.065","Text":"a and b are 2 and 4,"},{"Start":"06:55.065 ","End":"07:00.870","Text":"so f of x is x/x minus 1dx, is going to equal."},{"Start":"07:00.870 ","End":"07:07.330","Text":"Now Delta x/2 is the 1/2 over 2 which is 1/4."},{"Start":"07:09.550 ","End":"07:13.445","Text":"Notice the 2s here,"},{"Start":"07:13.445 ","End":"07:17.989","Text":"so we get these values, which is 2,"},{"Start":"07:17.989 ","End":"07:23.960","Text":"but we need twice 5/3, twice 3/2,"},{"Start":"07:23.960 ","End":"07:32.660","Text":"twice 7/5, and at the end, only once 4/3."},{"Start":"07:32.660 ","End":"07:38.910","Text":"I\u0027ll just spare you the tedium of this last calculation."},{"Start":"07:39.970 ","End":"07:43.490","Text":"To 8 decimal places it comes out as this,"},{"Start":"07:43.490 ","End":"07:46.220","Text":"actually it\u0027s a recurring 6 at the end,"},{"Start":"07:46.220 ","End":"07:49.490","Text":"rounded up to 7 in the last place."},{"Start":"07:49.490 ","End":"07:53.570","Text":"That does part b and now we need the 3rd 1,"},{"Start":"07:53.570 ","End":"07:55.730","Text":"which is Simpson\u0027s rule."},{"Start":"07:55.730 ","End":"07:59.120","Text":"This 1 was the mid-point trapezoid,"},{"Start":"07:59.120 ","End":"08:01.850","Text":"and now make room for Simpson\u0027s rule."},{"Start":"08:01.850 ","End":"08:06.680","Text":"But this time, if you remember,"},{"Start":"08:06.680 ","End":"08:09.650","Text":"we took n equals 8,"},{"Start":"08:09.650 ","End":"08:14.220","Text":"so it\u0027s a little bit different that the values"},{"Start":"08:14.220 ","End":"08:21.250","Text":"the interval 2,4 is split into 8 pieces so we get 2,"},{"Start":"08:21.250 ","End":"08:26.885","Text":"and then Delta x is different here,"},{"Start":"08:26.885 ","End":"08:30.050","Text":"it\u0027s 4 minus 2/8,"},{"Start":"08:30.050 ","End":"08:31.715","Text":"which is a 1/4."},{"Start":"08:31.715 ","End":"08:34.624","Text":"So we go in jumps of a quarter, 2,"},{"Start":"08:34.624 ","End":"08:51.560","Text":"2 1/4, 2 1/2, 2 3/4, 3, 3 1/4, 3 1/2, 3 3/4, 4."},{"Start":"08:51.560 ","End":"08:54.320","Text":"These are the 8 intervals."},{"Start":"08:54.320 ","End":"08:56.825","Text":"This is the first interval, second interval,"},{"Start":"08:56.825 ","End":"09:00.395","Text":"and so on up to the last interval."},{"Start":"09:00.395 ","End":"09:07.775","Text":"This is x_naught and this is x_n and these are the x_1,"},{"Start":"09:07.775 ","End":"09:09.755","Text":"x_2, x_3, and so on."},{"Start":"09:09.755 ","End":"09:13.055","Text":"Now the formula for Simpson\u0027s,"},{"Start":"09:13.055 ","End":"09:16.730","Text":"so here\u0027s the Simpson rule, nice and big."},{"Start":"09:16.730 ","End":"09:19.760","Text":"I\u0027m also going to do it with a table."},{"Start":"09:19.760 ","End":"09:22.655","Text":"The table will be a bit bigger this time."},{"Start":"09:22.655 ","End":"09:26.840","Text":"Here I\u0027ll write x_naught, x_1,"},{"Start":"09:26.840 ","End":"09:28.340","Text":"and here is up to x_n,"},{"Start":"09:28.340 ","End":"09:32.540","Text":"n is 8, that\u0027s x_8."},{"Start":"09:32.540 ","End":"09:36.860","Text":"The values, I\u0027ll just copy them from here."},{"Start":"09:36.860 ","End":"09:44.700","Text":"2, 2 1/4, and so on up to the last 1 which is 4."},{"Start":"09:44.700 ","End":"09:47.650","Text":"This is x, and here we want f of x,"},{"Start":"09:47.650 ","End":"09:52.450","Text":"might as well just write it as x/x minus 1."},{"Start":"09:52.450 ","End":"09:57.995","Text":"So 2/2 minus 1, 2/1 is 2."},{"Start":"09:57.995 ","End":"10:01.080","Text":"I\u0027ll do the whole ones first, they\u0027re easier."},{"Start":"10:01.570 ","End":"10:07.700","Text":"3/2, 4/3, the ones with 1/2 are not so bad."},{"Start":"10:07.700 ","End":"10:12.485","Text":"2 1/2 over 1 1/2 because I can just double so that\u0027s 5/3."},{"Start":"10:12.485 ","End":"10:21.800","Text":"3 1/2 over 2 1/2 is 7/5."},{"Start":"10:21.800 ","End":"10:27.050","Text":"Here I get 2 1/4/1 1/4, 9/5."},{"Start":"10:27.050 ","End":"10:34.940","Text":"This comes out to be 2 3/4/1 3/4 is 11/7,"},{"Start":"10:34.940 ","End":"10:38.690","Text":"here 13/9, here 15/11."},{"Start":"10:38.690 ","End":"10:42.650","Text":"Now it\u0027s time to substitute in the formula."