[{"Name":"Double Integrals, Applications","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Computing Area as a Double Integral","Duration":"13m 38s","ChapterTopicVideoID":8459,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8459.jpeg","UploadDate":"2020-02-26T12:12:05.5670000","DurationForVideoObject":"PT13M38S","Description":null,"MetaTitle":"Computing Area as a Double Integral: Video + Workbook | Proprep","MetaDescription":"Double Integrals - Applications - Double Integrals, Applications. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/double-integrals-_-applications/double-integrals%2c-applications/vid8675","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.340","Text":"We\u0027re in the chapter on applications of double integrals,"},{"Start":"00:03.340 ","End":"00:07.800","Text":"and the first 1 will be how to compute an area using a double integral."},{"Start":"00:07.800 ","End":"00:09.420","Text":"I\u0027ll start with a sketch."},{"Start":"00:09.420 ","End":"00:11.325","Text":"I have an x and a y-axis."},{"Start":"00:11.325 ","End":"00:19.410","Text":"Then I\u0027m going to maybe take 2 lines like this and join this to this,"},{"Start":"00:19.410 ","End":"00:21.195","Text":"and this to this,"},{"Start":"00:21.195 ","End":"00:23.280","Text":"and we have here a region,"},{"Start":"00:23.280 ","End":"00:28.020","Text":"let\u0027s call it R. What we would like"},{"Start":"00:28.020 ","End":"00:34.020","Text":"is to find the area of R. What is this equal to?"},{"Start":"00:34.020 ","End":"00:38.670","Text":"Now, you might justifiably say that you don\u0027t need double integrals for this,"},{"Start":"00:38.670 ","End":"00:42.385","Text":"that you know how to compute areas just using single integrals."},{"Start":"00:42.385 ","End":"00:45.095","Text":"Nevertheless, bear with me."},{"Start":"00:45.095 ","End":"00:48.050","Text":"There are advantages for doing it as a double integral."},{"Start":"00:48.050 ","End":"00:55.400","Text":"The formula for doing it as a double integral is this integral over R of 1,"},{"Start":"00:55.400 ","End":"00:59.040","Text":"the function 1, the constant function 1, dA."},{"Start":"00:59.040 ","End":"01:03.330","Text":"I\u0027m writing dA because I haven\u0027t decided yet dxdy or dydx."},{"Start":"01:04.420 ","End":"01:08.600","Text":"I\u0027m going to show you at the end why this formula is true."},{"Start":"01:08.600 ","End":"01:10.720","Text":"Meanwhile, we just accept it."},{"Start":"01:10.720 ","End":"01:12.990","Text":"It\u0027s just a small remark on notation."},{"Start":"01:12.990 ","End":"01:15.579","Text":"I usually use R for region,"},{"Start":"01:15.579 ","End":"01:19.099","Text":"but many books like to use the letter D for domain."},{"Start":"01:19.099 ","End":"01:22.100","Text":"If I sometimes use D and sometimes R,"},{"Start":"01:22.100 ","End":"01:23.150","Text":"it shouldn\u0027t confuse you."},{"Start":"01:23.150 ","End":"01:24.995","Text":"Now, an example."},{"Start":"01:24.995 ","End":"01:27.769","Text":"In the example, let\u0027s say we\u0027re given 2 curves,"},{"Start":"01:27.769 ","End":"01:33.260","Text":"y equals 2x and y equals x squared,"},{"Start":"01:33.260 ","End":"01:37.085","Text":"and we\u0027re asked to find the area bounded by them."},{"Start":"01:37.085 ","End":"01:39.459","Text":"We start off with a sketch."},{"Start":"01:39.459 ","End":"01:42.404","Text":"Let\u0027s label this y equals 2x,"},{"Start":"01:42.404 ","End":"01:44.500","Text":"and then the other 1,"},{"Start":"01:44.540 ","End":"01:49.760","Text":"and this 1 will be y equals x squared."},{"Start":"01:49.760 ","End":"01:52.750","Text":"Yeah, it\u0027s a really bad picture, I know."},{"Start":"01:52.750 ","End":"01:56.510","Text":"They\u0027ll call this region R. This is the region between them"},{"Start":"01:56.510 ","End":"01:59.645","Text":"and I\u0027ll shade it. There we are."},{"Start":"01:59.645 ","End":"02:01.490","Text":"I don\u0027t need this."},{"Start":"02:01.490 ","End":"02:03.785","Text":"According to this formula,"},{"Start":"02:03.785 ","End":"02:12.020","Text":"the area is equal to the double integral over this region R of 1 dA."},{"Start":"02:12.020 ","End":"02:14.645","Text":"In our particular example,"},{"Start":"02:14.645 ","End":"02:20.500","Text":"I would like to do it as an integral of the type dydx,"},{"Start":"02:20.500 ","End":"02:25.110","Text":"meaning that I extract y as a function of x,"},{"Start":"02:25.110 ","End":"02:29.185","Text":"and the outer loop is with x and the inner loop with y."},{"Start":"02:29.185 ","End":"02:36.320","Text":"Now, we see that x goes from 0 and this point is 2. How did I get that?"},{"Start":"02:36.320 ","End":"02:38.510","Text":"Just compare 2x equals x squared,"},{"Start":"02:38.510 ","End":"02:40.880","Text":"you get an equation x with minus 2x is 0,"},{"Start":"02:40.880 ","End":"02:43.145","Text":"x is 0 or 2."},{"Start":"02:43.145 ","End":"02:45.785","Text":"Then for each x,"},{"Start":"02:45.785 ","End":"02:49.580","Text":"say I have a particular x in this 0-2 range,"},{"Start":"02:49.580 ","End":"02:56.319","Text":"what I do is I take this like a vertical arrow slice and I see where it hits,"},{"Start":"02:56.319 ","End":"03:05.870","Text":"here and here, and we get the integral from 0 to 2."},{"Start":"03:05.870 ","End":"03:09.065","Text":"That\u0027s the outer integral dx."},{"Start":"03:09.065 ","End":"03:10.820","Text":"For each particular x,"},{"Start":"03:10.820 ","End":"03:14.540","Text":"y goes from here to here."},{"Start":"03:14.540 ","End":"03:17.965","Text":"Well, here, y equals x squared,"},{"Start":"03:17.965 ","End":"03:20.130","Text":"so y is x squared,"},{"Start":"03:20.130 ","End":"03:24.395","Text":"and here, y is equal to 2x dy,"},{"Start":"03:24.395 ","End":"03:27.145","Text":"and the 1 just stays 1."},{"Start":"03:27.145 ","End":"03:30.330","Text":"Now, I start computing this integral."},{"Start":"03:30.330 ","End":"03:38.455","Text":"What we get is the integral from 0 to 2, dx,"},{"Start":"03:38.455 ","End":"03:47.490","Text":"and the inner integral is the integral of 1 is y. I\u0027ll write it in brackets."},{"Start":"03:47.490 ","End":"03:55.320","Text":"This y has to be evaluated from x squared to 2x as a definite integral."},{"Start":"03:58.190 ","End":"04:02.670","Text":"I just plug in y equals 2x,"},{"Start":"04:02.670 ","End":"04:07.110","Text":"so that\u0027s 2x, and then y equals x squared, so that\u0027s x squared."},{"Start":"04:07.110 ","End":"04:14.200","Text":"I subtract them and I still have the integral from 0 to 2 dx."},{"Start":"04:14.200 ","End":"04:16.080","Text":"Now, at this point, you would say,"},{"Start":"04:16.080 ","End":"04:19.010","Text":"aha, I didn\u0027t need double integrals for that."},{"Start":"04:19.010 ","End":"04:21.245","Text":"If I have this function and I have this function,"},{"Start":"04:21.245 ","End":"04:23.270","Text":"I just use a regular single integral."},{"Start":"04:23.270 ","End":"04:24.905","Text":"I go from 0 to 2,"},{"Start":"04:24.905 ","End":"04:28.165","Text":"and the top function minus the lower function."},{"Start":"04:28.165 ","End":"04:30.510","Text":"I\u0027m not going to complete the computation,"},{"Start":"04:30.510 ","End":"04:32.650","Text":"I\u0027ll leave you to do that."},{"Start":"04:32.650 ","End":"04:36.770","Text":"But let\u0027s go on to our next example where it\u0027s not so easy to do it as a"},{"Start":"04:36.770 ","End":"04:40.310","Text":"single integral and I\u0027ll show the advantage of a double integral."},{"Start":"04:40.310 ","End":"04:43.370","Text":"Also going to take the area between 2 curves."},{"Start":"04:43.370 ","End":"04:48.920","Text":"This time, I\u0027ll let the 2 curves be x plus 2y equals"},{"Start":"04:48.920 ","End":"04:56.000","Text":"4 and y squared equals x over 2."},{"Start":"04:56.000 ","End":"04:57.830","Text":"Let\u0027s do a quick sketch."},{"Start":"04:57.830 ","End":"04:59.180","Text":"Start with this 1."},{"Start":"04:59.180 ","End":"05:06.365","Text":"Easiest to do the intersection with the axis when x is 0, y is 2."},{"Start":"05:06.365 ","End":"05:10.790","Text":"Say this is 2, and when y is 0, x is 4."},{"Start":"05:10.790 ","End":"05:14.280","Text":"Let\u0027s say this is 4, and here\u0027s the line."},{"Start":"05:14.280 ","End":"05:17.839","Text":"Now, the next 1, let\u0027s substitute some easy to compute values."},{"Start":"05:17.839 ","End":"05:20.315","Text":"When x is 0,"},{"Start":"05:20.315 ","End":"05:23.630","Text":"y squared is 0 so we have this point."},{"Start":"05:23.630 ","End":"05:27.140","Text":"When x is 2, 2 over 2 is 1,"},{"Start":"05:27.140 ","End":"05:31.375","Text":"y squared is 1 so y is plus or minus 1."},{"Start":"05:31.375 ","End":"05:34.275","Text":"Perhaps here, that\u0027s 1, that\u0027s 1."},{"Start":"05:34.275 ","End":"05:36.405","Text":"This will be 2."},{"Start":"05:36.405 ","End":"05:39.720","Text":"Negative value would be 8."},{"Start":"05:39.720 ","End":"05:41.655","Text":"8 over 2 is 4,"},{"Start":"05:41.655 ","End":"05:43.830","Text":"so y squared is plus or minus 2,"},{"Start":"05:43.830 ","End":"05:47.205","Text":"something like this and this,"},{"Start":"05:47.205 ","End":"05:50.710","Text":"and we know that it\u0027s a parabola."},{"Start":"05:50.710 ","End":"05:55.120","Text":"Not the greatest sketch. We need to continue this line."},{"Start":"05:55.120 ","End":"05:57.080","Text":"The region we get is here,"},{"Start":"05:57.080 ","End":"06:03.470","Text":"I\u0027ll shade it there and we\u0027ll give it a name R. Now,"},{"Start":"06:03.470 ","End":"06:07.205","Text":"if we were just to use the classical calculus 1 tools,"},{"Start":"06:07.205 ","End":"06:09.770","Text":"this would be quite awkward to compute."},{"Start":"06:09.770 ","End":"06:13.250","Text":"We\u0027d have to divide the region R into 2 regions."},{"Start":"06:13.250 ","End":"06:15.755","Text":"Maybe draw a vertical line here,"},{"Start":"06:15.755 ","End":"06:21.500","Text":"compute this point, and then split it up into 2 integrals on here,"},{"Start":"06:21.500 ","End":"06:27.680","Text":"I\u0027d have to isolate y in terms of x from this function and the upper 1,"},{"Start":"06:27.680 ","End":"06:29.945","Text":"because this is just a curve, not a function."},{"Start":"06:29.945 ","End":"06:33.639","Text":"Then also to compute this 1."},{"Start":"06:34.040 ","End":"06:40.385","Text":"Here, I take the integral from the upper part minus the lower part of the parabola."},{"Start":"06:40.385 ","End":"06:44.210","Text":"Here, I take the integral of the line"},{"Start":"06:44.210 ","End":"06:49.250","Text":"minus the lower curve or from the lower curve to the line."},{"Start":"06:49.250 ","End":"06:52.295","Text":"We\u0027d be taking sections like this,"},{"Start":"06:52.295 ","End":"06:54.890","Text":"and here, we\u0027d be taking sections like this."},{"Start":"06:54.890 ","End":"06:57.470","Text":"1 time with this curve and this curve,"},{"Start":"06:57.470 ","End":"06:59.090","Text":"1 time with the line in the curve."},{"Start":"06:59.090 ","End":"07:02.135","Text":"In other words, it would be quite awkward."},{"Start":"07:02.135 ","End":"07:06.050","Text":"Here, that might be an advantage to do it as a double integral,"},{"Start":"07:06.050 ","End":"07:12.270","Text":"in which case I would get the double integral over R of 1,"},{"Start":"07:12.270 ","End":"07:15.590","Text":"and dA, previous example,"},{"Start":"07:15.590 ","End":"07:17.825","Text":"I did it as a dydx."},{"Start":"07:17.825 ","End":"07:21.300","Text":"This time, I\u0027ll do it as dxdy."},{"Start":"07:21.300 ","End":"07:22.620","Text":"That was a type 1 region,"},{"Start":"07:22.620 ","End":"07:25.250","Text":"this is a type 2 region or the other way around,"},{"Start":"07:25.250 ","End":"07:27.380","Text":"I sometimes get mixed up."},{"Start":"07:27.380 ","End":"07:32.105","Text":"This just means slicing it horizontally."},{"Start":"07:32.105 ","End":"07:40.350","Text":"So what we do is compute this point and this point where they both cut."},{"Start":"07:41.570 ","End":"07:44.010","Text":"This comes out to be 1,"},{"Start":"07:44.010 ","End":"07:45.515","Text":"this comes out to be minus 2."},{"Start":"07:45.515 ","End":"07:47.995","Text":"The way we would do it would be to extract x."},{"Start":"07:47.995 ","End":"07:52.270","Text":"From the top 1, we\u0027d get x equals 4 minus 2y."},{"Start":"07:52.270 ","End":"07:55.735","Text":"From the bottom 1, we would get x equals 2y squared."},{"Start":"07:55.735 ","End":"08:00.450","Text":"Then we would equate these 2 and we\u0027d get"},{"Start":"08:00.450 ","End":"08:05.470","Text":"a quadratic equation and you\u0027d get the y here is 1 or minus 2."},{"Start":"08:05.470 ","End":"08:08.555","Text":"Actually, we don\u0027t care what the x is at these points."},{"Start":"08:08.555 ","End":"08:12.090","Text":"For each y between minus 2 and 1,"},{"Start":"08:12.090 ","End":"08:14.925","Text":"let\u0027s say this is a typical y,"},{"Start":"08:14.925 ","End":"08:18.970","Text":"we take a slice or an arrow,"},{"Start":"08:19.670 ","End":"08:24.645","Text":"and here are the entry and exit points."},{"Start":"08:24.645 ","End":"08:29.130","Text":"What we get is the integral."},{"Start":"08:29.130 ","End":"08:34.990","Text":"Now, the outer 1 is minus 2 to 1 dy,"},{"Start":"08:34.990 ","End":"08:40.670","Text":"and the inner 1 is going to be the integral."},{"Start":"08:40.670 ","End":"08:43.415","Text":"Lower 1 is this curve,"},{"Start":"08:43.415 ","End":"08:48.500","Text":"is 2y squared, and the upper 1,"},{"Start":"08:48.500 ","End":"08:53.820","Text":"the rightmost 1, is the 4 minus 2y."},{"Start":"08:54.280 ","End":"09:00.555","Text":"All this is dx and the function is 1."},{"Start":"09:00.555 ","End":"09:01.875","Text":"So it\u0027s 1 dxdy,"},{"Start":"09:01.875 ","End":"09:03.945","Text":"and these are the limits."},{"Start":"09:03.945 ","End":"09:06.830","Text":"Now, if I do the inner integral,"},{"Start":"09:06.830 ","End":"09:10.550","Text":"I will get the integral from minus 2 to 1,"},{"Start":"09:10.550 ","End":"09:16.775","Text":"and then the inner integral is going to be the integral of 1 is x."},{"Start":"09:16.775 ","End":"09:25.080","Text":"But I need to take it between 2y squared and 4 minus 2y."},{"Start":"09:25.400 ","End":"09:33.185","Text":"That just gives me the integral from minus 2 to 1 of this minus this."},{"Start":"09:33.185 ","End":"09:40.115","Text":"I can write it as 4 minus 2y minus 2y squared dy,"},{"Start":"09:40.115 ","End":"09:45.485","Text":"and then continue just a single integral case."},{"Start":"09:45.485 ","End":"09:51.650","Text":"Of course, some of you study areas slice it horizontally as well as vertically."},{"Start":"09:51.650 ","End":"09:53.420","Text":"In other words, x as a function of y,"},{"Start":"09:53.420 ","End":"09:54.770","Text":"so that y is a function of x,"},{"Start":"09:54.770 ","End":"09:56.695","Text":"and many of you don\u0027t."},{"Start":"09:56.695 ","End":"09:58.790","Text":"For those of you that don\u0027t,"},{"Start":"09:58.790 ","End":"10:03.300","Text":"this is definitely an advantage to do it as a double integral,"},{"Start":"10:03.300 ","End":"10:05.510","Text":"and there are other advantages later on."},{"Start":"10:05.510 ","End":"10:09.200","Text":"In any event, it\u0027s something you should know that the area of"},{"Start":"10:09.200 ","End":"10:14.975","Text":"a region is the integral of 1 with respect to the element of area dA."},{"Start":"10:14.975 ","End":"10:19.100","Text":"That concludes the second example and also the clip except"},{"Start":"10:19.100 ","End":"10:22.730","Text":"that I want to show you where this formula comes from,"},{"Start":"10:22.730 ","End":"10:26.380","Text":"and you\u0027re welcome to stay or leave as you please."},{"Start":"10:26.380 ","End":"10:29.060","Text":"Here we are with the sketch from the beginning of"},{"Start":"10:29.060 ","End":"10:32.735","Text":"the clip and I\u0027m going to explain where this formula comes from."},{"Start":"10:32.735 ","End":"10:35.645","Text":"Let\u0027s label these 2 curves."},{"Start":"10:35.645 ","End":"10:39.830","Text":"Let\u0027s say this is y equals f of x,"},{"Start":"10:39.830 ","End":"10:43.780","Text":"and this 1 is y equals g of x,"},{"Start":"10:43.780 ","End":"10:45.710","Text":"and label the values."},{"Start":"10:45.710 ","End":"10:48.790","Text":"Here, it\u0027s a, and here, it\u0027s b."},{"Start":"10:48.790 ","End":"10:52.925","Text":"Now, if we were in calculus 1 with just a single integrals,"},{"Start":"10:52.925 ","End":"11:01.085","Text":"we would write this the area as the integral from a to b of g of x,"},{"Start":"11:01.085 ","End":"11:05.990","Text":"the upper 1 minus the lower 1, dx."},{"Start":"11:05.990 ","End":"11:08.095","Text":"Put brackets here."},{"Start":"11:08.095 ","End":"11:13.775","Text":"Now, consider this expression just what\u0027s inside the square brackets."},{"Start":"11:13.775 ","End":"11:16.610","Text":"I claim that this equals,"},{"Start":"11:16.610 ","End":"11:19.085","Text":"let me just write it first, g of x minus f of x,"},{"Start":"11:19.085 ","End":"11:23.825","Text":"that it equals the integral of 1 dy"},{"Start":"11:23.825 ","End":"11:29.630","Text":"where y goes from the lower 1 f of x to the upper 1 g of x."},{"Start":"11:29.630 ","End":"11:31.775","Text":"Now, why do I claim this?"},{"Start":"11:31.775 ","End":"11:35.075","Text":"Well, let\u0027s just compute this."},{"Start":"11:35.075 ","End":"11:39.335","Text":"The integral of 1 dy is just y,"},{"Start":"11:39.335 ","End":"11:40.710","Text":"and it\u0027s a definite integral,"},{"Start":"11:40.710 ","End":"11:45.920","Text":"so I plug in the upper value g of x and the lower value f of x."},{"Start":"11:45.920 ","End":"11:49.160","Text":"What this means is I substitute y equals g of x."},{"Start":"11:49.160 ","End":"11:55.730","Text":"This just gives me g of x. I subtract when I plug in y equals f of x, which is f of x."},{"Start":"11:55.730 ","End":"11:57.470","Text":"That\u0027s the same thing as this,"},{"Start":"11:57.470 ","End":"12:00.319","Text":"so this equality is true."},{"Start":"12:00.319 ","End":"12:06.280","Text":"If this is true, then I can substitute this by what it\u0027s equal to,"},{"Start":"12:06.280 ","End":"12:12.755","Text":"and what it\u0027s equal to is the integral from a to b."},{"Start":"12:12.755 ","End":"12:18.980","Text":"Instead of this, I can write this here,"},{"Start":"12:18.980 ","End":"12:21.545","Text":"which we just showed is equal to it."},{"Start":"12:21.545 ","End":"12:32.040","Text":"So the integral from f of x to g of x of 1 dy and dx,"},{"Start":"12:32.040 ","End":"12:34.155","Text":"of course, at the end."},{"Start":"12:34.155 ","End":"12:38.270","Text":"We\u0027ve got to this expression and I claim that"},{"Start":"12:38.270 ","End":"12:42.170","Text":"this expression is equal to this because remember,"},{"Start":"12:42.170 ","End":"12:44.149","Text":"when we do a double integral,"},{"Start":"12:44.149 ","End":"12:47.299","Text":"we can choose vertical or horizontal slices,"},{"Start":"12:47.299 ","End":"12:50.495","Text":"like a type 1 or type 2 region."},{"Start":"12:50.495 ","End":"12:52.954","Text":"If I choose it as the vertical slice,"},{"Start":"12:52.954 ","End":"12:57.440","Text":"then I see that the outer loop is x from a to b,"},{"Start":"12:57.440 ","End":"12:59.075","Text":"which is what I have here."