},{"Start":"10:42.650 ","End":"10:51.740","Text":"We get that this integral of ours from 2-4/x/x minus 1 dx is approximately equal to,"},{"Start":"10:51.740 ","End":"10:54.335","Text":"Delta x was 1/4."},{"Start":"10:54.335 ","End":"10:58.310","Text":"1/4/3 is 1/12."},{"Start":"10:58.310 ","End":"11:04.535","Text":"What we get is we add these, but with coefficients."},{"Start":"11:04.535 ","End":"11:06.664","Text":"The first and last are 1,"},{"Start":"11:06.664 ","End":"11:08.150","Text":"and then it alternates 4, 2,"},{"Start":"11:08.150 ","End":"11:09.755","Text":"4, 2, etc."},{"Start":"11:09.755 ","End":"11:12.350","Text":"So we get 2,"},{"Start":"11:12.350 ","End":"11:18.785","Text":"and then 4 times 9/5, twice 5/3."},{"Start":"11:18.785 ","End":"11:27.110","Text":"Again, 4 times 11/7, twice 3/2,"},{"Start":"11:27.110 ","End":"11:37.820","Text":"4 times 13/9, only twice 7/5,"},{"Start":"11:37.820 ","End":"11:42.440","Text":"then 4 times 15/11,"},{"Start":"11:42.440 ","End":"11:49.520","Text":"and the last 1 just 1 as is 4/3."},{"Start":"11:49.520 ","End":"11:51.410","Text":"Exercise in fractions."},{"Start":"11:51.410 ","End":"11:54.605","Text":"I\u0027ll just give you the answer,"},{"Start":"11:54.605 ","End":"11:59.690","Text":"you reduce to 8 decimal places and I\u0027ll highlight it."},{"Start":"11:59.690 ","End":"12:03.785","Text":"Maybe I\u0027ll collect our 3 values together."},{"Start":"12:03.785 ","End":"12:08.060","Text":"Here I collected them near the top, found some space."},{"Start":"12:08.060 ","End":"12:11.510","Text":"This was a, this was b, this was c,"},{"Start":"12:11.510 ","End":"12:19.970","Text":"mid-point trapezoid Simpson, and would be nice to know the actual value."},{"Start":"12:19.970 ","End":"12:23.570","Text":"Officially this lesson ends,"},{"Start":"12:23.570 ","End":"12:27.680","Text":"but if you\u0027re curious to how to do this and get it exactly,"},{"Start":"12:27.680 ","End":"12:29.360","Text":"you\u0027re welcome to stay."},{"Start":"12:29.360 ","End":"12:31.640","Text":"I\u0027ll just move this out of the way."},{"Start":"12:31.640 ","End":"12:35.165","Text":"We can actually find the anti-derivative of this,"},{"Start":"12:35.165 ","End":"12:39.230","Text":"and I did it and I got that it is x minus"},{"Start":"12:39.230 ","End":"12:45.965","Text":"1 plus natural log of x minus 1."},{"Start":"12:45.965 ","End":"12:50.630","Text":"The way I did it, I basically just rewrote the numerator"},{"Start":"12:50.630 ","End":"12:57.035","Text":"as x minus 1 plus 1/x minus 1."},{"Start":"12:57.035 ","End":"13:01.895","Text":"Then this bit came out to be 1,"},{"Start":"13:01.895 ","End":"13:07.580","Text":"and the other bit 1/x minus 1."},{"Start":"13:07.580 ","End":"13:10.640","Text":"There is no minus 1 there."},{"Start":"13:10.640 ","End":"13:15.200","Text":"You see the x minus 1/x minus 1 gives us just 1, and that gives us the x."},{"Start":"13:15.200 ","End":"13:18.320","Text":"1/x minus 1 gives us natural log."},{"Start":"13:18.320 ","End":"13:24.360","Text":"Then if we take all this from 2-4,"},{"Start":"13:24.880 ","End":"13:30.305","Text":"x from 2-4 gives us 4 minus 2 is 2."},{"Start":"13:30.305 ","End":"13:35.585","Text":"Natural log of 4 minus 1 is natural log of 3."},{"Start":"13:35.585 ","End":"13:40.145","Text":"Natural log of 2 minus 1 is natural log of 1, which is 0."},{"Start":"13:40.145 ","End":"13:43.625","Text":"So the answer is 2 plus natural log of 3."},{"Start":"13:43.625 ","End":"13:47.255","Text":"This now you can punch in on the calculator,"},{"Start":"13:47.255 ","End":"13:50.730","Text":"then we get 3.0986122"},{"Start":"13:58.180 ","End":"14:01.310","Text":"and round it to 9."},{"Start":"14:01.310 ","End":"14:07.310","Text":"Notice that it is very close to the Simpson."},{"Start":"14:07.310 ","End":"14:08.990","Text":"There\u0027s 2 reasons: first of all,"},{"Start":"14:08.990 ","End":"14:11.000","Text":"Simpson\u0027s generally more accurate,"},{"Start":"14:11.000 ","End":"14:16.625","Text":"but also we had n equals 8 and more points or more accuracy."},{"Start":"14:16.625 ","End":"14:21.230","Text":"Anyway, so this is 1 we could have done without numerical,"},{"Start":"14:21.230 ","End":"14:23.225","Text":"but we needed the practice."},{"Start":"14:23.225 ","End":"14:26.040","Text":"Now we\u0027re done."}],"Thumbnail":null,"ID":8504}],"ID":6167}]

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