},{"Start":"12:59.075 ","End":"13:01.545","Text":"For each such x,"},{"Start":"13:01.545 ","End":"13:03.270","Text":"let\u0027s say this is my typical x,"},{"Start":"13:03.270 ","End":"13:06.860","Text":"then I take this vertical slice."},{"Start":"13:06.860 ","End":"13:10.130","Text":"The arrow goes in here, goes out here,"},{"Start":"13:10.130 ","End":"13:14.605","Text":"and so the outer loop from a to b,"},{"Start":"13:14.605 ","End":"13:16.050","Text":"let me just emphasize that,"},{"Start":"13:16.050 ","End":"13:19.380","Text":"from a to b as the outer loop here."},{"Start":"13:19.380 ","End":"13:24.185","Text":"For each x, we take y from the lower to the upper."},{"Start":"13:24.185 ","End":"13:27.650","Text":"This is exactly how we do the double"},{"Start":"13:27.650 ","End":"13:31.835","Text":"integral if we slice it vertically and so these 2 are equal."},{"Start":"13:31.835 ","End":"13:36.605","Text":"I just wanted to give you some reasoning behind this rule,"},{"Start":"13:36.605 ","End":"13:39.630","Text":"and that concludes this clip."}],"ID":8675},{"Watched":false,"Name":"Computing Volume as a Double Integral","Duration":"16m 42s","ChapterTopicVideoID":8460,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"We\u0027re still in the chapter on applications of the double"},{"Start":"00:03.240 ","End":"00:07.815","Text":"integral and 1 of the applications is for computing volumes."},{"Start":"00:07.815 ","End":"00:11.235","Text":"You might ask what volumes am I talking about."},{"Start":"00:11.235 ","End":"00:14.250","Text":"I\u0027ll get to that in a moment and there\u0027s a diagram here."},{"Start":"00:14.250 ","End":"00:17.190","Text":"But before that I want to give an analogy."},{"Start":"00:17.190 ","End":"00:20.940","Text":"In the case of a function of a single variable,"},{"Start":"00:20.940 ","End":"00:25.515","Text":"we defined area as the definite integral."},{"Start":"00:25.515 ","End":"00:27.885","Text":"If I have a curve y equals f of x,"},{"Start":"00:27.885 ","End":"00:32.100","Text":"the area under this curve over a certain interval from a to b,"},{"Start":"00:32.100 ","End":"00:34.184","Text":"this might be the interval,"},{"Start":"00:34.184 ","End":"00:37.260","Text":"or I could even call it I for interval."},{"Start":"00:37.260 ","End":"00:40.550","Text":"Then this is the area,"},{"Start":"00:40.550 ","End":"00:43.130","Text":"the integral from a to b of f of x dx."},{"Start":"00:43.130 ","End":"00:46.775","Text":"Now a similar thing works with volume."},{"Start":"00:46.775 ","End":"00:53.600","Text":"If we generalize from a function of a single variable to a function of 2 variables,"},{"Start":"00:53.600 ","End":"01:04.055","Text":"then we get a surface and the simplest scenario is the volume that under a surface,"},{"Start":"01:04.055 ","End":"01:09.110","Text":"or more specifically, just like we had this interval here,"},{"Start":"01:09.110 ","End":"01:14.690","Text":"in 2 dimensions, we might have a region R,"},{"Start":"01:14.690 ","End":"01:16.430","Text":"although R could be rectangle."},{"Start":"01:16.430 ","End":"01:18.185","Text":"In fact, here it\u0027s a rectangle,"},{"Start":"01:18.185 ","End":"01:20.450","Text":"but in general not."},{"Start":"01:20.450 ","End":"01:24.455","Text":"What we do is just like we raise parallel lines here,"},{"Start":"01:24.455 ","End":"01:29.794","Text":"we take vertical lines from this region until we hit the surface,"},{"Start":"01:29.794 ","End":"01:33.890","Text":"and the volume between the surface and"},{"Start":"01:33.890 ","End":"01:38.870","Text":"the region or domain is the volume I\u0027m talking about."},{"Start":"01:38.870 ","End":"01:41.720","Text":"It\u0027s a bit hard to shade 3D."},{"Start":"01:41.720 ","End":"01:47.480","Text":"You might imagine maybe a room with straight floor and vertical sides,"},{"Start":"01:47.480 ","End":"01:49.165","Text":"but with a curved roof."},{"Start":"01:49.165 ","End":"01:54.810","Text":"The curved roof is just this part here and it\u0027s the volume of the room."},{"Start":"01:54.810 ","End":"01:58.220","Text":"The analogy is that if this is the formula for area,"},{"Start":"01:58.220 ","End":"02:00.755","Text":"the volume, instead of being a single integral,"},{"Start":"02:00.755 ","End":"02:01.850","Text":"is a double integral."},{"Start":"02:01.850 ","End":"02:03.439","Text":"Instead of over an interval,"},{"Start":"02:03.439 ","End":"02:05.870","Text":"it\u0027s going to be over a region."},{"Start":"02:05.870 ","End":"02:11.085","Text":"Function of a single variable will be a function of 2 variables."},{"Start":"02:11.085 ","End":"02:13.620","Text":"Could be dx, dy, dy, dx."},{"Start":"02:13.620 ","End":"02:17.105","Text":"But usually we would just say dA and we\u0027re not committing."},{"Start":"02:17.105 ","End":"02:18.830","Text":"It\u0027s just by the way,"},{"Start":"02:18.830 ","End":"02:21.875","Text":"that if it is a rectangle which it needn\u0027t be,"},{"Start":"02:21.875 ","End":"02:25.055","Text":"then this thing turns out to be even more like this."},{"Start":"02:25.055 ","End":"02:31.920","Text":"It just comes out to be the integral from a to b."},{"Start":"02:31.920 ","End":"02:36.830","Text":"This is a to b in the x-direction and the integral from c to d,"},{"Start":"02:36.830 ","End":"02:40.085","Text":"f of x, y,"},{"Start":"02:40.085 ","End":"02:45.499","Text":"and then the inner integral is dy and the outer integral is dx."},{"Start":"02:45.499 ","End":"02:50.360","Text":"I\u0027m going to get rid of this picture and I\u0027m going to highlight this formula."},{"Start":"02:50.360 ","End":"02:53.600","Text":"It\u0027s important. This we\u0027ll see later,"},{"Start":"02:53.600 ","End":"02:55.130","Text":"I\u0027m going to delete it now."},{"Start":"02:55.130 ","End":"02:59.030","Text":"By the way, note that if I have a region in the xy plane,"},{"Start":"02:59.030 ","End":"03:02.450","Text":"the xy plane has another description."},{"Start":"03:02.450 ","End":"03:06.930","Text":"If I give the equation z equals naught,"},{"Start":"03:06.930 ","End":"03:12.680","Text":"then that is exactly the xy plane because everywhere where z is 0 is this."},{"Start":"03:12.680 ","End":"03:14.660","Text":"That\u0027s where the region is given."},{"Start":"03:14.660 ","End":"03:19.110","Text":"Also sometimes we use the letter D instead of R,"},{"Start":"03:19.110 ","End":"03:21.290","Text":"D for domain instead of R for region."},{"Start":"03:21.290 ","End":"03:24.920","Text":"Don\u0027t worry if you see D instead of R. Now let me"},{"Start":"03:24.920 ","End":"03:29.000","Text":"point out that when we are given problems to solve,"},{"Start":"03:29.000 ","End":"03:34.080","Text":"we\u0027re usually almost always not given a sketch."},{"Start":"03:34.190 ","End":"03:41.270","Text":"I\u0027m not even good at drawing these 3D sketches so that\u0027s the bad news."},{"Start":"03:41.270 ","End":"03:44.360","Text":"But the good news is that we don\u0027t need sketches,"},{"Start":"03:44.360 ","End":"03:47.900","Text":"because if you look at this integral,"},{"Start":"03:47.900 ","End":"03:50.495","Text":"we see that f appears there,"},{"Start":"03:50.495 ","End":"03:54.710","Text":"but the main thing is that we have the region R here,"},{"Start":"03:54.710 ","End":"04:00.485","Text":"and it\u0027s the region that we need to sketch when we do double integrals."},{"Start":"04:00.485 ","End":"04:03.470","Text":"In this example, our sketch might look like this."},{"Start":"04:03.470 ","End":"04:05.750","Text":"As I said here, it\u0027s a rectangle,"},{"Start":"04:05.750 ","End":"04:08.645","Text":"but in general, it could be any funny shape."},{"Start":"04:08.645 ","End":"04:10.250","Text":"The shape that we compute,"},{"Start":"04:10.250 ","End":"04:14.480","Text":"the volume of that\u0027s between this region and the surface,"},{"Start":"04:14.480 ","End":"04:16.715","Text":"here it looks like it\u0027s all air,"},{"Start":"04:16.715 ","End":"04:18.765","Text":"but we imagine it to be solid."},{"Start":"04:18.765 ","End":"04:22.160","Text":"The usual term for this thing is a body,"},{"Start":"04:22.160 ","End":"04:26.630","Text":"or sometimes we use the term a solid,"},{"Start":"04:26.630 ","End":"04:28.970","Text":"a 3D solid, a 3D body."},{"Start":"04:28.970 ","End":"04:35.180","Text":"Also don\u0027t confuse this with the solid or body of revolution or rotation."},{"Start":"04:35.180 ","End":"04:37.645","Text":"That\u0027s a different topic."},{"Start":"04:37.645 ","End":"04:41.750","Text":"Now the example, the general example,"},{"Start":"04:41.750 ","End":"04:44.780","Text":"most of them are very similar."},{"Start":"04:44.780 ","End":"04:54.800","Text":"We\u0027ll be asked to compute the volume of the body bound above by f of x,"},{"Start":"04:54.800 ","End":"04:59.450","Text":"y equals and here I\u0027ll give you the example in a moment."},{"Start":"04:59.450 ","End":"05:02.210","Text":"This is just a template for the example."},{"Start":"05:02.210 ","End":"05:09.690","Text":"They would say, and bounded below by the region R,"},{"Start":"05:09.690 ","End":"05:12.025","Text":"and sometimes they might say, though,"},{"Start":"05:12.025 ","End":"05:16.805","Text":"it\u0027s usually understood in the xy plane."},{"Start":"05:16.805 ","End":"05:20.785","Text":"Then you\u0027d get some description of what R is,"},{"Start":"05:20.785 ","End":"05:24.890","Text":"so these are the things that we have to fill in basically."},{"Start":"05:24.890 ","End":"05:29.965","Text":"That\u0027s in general now let me do a specific example."},{"Start":"05:29.965 ","End":"05:40.405","Text":"Here I\u0027ll take x squared plus y squared plus 1 and the region R."},{"Start":"05:40.405 ","End":"05:46.920","Text":"Here I\u0027m going to describe it as bounded by or between"},{"Start":"05:48.280 ","End":"05:59.225","Text":"the curves y equals x squared and y equals x. I\u0027m going to change this to enclosed."},{"Start":"05:59.225 ","End":"06:05.165","Text":"As I said, the main thing in sketching is just the region R. We don\u0027t do anything 3D,"},{"Start":"06:05.165 ","End":"06:08.769","Text":"so I can reuse part of this sketch."},{"Start":"06:08.769 ","End":"06:11.180","Text":"Let\u0027s see, y equals x squared."},{"Start":"06:11.180 ","End":"06:14.720","Text":"This really just has to be a very rough sketch just to get an idea,"},{"Start":"06:14.720 ","End":"06:16.880","Text":"it might be something like this,"},{"Start":"06:16.880 ","End":"06:21.025","Text":"and y equals x is a straight line through the origin."},{"Start":"06:21.025 ","End":"06:23.760","Text":"This might be, this."},{"Start":"06:23.760 ","End":"06:28.760","Text":"I straightened it out a bit and we see they intersect in 2 points."},{"Start":"06:28.760 ","End":"06:34.160","Text":"The region enclosed by the curves is going to be this and I\u0027ll"},{"Start":"06:34.160 ","End":"06:39.439","Text":"label it R. Now according to this formula,"},{"Start":"06:39.439 ","End":"06:49.390","Text":"our volume will just be the double integral over R of f of x, y."},{"Start":"06:49.390 ","End":"06:50.680","Text":"But we have it here,"},{"Start":"06:50.680 ","End":"06:57.290","Text":"which is x squared plus y squared plus 1 dA."},{"Start":"06:57.290 ","End":"07:00.205","Text":"Now to evaluate this,"},{"Start":"07:00.205 ","End":"07:04.779","Text":"we want to do it as a double iterated integral."},{"Start":"07:04.779 ","End":"07:08.600","Text":"We can slice it horizontally or vertically."},{"Start":"07:08.600 ","End":"07:11.020","Text":"I\u0027ll go for vertical slices."},{"Start":"07:11.020 ","End":"07:12.700","Text":"I think that\u0027s a type 1 region,"},{"Start":"07:12.700 ","End":"07:19.630","Text":"so use this arrow and see it cuts here and here."},{"Start":"07:19.630 ","End":"07:21.610","Text":"This is my typical x."},{"Start":"07:21.610 ","End":"07:24.740","Text":"Now where does x go from and to?"},{"Start":"07:24.740 ","End":"07:27.720","Text":"Well, we can mentally do that."},{"Start":"07:27.720 ","End":"07:29.940","Text":"If x squared equals x,"},{"Start":"07:29.940 ","End":"07:34.250","Text":"which is what is equal when the curves intersect,"},{"Start":"07:34.250 ","End":"07:36.830","Text":"then x squared equals x gives 2 solutions,"},{"Start":"07:36.830 ","End":"07:39.260","Text":"x equals 0 and x equals 1."},{"Start":"07:39.260 ","End":"07:41.930","Text":"This is 0 as we can see,"},{"Start":"07:41.930 ","End":"07:45.510","Text":"and this will be 1."},{"Start":"07:49.140 ","End":"07:52.210","Text":"The outer integral is,"},{"Start":"07:52.210 ","End":"07:54.460","Text":"with respect to x,"},{"Start":"07:54.460 ","End":"07:59.410","Text":"from 0-1, and that\u0027s going to be dx."},{"Start":"07:59.410 ","End":"08:05.950","Text":"The inner integral will be from the lower curve to the upper curve."},{"Start":"08:05.950 ","End":"08:12.400","Text":"Here we can see that the parabola is lower than the line."},{"Start":"08:12.400 ","End":"08:14.215","Text":"But if you didn\u0027t have a sketch,"},{"Start":"08:14.215 ","End":"08:19.030","Text":"you could still figure out which was on top because you could just take"},{"Start":"08:19.030 ","End":"08:23.845","Text":"any value between 0 and 1 once you know where they intersect in 0 and 1,"},{"Start":"08:23.845 ","End":"08:25.060","Text":"you could plug in, say,"},{"Start":"08:25.060 ","End":"08:27.175","Text":"a 1/2 into each of them,"},{"Start":"08:27.175 ","End":"08:28.750","Text":"and here you\u0027d get a quarter,"},{"Start":"08:28.750 ","End":"08:30.310","Text":"here you\u0027d get a 1/2."},{"Start":"08:30.310 ","End":"08:32.560","Text":"You\u0027d know that this is below this,"},{"Start":"08:32.560 ","End":"08:34.360","Text":"and that\u0027s how you would know."},{"Start":"08:34.360 ","End":"08:36.370","Text":"In any case between 0 and 1,"},{"Start":"08:36.370 ","End":"08:41.140","Text":"the x squared is the lower one and the x is the upper one,"},{"Start":"08:41.140 ","End":"08:44.515","Text":"and that\u0027s going to be dy."},{"Start":"08:44.515 ","End":"08:53.410","Text":"Here\u0027s the function which is x squared plus y squared plus 1."},{"Start":"08:53.410 ","End":"08:56.080","Text":"I\u0027m not going to do the actual computation of this."},{"Start":"08:56.080 ","End":"09:00.190","Text":"It\u0027s not difficult. What I want to do is move to a different scenario."},{"Start":"09:00.190 ","End":"09:06.040","Text":"Sometimes we\u0027re not actually given the region R. I\u0027ll show you what I mean,"},{"Start":"09:06.040 ","End":"09:08.005","Text":"we\u0027ll start on another page."},{"Start":"09:08.005 ","End":"09:11.290","Text":"We\u0027ll keep this formula and I\u0027ll bring in another picture."},{"Start":"09:11.290 ","End":"09:13.675","Text":"Here it is, it\u0027s a pyramid."},{"Start":"09:13.675 ","End":"09:15.294","Text":"It\u0027s meant to be a pyramid."},{"Start":"09:15.294 ","End":"09:17.620","Text":"It\u0027s got 4 sides."},{"Start":"09:17.620 ","End":"09:21.220","Text":"It\u0027s a triangular pyramid but we want to compute the volume of"},{"Start":"09:21.220 ","End":"09:26.500","Text":"the pyramid because we need some information, some data."},{"Start":"09:26.500 ","End":"09:29.830","Text":"This is 4, this is 4,"},{"Start":"09:29.830 ","End":"09:31.900","Text":"and this is 2."},{"Start":"09:31.900 ","End":"09:35.110","Text":"Yeah, this arrow here is where I meant to write."},{"Start":"09:35.110 ","End":"09:37.225","Text":"Z equals 0."},{"Start":"09:37.225 ","End":"09:41.740","Text":"That\u0027s the triangle in the x, y plane,"},{"Start":"09:41.740 ","End":"09:47.680","Text":"and this is the sloping triangle,"},{"Start":"09:47.680 ","End":"09:50.485","Text":"and I\u0027ll tell you its formula."},{"Start":"09:50.485 ","End":"09:52.120","Text":"This is the plane,"},{"Start":"09:52.120 ","End":"09:58.164","Text":"x plus 2y plus z equals 4."},{"Start":"09:58.164 ","End":"10:01.750","Text":"Actually, I was given the equation of the plane,"},{"Start":"10:01.750 ","End":"10:05.065","Text":"I just found the intersection with the axes."},{"Start":"10:05.065 ","End":"10:10.000","Text":"For example, on the z-axis I\u0027d let x and y equal 0 because z equals 4."},{"Start":"10:10.000 ","End":"10:12.745","Text":"Here, x and z would be 0,"},{"Start":"10:12.745 ","End":"10:14.665","Text":"you get 2y is 4, y is 2."},{"Start":"10:14.665 ","End":"10:17.710","Text":"Similarly here y and z is 0, x is 4."},{"Start":"10:17.710 ","End":"10:22.150","Text":"I was given this plane and what I want to do is"},{"Start":"10:22.150 ","End":"10:27.430","Text":"write the volume of the pyramid as a double integral."},{"Start":"10:27.430 ","End":"10:31.690","Text":"I\u0027m going to call this lower triangle R. If you think about it,"},{"Start":"10:31.690 ","End":"10:34.210","Text":"that\u0027s the region in the x,"},{"Start":"10:34.210 ","End":"10:38.650","Text":"y plane, and we have a function above it."},{"Start":"10:38.650 ","End":"10:40.570","Text":"This is not written as a function,"},{"Start":"10:40.570 ","End":"10:43.720","Text":"but I could write it as z in terms of x and y."},{"Start":"10:43.720 ","End":"10:45.505","Text":"Then if I did that,"},{"Start":"10:45.505 ","End":"10:49.000","Text":"this pyramid would be exactly the volume,"},{"Start":"10:49.000 ","End":"10:53.005","Text":"the solid, below the plane and above the region."},{"Start":"10:53.005 ","End":"10:56.169","Text":"We could do it as a double integral."},{"Start":"10:56.169 ","End":"10:59.995","Text":"To extract z as a function of x and y,"},{"Start":"10:59.995 ","End":"11:03.310","Text":"we could say z or f of x,"},{"Start":"11:03.310 ","End":"11:07.060","Text":"y, well, call it both, is going to equal."},{"Start":"11:07.060 ","End":"11:08.710","Text":"If I just bring everything to the side,"},{"Start":"11:08.710 ","End":"11:12.700","Text":"I get 4 minus x minus 2y."},{"Start":"11:12.700 ","End":"11:14.980","Text":"That\u0027s the function."},{"Start":"11:14.980 ","End":"11:20.800","Text":"We get the double integral over the region R,"},{"Start":"11:20.800 ","End":"11:25.120","Text":"of this function 4 minus x minus 2y,"},{"Start":"11:25.120 ","End":"11:27.939","Text":"da, using this formula."},{"Start":"11:27.939 ","End":"11:30.460","Text":"Now how would I go about computing this?"},{"Start":"11:30.460 ","End":"11:34.194","Text":"Well, I have to describe the region R in more detail."},{"Start":"11:34.194 ","End":"11:36.985","Text":"Let me sketch it in the x, y plane,"},{"Start":"11:36.985 ","End":"11:42.160","Text":"and here\u0027s what the region R looks like in the x, y plane."},{"Start":"11:42.160 ","End":"11:46.870","Text":"We know the equation of this line because that\u0027s"},{"Start":"11:46.870 ","End":"11:50.995","Text":"where this plane cuts the z equals naught plane."},{"Start":"11:50.995 ","End":"11:53.380","Text":"If I just put z equals naught here,"},{"Start":"11:53.380 ","End":"11:55.360","Text":"I get x plus 2y equals 4,"},{"Start":"11:55.360 ","End":"12:00.175","Text":"and I\u0027ll write that x plus 2y equals 4."},{"Start":"12:00.175 ","End":"12:07.910","Text":"Note that I have the equation of the other 2 planes, x equals 0."},{"Start":"12:07.920 ","End":"12:11.845","Text":"Not just the triangle, but the whole plane."},{"Start":"12:11.845 ","End":"12:14.110","Text":"Just like this is z equals 0,"},{"Start":"12:14.110 ","End":"12:16.870","Text":"it\u0027s the whole plane of x, y."},{"Start":"12:16.870 ","End":"12:23.740","Text":"The other triangle is in the z,"},{"Start":"12:23.740 ","End":"12:25.825","Text":"x plane where y is 0,"},{"Start":"12:25.825 ","End":"12:28.930","Text":"so this, we write as y equals 0."},{"Start":"12:28.930 ","End":"12:30.640","Text":"Now why am I doing all this?"},{"Start":"12:30.640 ","End":"12:35.560","Text":"Because I just said before that normally you would not be given a sketch."},{"Start":"12:35.560 ","End":"12:40.270","Text":"What I\u0027d like to do is start again without the sketch."},{"Start":"12:40.270 ","End":"12:43.510","Text":"I just left the equations."},{"Start":"12:43.510 ","End":"12:45.850","Text":"Now how would the problem normally be phrased?"},{"Start":"12:45.850 ","End":"12:54.350","Text":"It would say, find the volume of the solid bounded above by this,"},{"Start":"12:55.170 ","End":"13:00.970","Text":"bounded below by this."},{"Start":"13:00.970 ","End":"13:05.335","Text":"But this is not enough because these 2 planes don\u0027t enclose any solid,"},{"Start":"13:05.335 ","End":"13:07.165","Text":"so we would also say,"},{"Start":"13:07.165 ","End":"13:12.740","Text":"and bounded by, yeah, and by."},{"Start":"13:14.580 ","End":"13:23.230","Text":"Well, 1 thing I would do would be to find this as z as a function of x and y."},{"Start":"13:23.230 ","End":"13:26.410","Text":"This is what we had before,"},{"Start":"13:26.410 ","End":"13:30.955","Text":"z equals 4 minus x minus 2y."},{"Start":"13:30.955 ","End":"13:33.985","Text":"This is going to be our f of x, y."},{"Start":"13:33.985 ","End":"13:39.370","Text":"Next, I\u0027d need to find the region in the x,"},{"Start":"13:39.370 ","End":"13:44.725","Text":"y plane, I\u0027d let this equal 0."},{"Start":"13:44.725 ","End":"13:46.735","Text":"In other words, this with this,"},{"Start":"13:46.735 ","End":"13:50.575","Text":"z equals 0 and z equals this would give me this line."},{"Start":"13:50.575 ","End":"13:53.605","Text":"I just plug in 0,"},{"Start":"13:53.605 ","End":"13:56.080","Text":"here I got x plus 2y equals 4,"},{"Start":"13:56.080 ","End":"13:58.750","Text":"and then I have the x equals 0,"},{"Start":"13:58.750 ","End":"14:00.370","Text":"which is the y-axis."},{"Start":"14:00.370 ","End":"14:01.600","Text":"This is the x equals 0,"},{"Start":"14:01.600 ","End":"14:04.015","Text":"that closes it on this side."},{"Start":"14:04.015 ","End":"14:07.570","Text":"Y equals 0, which is the x-axis,"},{"Start":"14:07.570 ","End":"14:09.505","Text":"closes it on this side."},{"Start":"14:09.505 ","End":"14:12.490","Text":"Actually y equals 0 and x equals 0 are not lines,"},{"Start":"14:12.490 ","End":"14:15.970","Text":"they\u0027re actually planes vertical to what we\u0027re viewing."},{"Start":"14:15.970 ","End":"14:18.370","Text":"But in the x, y plane, this is what they are."},{"Start":"14:18.370 ","End":"14:20.545","Text":"This gives us the region."},{"Start":"14:20.545 ","End":"14:22.644","Text":"Now that we have the region,"},{"Start":"14:22.644 ","End":"14:24.370","Text":"we could then say,"},{"Start":"14:24.370 ","End":"14:35.920","Text":"we need the double integral over this region of 4 minus x minus 2y, da."},{"Start":"14:35.920 ","End":"14:39.925","Text":"Then we\u0027d go about computing it."},{"Start":"14:39.925 ","End":"14:41.410","Text":"In our usual methods,"},{"Start":"14:41.410 ","End":"14:45.160","Text":"we might decide to slice it vertically or horizontally,"},{"Start":"14:45.160 ","End":"14:47.740","Text":"type 1 or type 2 region and then,"},{"Start":"14:47.740 ","End":"14:48.940","Text":"say we did it vertically,"},{"Start":"14:48.940 ","End":"14:51.460","Text":"we\u0027d get y as a function of x."},{"Start":"14:51.460 ","End":"14:56.350","Text":"Maybe we\u0027d say y equals 4 minus x over 2 and so on."},{"Start":"14:56.350 ","End":"15:02.260","Text":"I don\u0027t want to get into the computation of the actual integral,"},{"Start":"15:02.260 ","End":"15:08.500","Text":"just wanted to show you how sometimes not only we\u0027re not given the sketch,"},{"Start":"15:08.500 ","End":"15:12.475","Text":"we\u0027re not given the region even and we have to figure out the region,"},{"Start":"15:12.475 ","End":"15:21.325","Text":"and here we got the region using 3 lines which outlined a triangle and that\u0027s that."},{"Start":"15:21.325 ","End":"15:27.580","Text":"Just by way of analogy in the case of single integrals,"},{"Start":"15:27.580 ","End":"15:32.675","Text":"sometimes we\u0027re asked to compute an integral where we\u0027re given"},{"Start":"15:32.675 ","End":"15:38.180","Text":"the function f and we\u0027re given from where to where it goes."},{"Start":"15:38.180 ","End":"15:41.195","Text":"Then say this is a, and this is b, this is f of x."},{"Start":"15:41.195 ","End":"15:45.660","Text":"But sometimes we\u0027re given a situation where we say,"},{"Start":"15:45.660 ","End":"15:54.610","Text":"the function is y equals 4 minus x squared,"},{"Start":"15:54.610 ","End":"15:59.420","Text":"and we have to compute the area"},{"Start":"15:59.420 ","End":"16:06.535","Text":"between this function and between y equals 0."},{"Start":"16:06.535 ","End":"16:12.645","Text":"Then we\u0027d have to by ourselves draw the picture of this is one upside down parabola,"},{"Start":"16:12.645 ","End":"16:15.490","Text":"and then we\u0027d find this point and"},{"Start":"16:15.490 ","End":"16:19.040","Text":"this point and then we\u0027d know which area we\u0027re talking about."},{"Start":"16:19.040 ","End":"16:21.395","Text":"Right here, it\u0027s given explicitly."},{"Start":"16:21.395 ","End":"16:26.705","Text":"Similarly in 3D, sometimes we\u0027re given this function and the region,"},{"Start":"16:26.705 ","End":"16:33.000","Text":"and sometimes we have to figure out the region by the methods I just showed you."},{"Start":"16:33.130 ","End":"16:35.420","Text":"That\u0027s it for this clip,"},{"Start":"16:35.420 ","End":"16:42.840","Text":"there will be more examples in the following clip as well as in the exercises. That\u0027s it."}],"ID":8676},{"Watched":false,"Name":"Computing Volume - an Example","Duration":"3m 40s","ChapterTopicVideoID":8461,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.004","Text":"We\u0027re continuing with the volume as a double integral,"},{"Start":"00:04.004 ","End":"00:08.445","Text":"and what I want to present now is another example."},{"Start":"00:08.445 ","End":"00:11.115","Text":"It goes as follows."},{"Start":"00:11.115 ","End":"00:13.140","Text":"Here\u0027s the example."},{"Start":"00:13.140 ","End":"00:18.975","Text":"We want to find the volume bounded by these all surfaces,"},{"Start":"00:18.975 ","End":"00:21.810","Text":"and there\u0027s an extra condition,"},{"Start":"00:21.810 ","End":"00:26.015","Text":"and we\u0027ll see why this extra condition is here."},{"Start":"00:26.015 ","End":"00:28.490","Text":"Sometimes, this question would be phrased as,"},{"Start":"00:28.490 ","End":"00:34.265","Text":"find the volume of the solid or of the body."},{"Start":"00:34.265 ","End":"00:36.860","Text":"Actually, this is also sometimes called a region,"},{"Start":"00:36.860 ","End":"00:38.665","Text":"so a 3D region."},{"Start":"00:38.665 ","End":"00:40.675","Text":"We have several names for this,"},{"Start":"00:40.675 ","End":"00:43.970","Text":"but sometimes, we just say, find the volume bounded by."},{"Start":"00:43.970 ","End":"00:49.355","Text":"I got rid of the diagram because we\u0027re given this problem without a picture."},{"Start":"00:49.355 ","End":"00:52.655","Text":"If I look at these 2 equations,"},{"Start":"00:52.655 ","End":"00:54.800","Text":"this is the x,y plane,"},{"Start":"00:54.800 ","End":"00:56.690","Text":"this is a surface,"},{"Start":"00:56.690 ","End":"00:59.120","Text":"this is like a function of x and y."},{"Start":"00:59.120 ","End":"01:05.135","Text":"It looks like I want the volume below a surface and above the x,y plane."},{"Start":"01:05.135 ","End":"01:15.140","Text":"I would write something like the double integral of 1 minus x squared minus y squared dA."},{"Start":"01:15.140 ","End":"01:21.500","Text":"But it has to be over some region or domain R. This is what we have to find now,"},{"Start":"01:21.500 ","End":"01:23.845","Text":"and this is in the x,y plane."},{"Start":"01:23.845 ","End":"01:29.570","Text":"We want to find this region R. We have these 2 equations,"},{"Start":"01:29.570 ","End":"01:31.550","Text":"and they are equations of lines,"},{"Start":"01:31.550 ","End":"01:34.260","Text":"at least in the x,y plane there."},{"Start":"01:34.550 ","End":"01:39.120","Text":"Let\u0027s say this is y equals x,"},{"Start":"01:39.120 ","End":"01:43.155","Text":"and here\u0027s y equals 2x."},{"Start":"01:43.155 ","End":"01:46.280","Text":"I said they\u0027re lines but only in the plane."},{"Start":"01:46.280 ","End":"01:50.030","Text":"They\u0027re really actually equations of planes."},{"Start":"01:50.030 ","End":"01:51.710","Text":"The z is missing,"},{"Start":"01:51.710 ","End":"01:55.865","Text":"so it just means that you can extend them vertically."},{"Start":"01:55.865 ","End":"01:59.375","Text":"Well, looking above from the z-axis they\u0027re planes,"},{"Start":"01:59.375 ","End":"02:03.580","Text":"but they cut the x,y plane in these lines."},{"Start":"02:03.580 ","End":"02:05.689","Text":"What about the region?"},{"Start":"02:05.689 ","End":"02:09.545","Text":"I mean, I can go, here is one border of the region,"},{"Start":"02:09.545 ","End":"02:11.960","Text":"here is another, but it\u0027s not closed."},{"Start":"02:11.960 ","End":"02:14.959","Text":"I need something extra."},{"Start":"02:14.959 ","End":"02:17.165","Text":"By the way, I forgot to say here,"},{"Start":"02:17.165 ","End":"02:18.935","Text":"because x is bigger or equal to 0,"},{"Start":"02:18.935 ","End":"02:23.660","Text":"y bigger or equal to 0, I\u0027m just concerned with the first quadrant here."},{"Start":"02:23.660 ","End":"02:30.615","Text":"This surface might just cut the x,y plane in this surface,"},{"Start":"02:30.615 ","End":"02:32.925","Text":"and if so, that will give me an extra line,"},{"Start":"02:32.925 ","End":"02:34.470","Text":"and in fact, that\u0027s what we do."},{"Start":"02:34.470 ","End":"02:36.500","Text":"If we compare this to this,"},{"Start":"02:36.500 ","End":"02:37.880","Text":"we get the equation,"},{"Start":"02:37.880 ","End":"02:42.379","Text":"1 minus x squared minus y squared equals 0,"},{"Start":"02:42.379 ","End":"02:48.035","Text":"which is the same as x squared plus y squared equals 1 or 1 squared,"},{"Start":"02:48.035 ","End":"02:52.045","Text":"so it\u0027s a circle of radius 1 centered at the origin."},{"Start":"02:52.045 ","End":"02:54.585","Text":"Here\u0027s that circle."},{"Start":"02:54.585 ","End":"02:56.775","Text":"That gives us the extra border."},{"Start":"02:56.775 ","End":"02:59.985","Text":"What we\u0027re talking about is this region here,"},{"Start":"02:59.985 ","End":"03:06.460","Text":"and we\u0027ll call it R. Let me label this,"},{"Start":"03:06.460 ","End":"03:09.200","Text":"x squared plus y squared equals 1."},{"Start":"03:09.200 ","End":"03:11.290","Text":"Let\u0027s clean up a bit."},{"Start":"03:11.290 ","End":"03:13.760","Text":"We\u0027ve expressed our problem,"},{"Start":"03:13.760 ","End":"03:16.685","Text":"the volume as a double integral,"},{"Start":"03:16.685 ","End":"03:21.454","Text":"and we have R precisely defined by this picture,"},{"Start":"03:21.454 ","End":"03:24.725","Text":"and how we actually compute it,"},{"Start":"03:24.725 ","End":"03:27.665","Text":"I\u0027m not going to do that now."},{"Start":"03:27.665 ","End":"03:31.930","Text":"The idea would be to convert into polar coordinates,"},{"Start":"03:31.930 ","End":"03:34.220","Text":"something you have yet to learn,"},{"Start":"03:34.220 ","End":"03:40.890","Text":"but the point was just how to set it up as a double integral. That\u0027s it."}],"ID":8677},{"Watched":false,"Name":"Computing Volume of a Body between Two Surfaces","Duration":"10m 5s","ChapterTopicVideoID":8462,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.969","Text":"Continuing with the topic of applications of the double integral computing volume,"},{"Start":"00:05.969 ","End":"00:10.020","Text":"I\u0027m going to take a bit of a different paradigm."},{"Start":"00:10.020 ","End":"00:13.245","Text":"The topic will now be called,"},{"Start":"00:13.245 ","End":"00:16.530","Text":"I\u0027ll just add the words between two surfaces,"},{"Start":"00:16.530 ","End":"00:18.450","Text":"and I\u0027ll get to that in a minute."},{"Start":"00:18.450 ","End":"00:24.310","Text":"But first I want to go back to area as a single integral."},{"Start":"00:25.790 ","End":"00:30.525","Text":"If we had a function f of a single variable,"},{"Start":"00:30.525 ","End":"00:36.000","Text":"and we wanted the area under the curve f between 2 points,"},{"Start":"00:36.000 ","End":"00:42.845","Text":"then we had this formula that the definite integral is the area under the curve."},{"Start":"00:42.845 ","End":"00:49.880","Text":"Then there was a slightly more general problem of finding the area between 2 curves,"},{"Start":"00:49.880 ","End":"00:57.050","Text":"column f and g. Let\u0027s say f is the 1 above g. Then we had a different formula,"},{"Start":"00:57.050 ","End":"01:03.215","Text":"which was the integral of f minus g between a and b."},{"Start":"01:03.215 ","End":"01:08.760","Text":"Now, a similar thing happens in double integrals without"},{"Start":"01:08.760 ","End":"01:15.005","Text":"the volume below a surface or between 2 surfaces."},{"Start":"01:15.005 ","End":"01:17.475","Text":"I better get rid of this one."},{"Start":"01:17.475 ","End":"01:19.840","Text":"In this case, we have two surfaces,"},{"Start":"01:19.840 ","End":"01:22.150","Text":"f above, g below,"},{"Start":"01:22.150 ","End":"01:27.039","Text":"and they\u0027re both over a domain."},{"Start":"01:27.039 ","End":"01:28.585","Text":"We had region here I told you,"},{"Start":"01:28.585 ","End":"01:35.440","Text":"sometimes called D. Then the formula becomes the double"},{"Start":"01:35.440 ","End":"01:42.730","Text":"integral over the region D of f of x,"},{"Start":"01:42.730 ","End":"01:48.220","Text":"y minus g of x, y."},{"Start":"01:48.220 ","End":"01:51.140","Text":"Put that in brackets, dA."},{"Start":"01:51.140 ","End":"01:53.274","Text":"Instead of this formula,"},{"Start":"01:53.274 ","End":"01:55.359","Text":"we have this formula."},{"Start":"01:55.359 ","End":"02:03.260","Text":"I just want to emphasize that D is the projection onto the x-y plane of these 2 surfaces,"},{"Start":"02:03.260 ","End":"02:08.840","Text":"they\u0027re both directly above D. We could use just this 1 always."},{"Start":"02:08.840 ","End":"02:14.730","Text":"If the lowest surface is the x-y plane,"},{"Start":"02:14.730 ","End":"02:16.605","Text":"then the x-y plane,"},{"Start":"02:16.605 ","End":"02:19.310","Text":"if we took g of x,"},{"Start":"02:19.310 ","End":"02:22.010","Text":"y to be equal to 0,"},{"Start":"02:22.010 ","End":"02:27.465","Text":"then we would get this minus 0 and it would be the same as this formula."},{"Start":"02:27.465 ","End":"02:30.650","Text":"Whatever you can use for this case, this formula,"},{"Start":"02:30.650 ","End":"02:33.975","Text":"or you could use this formula with g equals 0."},{"Start":"02:33.975 ","End":"02:36.455","Text":"Now an example."},{"Start":"02:36.455 ","End":"02:40.830","Text":"I think I don\u0027t need this picture anymore."},{"Start":"02:41.450 ","End":"02:44.885","Text":"The example is as follows."},{"Start":"02:44.885 ","End":"02:55.085","Text":"We have to find the volume between the surfaces z equals x squared plus y squared,"},{"Start":"02:55.085 ","End":"02:56.689","Text":"and the other surface,"},{"Start":"02:56.689 ","End":"03:01.820","Text":"z equals 1 minus x squared minus y squared."},{"Start":"03:01.820 ","End":"03:05.060","Text":"There\u0027s a couple of things that we need to know,"},{"Start":"03:05.060 ","End":"03:07.940","Text":"which is above and which is below,"},{"Start":"03:07.940 ","End":"03:13.565","Text":"which is f and which is g. Also we\u0027re not given D explicitly."},{"Start":"03:13.565 ","End":"03:20.195","Text":"I mentioned in an earlier clip that when we\u0027re missing all or part of the borders for D,"},{"Start":"03:20.195 ","End":"03:22.550","Text":"we intersect the two surfaces."},{"Start":"03:22.550 ","End":"03:31.580","Text":"If I compare x squared plus y squared to 1 minus x squared minus y squared,"},{"Start":"03:31.580 ","End":"03:33.590","Text":"I\u0027ll get an equation in x and y,"},{"Start":"03:33.590 ","End":"03:37.550","Text":"and that will be the boundary in the x-y plane."},{"Start":"03:37.550 ","End":"03:42.905","Text":"If we bring the x squared plus y squared to the other side, divide by 2,"},{"Start":"03:42.905 ","End":"03:50.545","Text":"this thing comes out to be x squared plus y squared equals 1/2."},{"Start":"03:50.545 ","End":"03:57.035","Text":"That\u0027s the equation of a circle with radius square root of 1/2."},{"Start":"03:57.035 ","End":"03:58.850","Text":"This is what it looks like."},{"Start":"03:58.850 ","End":"04:05.260","Text":"We\u0027ll call it D. We have the double integral over D,"},{"Start":"04:05.260 ","End":"04:11.790","Text":"now which is f and which is g. Just like in the case of 1 dimension,"},{"Start":"04:11.790 ","End":"04:15.740","Text":"a single variable, we find"},{"Start":"04:15.740 ","End":"04:20.045","Text":"which is above which by taking a sample value and substituting."},{"Start":"04:20.045 ","End":"04:24.975","Text":"The easiest point to pick as a sample point would be say 0, 0."},{"Start":"04:24.975 ","End":"04:27.810","Text":"If I substituted in here, 0,"},{"Start":"04:27.810 ","End":"04:31.245","Text":"0, I will get 0."},{"Start":"04:31.245 ","End":"04:32.750","Text":"If I substituted here,"},{"Start":"04:32.750 ","End":"04:35.420","Text":"I get 1 minus 0 minus 0 is 1."},{"Start":"04:35.420 ","End":"04:38.135","Text":"This is bigger than this here,"},{"Start":"04:38.135 ","End":"04:41.030","Text":"so it will be true in the whole disk."},{"Start":"04:41.030 ","End":"04:47.090","Text":"This will be f and this 1 will be the g. Is that right?"},{"Start":"04:47.090 ","End":"04:48.665","Text":"Yeah, f is the upper 1."},{"Start":"04:48.665 ","End":"04:50.740","Text":"I need the upper 1,"},{"Start":"04:50.740 ","End":"04:56.690","Text":"1 minus x squared minus y squared minus lower 1,"},{"Start":"04:56.690 ","End":"05:02.165","Text":"g, which is x squared plus y squared."},{"Start":"05:02.165 ","End":"05:04.960","Text":"All this dA."},{"Start":"05:04.960 ","End":"05:08.155","Text":"Sure I can simplify it."},{"Start":"05:08.155 ","End":"05:10.670","Text":"But that\u0027s not my point here."},{"Start":"05:10.670 ","End":"05:13.360","Text":"I\u0027m not even going to evaluate it."},{"Start":"05:13.360 ","End":"05:19.685","Text":"I just want to get to this step where we set up the volume as a double integral."},{"Start":"05:19.685 ","End":"05:21.970","Text":"How about another example?"},{"Start":"05:21.970 ","End":"05:25.895","Text":"The next example is going to be what I call a hybrid example."},{"Start":"05:25.895 ","End":"05:27.365","Text":"I\u0027ll tell you what I mean."},{"Start":"05:27.365 ","End":"05:31.455","Text":"In an earlier clip, we had the D,"},{"Start":"05:31.455 ","End":"05:36.435","Text":"we called it there R was completely given."},{"Start":"05:36.435 ","End":"05:38.270","Text":"Here it wasn\u0027t given at all."},{"Start":"05:38.270 ","End":"05:41.120","Text":"We found it as the intersection between the two surfaces,"},{"Start":"05:41.120 ","End":"05:44.090","Text":"but sometimes the region or"},{"Start":"05:44.090 ","End":"05:48.455","Text":"the border is partly given and partly form the intersection of the surfaces."},{"Start":"05:48.455 ","End":"05:50.570","Text":"That\u0027s what\u0027s going to be next."},{"Start":"05:50.570 ","End":"05:59.935","Text":"This time we\u0027re going to take z equals x squared minus y squared as one surface."},{"Start":"05:59.935 ","End":"06:02.965","Text":"The other surface z equals 0."},{"Start":"06:02.965 ","End":"06:05.035","Text":"That\u0027s the x-y plane."},{"Start":"06:05.035 ","End":"06:07.555","Text":"I need an extra surface,"},{"Start":"06:07.555 ","End":"06:10.665","Text":"and that is x equals 1."},{"Start":"06:10.665 ","End":"06:13.535","Text":"Maybe not the word between."},{"Start":"06:13.535 ","End":"06:17.540","Text":"Probably better to say bounded by."},{"Start":"06:18.140 ","End":"06:21.375","Text":"You\u0027ll see this more appropriate,"},{"Start":"06:21.375 ","End":"06:24.605","Text":"will be between these two surfaces,"},{"Start":"06:24.605 ","End":"06:26.710","Text":"but also bounded by this."},{"Start":"06:26.710 ","End":"06:28.925","Text":"Let\u0027s see what happens here."},{"Start":"06:28.925 ","End":"06:33.265","Text":"This time if I intersect the 2 functions,"},{"Start":"06:33.265 ","End":"06:35.665","Text":"I\u0027ll get that this equals 0,"},{"Start":"06:35.665 ","End":"06:38.595","Text":"x squared minus y squared equals 0."},{"Start":"06:38.595 ","End":"06:42.050","Text":"This will give me that x squared equals y squared."},{"Start":"06:42.050 ","End":"06:45.835","Text":"There are 2 solutions."},{"Start":"06:45.835 ","End":"06:48.310","Text":"X equals plus or minus y."},{"Start":"06:48.310 ","End":"06:50.060","Text":"Perhaps if I had written it the other way round,"},{"Start":"06:50.060 ","End":"06:54.320","Text":"I prefer to have it as y equals plus or minus x. Let\u0027s write them separately."},{"Start":"06:54.320 ","End":"06:56.225","Text":"It could be y equals x,"},{"Start":"06:56.225 ","End":"07:03.925","Text":"or it could be y equals minus x as the square root we get plus or minus. Now y equals x."},{"Start":"07:03.925 ","End":"07:07.230","Text":"Look something like this, y equals x,"},{"Start":"07:07.230 ","End":"07:10.250","Text":"and the other 1, something like this,"},{"Start":"07:10.250 ","End":"07:15.140","Text":"y equals minus x and we don\u0027t have a closed region here,"},{"Start":"07:15.140 ","End":"07:18.030","Text":"but if we add x equals 1,"},{"Start":"07:18.710 ","End":"07:22.020","Text":"label it x equals 1."},{"Start":"07:22.020 ","End":"07:27.160","Text":"Let\u0027s say this is the point 1, 0."},{"Start":"07:27.460 ","End":"07:30.440","Text":"Then, well, this is not really aligned."},{"Start":"07:30.440 ","End":"07:32.195","Text":"It\u0027s a vertical plane,"},{"Start":"07:32.195 ","End":"07:36.905","Text":"but it cuts the x-y plane at this line x equals 1."},{"Start":"07:36.905 ","End":"07:41.595","Text":"Now we do have a region, and I\u0027ll label it."},{"Start":"07:41.595 ","End":"07:43.260","Text":"I\u0027ll use D again."},{"Start":"07:43.260 ","End":"07:47.035","Text":"D or R doesn\u0027t really matter, domain or region."},{"Start":"07:47.035 ","End":"07:49.550","Text":"If we use this formula,"},{"Start":"07:49.550 ","End":"07:52.840","Text":"but we have to know first which is upper and which is lower."},{"Start":"07:52.840 ","End":"07:55.980","Text":"We take a sample point from the region in a while,"},{"Start":"07:55.980 ","End":"07:57.575","Text":"I\u0027ll take this point here."},{"Start":"07:57.575 ","End":"07:59.450","Text":"It looks pretty much in the middle."},{"Start":"07:59.450 ","End":"08:04.275","Text":"Let\u0027s say it\u0027s 1/5, 0."},{"Start":"08:04.275 ","End":"08:07.490","Text":"If x is a 1/2 and y is 0,"},{"Start":"08:07.490 ","End":"08:11.030","Text":"then if I plug it in here,"},{"Start":"08:11.030 ","End":"08:15.960","Text":"I\u0027ll get 1/2 squared minus 0 squared it will be 1/4."},{"Start":"08:15.960 ","End":"08:18.960","Text":"Now, if I plug it in here, there\u0027s nothing to plug in at 0."},{"Start":"08:18.960 ","End":"08:20.850","Text":"This one is the bigger 1."},{"Start":"08:20.850 ","End":"08:27.980","Text":"This is my f and this is my g. We get the double integral"},{"Start":"08:27.980 ","End":"08:35.885","Text":"over D of x squared minus y squared minus 0."},{"Start":"08:35.885 ","End":"08:40.410","Text":"I won\u0027t bother with the minus 0, dA."},{"Start":"08:42.050 ","End":"08:45.585","Text":"You know what? I at least start computing it."},{"Start":"08:45.585 ","End":"08:49.135","Text":"I\u0027ll at least bring it to the form of an iterated integral."},{"Start":"08:49.135 ","End":"08:54.760","Text":"I just have to decide if it\u0027s more convenient to slice it vertically or horizontally."},{"Start":"08:54.760 ","End":"08:58.840","Text":"It seems to me that vertically would be much"},{"Start":"08:58.840 ","End":"09:03.520","Text":"better because we have it between the same 2 upper and lower functions."},{"Start":"09:03.520 ","End":"09:04.780","Text":"If I did it horizontally,"},{"Start":"09:04.780 ","End":"09:08.019","Text":"I have to do this triangle separately and this triangle separately."},{"Start":"09:08.019 ","End":"09:10.395","Text":"We\u0027ll go for vertical slices."},{"Start":"09:10.395 ","End":"09:14.675","Text":"What we\u0027ll get, let me just draw 1 of"},{"Start":"09:14.675 ","End":"09:21.610","Text":"these vertical arrows slices to indicate here it goes in,"},{"Start":"09:21.610 ","End":"09:24.445","Text":"here it goes out and this is x and this is minus x."},{"Start":"09:24.445 ","End":"09:28.620","Text":"We get the outer integral,"},{"Start":"09:28.620 ","End":"09:36.720","Text":"that\u0027s the dx integral x goes from 0-1, and that\u0027s dx."},{"Start":"09:36.720 ","End":"09:38.915","Text":"Then for each particular x,"},{"Start":"09:38.915 ","End":"09:40.475","Text":"this is my typical x,"},{"Start":"09:40.475 ","End":"09:44.480","Text":"y will go from minus x to plus x."},{"Start":"09:44.480 ","End":"09:48.340","Text":"From minus x to x dy."},{"Start":"09:48.340 ","End":"09:52.084","Text":"Now just the function or the difference of the functions,"},{"Start":"09:52.084 ","End":"09:56.345","Text":"x squared minus y squared dy, dx."},{"Start":"09:56.345 ","End":"10:02.195","Text":"For this point, it\u0027s pretty routine and I\u0027m going to just leave it at that."},{"Start":"10:02.195 ","End":"10:05.790","Text":"That concludes this clip."}],"ID":8678},{"Watched":false,"Name":"Exercise 1 part a","Duration":"12m ","ChapterTopicVideoID":8463,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"In this exercise, we have to compute the area bounded by the curves."},{"Start":"00:04.740 ","End":"00:06.960","Text":"X plus y equals 2,"},{"Start":"00:06.960 ","End":"00:10.440","Text":"and the other curve x squared minus 4 y equals 4."},{"Start":"00:10.440 ","End":"00:12.960","Text":"Now, pretend you haven\u0027t seen the picture yet."},{"Start":"00:12.960 ","End":"00:17.490","Text":"I just drew it here in advance to be helpful."},{"Start":"00:17.490 ","End":"00:19.755","Text":"Wouldn\u0027t have to draw it on the fly."},{"Start":"00:19.755 ","End":"00:21.720","Text":"But let\u0027s see how did I get to this picture."},{"Start":"00:21.720 ","End":"00:25.620","Text":"Really that\u0027s the most important thing because we want to know what is this region?"},{"Start":"00:25.620 ","End":"00:27.150","Text":"What does it look like?"},{"Start":"00:27.150 ","End":"00:31.020","Text":"Then describe it as a double integral and then"},{"Start":"00:31.020 ","End":"00:35.190","Text":"decide whether we\u0027re going to do it horizontal slicing or vertical slicing."},{"Start":"00:35.190 ","End":"00:37.010","Text":"D x d y or d y, d x."},{"Start":"00:37.010 ","End":"00:39.420","Text":"Let\u0027s start with this 1."},{"Start":"00:39.420 ","End":"00:41.610","Text":"It\u0027s obviously a straight line function."},{"Start":"00:41.610 ","End":"00:46.595","Text":"If we start with the x plus y equals 2,"},{"Start":"00:46.595 ","End":"00:48.755","Text":"we could draw it by,"},{"Start":"00:48.755 ","End":"00:52.190","Text":"for example, plotting a few points."},{"Start":"00:52.190 ","End":"00:53.734","Text":"I\u0027d like to take the intercepts."},{"Start":"00:53.734 ","End":"00:56.360","Text":"For example, if x equals 0,"},{"Start":"00:56.360 ","End":"00:59.030","Text":"then y equals 2,"},{"Start":"00:59.030 ","End":"01:01.190","Text":"and if y equals 0,"},{"Start":"01:01.190 ","End":"01:03.505","Text":"then x equals 2,"},{"Start":"01:03.505 ","End":"01:12.770","Text":"and so I know that the points 0 comma 2 and 2 comma 0 are on the straight line,"},{"Start":"01:12.770 ","End":"01:15.020","Text":"I can draw the line through those."},{"Start":"01:15.020 ","End":"01:19.430","Text":"Let me write this is 2 and this is 2 also."},{"Start":"01:19.430 ","End":"01:21.500","Text":"Next, the other 1,"},{"Start":"01:21.500 ","End":"01:25.505","Text":"the other 1 is a parabola."},{"Start":"01:25.505 ","End":"01:27.800","Text":"If I extract y,"},{"Start":"01:27.800 ","End":"01:35.225","Text":"if I bring the 4 y to this side and the 4 to this side and then divide by 4."},{"Start":"01:35.225 ","End":"01:38.940","Text":"Then instead of, I\u0027ll write it first of all,"},{"Start":"01:38.940 ","End":"01:43.680","Text":"4 equals x squared minus 4 y."},{"Start":"01:43.680 ","End":"01:47.480","Text":"Then I\u0027ll bring the 4 y over here and then divide by 4."},{"Start":"01:47.480 ","End":"01:51.485","Text":"Basically, I\u0027ll get y equals x squared over 4,"},{"Start":"01:51.485 ","End":"01:53.539","Text":"say a quarter x squared,"},{"Start":"01:53.539 ","End":"01:57.950","Text":"and then the minus 4 after I divide it by 4 is minus 1."},{"Start":"01:57.950 ","End":"02:05.930","Text":"For a parabola, what I like to do is to find the intersections with the axes,"},{"Start":"02:05.930 ","End":"02:07.340","Text":"the x-intercept, the y-intercept,"},{"Start":"02:07.340 ","End":"02:10.560","Text":"and they\u0027re usually like the vertex also."},{"Start":"02:10.900 ","End":"02:16.010","Text":"There\u0027s a formula for the vertex minus b over 2 a which comes out 0."},{"Start":"02:16.010 ","End":"02:19.415","Text":"Or you can differentiate the right-hand side and set to 0."},{"Start":"02:19.415 ","End":"02:25.890","Text":"We know that the vertex is when x equals 0 and when x is 0,"},{"Start":"02:26.710 ","End":"02:29.450","Text":"which is also an intercept,"},{"Start":"02:29.450 ","End":"02:32.320","Text":"then y equals minus 1."},{"Start":"02:32.320 ","End":"02:36.470","Text":"I know that this point is the point where y is minus"},{"Start":"02:36.470 ","End":"02:43.070","Text":"1 and intersection with the x-axis."},{"Start":"02:43.070 ","End":"02:44.825","Text":"Let y equal 0."},{"Start":"02:44.825 ","End":"02:51.150","Text":"If y is 0, I get 1/4 x squared minus 1 equals 0."},{"Start":"02:51.150 ","End":"02:56.575","Text":"I get 1/4 x squared equals 1."},{"Start":"02:56.575 ","End":"03:04.000","Text":"Multiply both sides by 4 x squared equals 4 x equals plus or minus 2,"},{"Start":"03:04.000 ","End":"03:07.545","Text":"0 comma 2 I already have,"},{"Start":"03:07.545 ","End":"03:09.470","Text":"so no point writing it again."},{"Start":"03:09.470 ","End":"03:12.605","Text":"It\u0027s on the parabola and on the line, and the other 1,"},{"Start":"03:12.605 ","End":"03:16.950","Text":"this 1 here is where x is minus 2,"},{"Start":"03:16.950 ","End":"03:20.510","Text":"so I have these and these and I draw a parabola,"},{"Start":"03:20.510 ","End":"03:28.640","Text":"and the region that\u0027s bounded by the 2 of them"},{"Start":"03:28.640 ","End":"03:32.180","Text":"is what I\u0027ve highlighted here in yellow and I\u0027ve called"},{"Start":"03:32.180 ","End":"03:36.844","Text":"it D. We might also need this point."},{"Start":"03:36.844 ","End":"03:40.400","Text":"I think either way we do the integral,"},{"Start":"03:40.400 ","End":"03:44.525","Text":"whichever way we slice it we could need this point."},{"Start":"03:44.525 ","End":"03:49.340","Text":"This point is where the parabola cuts the straight line."},{"Start":"03:49.340 ","End":"03:57.820","Text":"I can take from here 1/4 x squared minus 1."},{"Start":"03:57.820 ","End":"04:00.570","Text":"In fact, I\u0027ll do it over here,"},{"Start":"04:00.570 ","End":"04:04.370","Text":"equals, well, here I\u0027ve got y in terms of x here."},{"Start":"04:04.370 ","End":"04:06.305","Text":"If I put y in terms of x,"},{"Start":"04:06.305 ","End":"04:09.975","Text":"y equals 2 minus x."},{"Start":"04:09.975 ","End":"04:12.945","Text":"This is equal to 2 minus x,"},{"Start":"04:12.945 ","End":"04:16.775","Text":"and then if I bring everything to the left-hand side,"},{"Start":"04:16.775 ","End":"04:25.740","Text":"I\u0027ve got 1/4 x squared plus x minus 1 minus 2 equals 0,"},{"Start":"04:25.740 ","End":"04:33.410","Text":"multiply by 4 x squared plus 4 x minus 12 equals 0."},{"Start":"04:33.410 ","End":"04:35.390","Text":"I\u0027m not going to solve this for you,"},{"Start":"04:35.390 ","End":"04:36.770","Text":"I\u0027ll just tell you the solutions,"},{"Start":"04:36.770 ","End":"04:38.555","Text":"we have 2 solutions."},{"Start":"04:38.555 ","End":"04:43.790","Text":"X equals 2 or x equals minus 6."},{"Start":"04:43.790 ","End":"04:48.755","Text":"X equals 2 is no surprise because we already noticed that they cut each other here."},{"Start":"04:48.755 ","End":"04:51.290","Text":"The other 1 is x equals minus 6."},{"Start":"04:51.290 ","End":"04:53.675","Text":"When x equals minus 6,"},{"Start":"04:53.675 ","End":"05:01.085","Text":"this 1 will give me if I substitute into y in either 1 of them."},{"Start":"05:01.085 ","End":"05:04.730","Text":"Let\u0027s say here, y is 2 minus minus 6."},{"Start":"05:04.730 ","End":"05:07.775","Text":"That gives me the y equals 8."},{"Start":"05:07.775 ","End":"05:10.265","Text":"This would be the point."},{"Start":"05:10.265 ","End":"05:11.930","Text":"Well, I can just write it here."},{"Start":"05:11.930 ","End":"05:14.285","Text":"Here I have 8,"},{"Start":"05:14.285 ","End":"05:18.905","Text":"and if I draw dotted line here,"},{"Start":"05:18.905 ","End":"05:21.755","Text":"this would be minus 6."},{"Start":"05:21.755 ","End":"05:24.510","Text":"We have everything we need."},{"Start":"05:25.490 ","End":"05:27.735","Text":"We want the area,"},{"Start":"05:27.735 ","End":"05:32.720","Text":"the area of a region is simply the double"},{"Start":"05:32.720 ","End":"05:38.620","Text":"integral of 1 and sometimes you don\u0027t even write the 1 of 1,"},{"Start":"05:38.620 ","End":"05:42.890","Text":"and then I don\u0027t know if I\u0027m going to do it d x d y or d y d x ciphers"},{"Start":"05:42.890 ","End":"05:47.360","Text":"just to say d a over the region D. Now,"},{"Start":"05:47.360 ","End":"05:48.740","Text":"this is the big question."},{"Start":"05:48.740 ","End":"05:52.100","Text":"Are we going to slice it horizontally as"},{"Start":"05:52.100 ","End":"05:56.150","Text":"a type 2 region or vertically as a type 1 region?"},{"Start":"05:56.150 ","End":"06:00.875","Text":"I\u0027m going to argue that it\u0027s better to take vertical slices."},{"Start":"06:00.875 ","End":"06:04.880","Text":"Because if I take vertical slices wherever I take my x,"},{"Start":"06:04.880 ","End":"06:06.785","Text":"if I took my x here,"},{"Start":"06:06.785 ","End":"06:15.135","Text":"then the vertical slice through x would cut the line and the parabola,"},{"Start":"06:15.135 ","End":"06:19.135","Text":"and even if I did it over here,"},{"Start":"06:19.135 ","End":"06:22.945","Text":"it would still be from the parabola to the line."},{"Start":"06:22.945 ","End":"06:24.670","Text":"From the parabola to the line,"},{"Start":"06:24.670 ","End":"06:27.695","Text":"there is no different cases."},{"Start":"06:27.695 ","End":"06:33.595","Text":"On the other hand, if I made horizontal slices through a typical y,"},{"Start":"06:33.595 ","End":"06:35.995","Text":"sometimes for some y,"},{"Start":"06:35.995 ","End":"06:40.000","Text":"I would cross the parabola and the line."},{"Start":"06:40.000 ","End":"06:41.755","Text":"But if I did it down here,"},{"Start":"06:41.755 ","End":"06:44.350","Text":"I\u0027d be crossing the parabola twice,"},{"Start":"06:44.350 ","End":"06:46.630","Text":"so I have to separate into cases,"},{"Start":"06:46.630 ","End":"06:51.500","Text":"so I\u0027d rather do it with vertical slices."},{"Start":"06:51.500 ","End":"06:54.045","Text":"I just put this out the way."},{"Start":"06:54.045 ","End":"07:00.080","Text":"The outer integral we\u0027re going to do with x from minus 6-2."},{"Start":"07:00.080 ","End":"07:05.960","Text":"I\u0027ll just write that minus equal even emphasize x equals minus 6 to 2,"},{"Start":"07:05.960 ","End":"07:08.900","Text":"and that will be d x, and for a given x,"},{"Start":"07:08.900 ","End":"07:12.275","Text":"y will go from the parabola to the line."},{"Start":"07:12.275 ","End":"07:17.910","Text":"I\u0027ve got the integral d y from this point to this point."},{"Start":"07:17.910 ","End":"07:20.155","Text":"Now, I have these points."},{"Start":"07:20.155 ","End":"07:22.790","Text":"I have, for the parabola,"},{"Start":"07:22.790 ","End":"07:27.535","Text":"I have this, and for the line I have this,"},{"Start":"07:27.535 ","End":"07:33.470","Text":"and so the limits for y are the parabola 1/4 x"},{"Start":"07:33.470 ","End":"07:40.070","Text":"squared minus 1 up to the line which we said was here 2 minus x,"},{"Start":"07:40.070 ","End":"07:46.820","Text":"and the function is just 1,1 d y d x."},{"Start":"07:46.820 ","End":"07:49.505","Text":"Let\u0027s start with the inner integral."},{"Start":"07:49.505 ","End":"07:53.390","Text":"Shade it so we can see this integral."},{"Start":"07:53.390 ","End":"07:56.570","Text":"Maybe I\u0027ll label it asterisk and do it at the side."},{"Start":"07:56.570 ","End":"08:01.970","Text":"The asterisk is the integral from"},{"Start":"08:01.970 ","End":"08:10.670","Text":"1/4 x squared minus 1 to 2 minus x of 1 d y,"},{"Start":"08:10.670 ","End":"08:14.869","Text":"which is, and the integral of 1 is just y,"},{"Start":"08:14.869 ","End":"08:24.725","Text":"so I have y taken from 1 quarter x squared minus 1 to 2 minus x,"},{"Start":"08:24.725 ","End":"08:26.765","Text":"put it in brackets for emphasis,"},{"Start":"08:26.765 ","End":"08:30.790","Text":"which means that a substitute y equals this and then y equals this and subtract."},{"Start":"08:30.790 ","End":"08:40.680","Text":"What I get is just this minus this 2 minus x minus this in bracket,"},{"Start":"08:40.680 ","End":"08:43.520","Text":"so it\u0027s minus 1, oh, just copy it then."},{"Start":"08:43.520 ","End":"08:46.115","Text":"1/4 x squared minus 1."},{"Start":"08:46.115 ","End":"08:49.475","Text":"But when I put it back here,"},{"Start":"08:49.475 ","End":"08:51.230","Text":"this is the asterisk."},{"Start":"08:51.230 ","End":"08:53.300","Text":"I\u0027ll just simplify it,"},{"Start":"08:53.300 ","End":"08:58.790","Text":"and so we get the integral from minus 6-2."},{"Start":"08:58.790 ","End":"09:02.370","Text":"Let\u0027s see, let\u0027s do it in order."},{"Start":"09:02.370 ","End":"09:05.160","Text":"I\u0027ll do the numbers first,"},{"Start":"09:05.160 ","End":"09:08.100","Text":"2 minus minus 1 is"},{"Start":"09:08.100 ","End":"09:14.550","Text":"3 minus x minus"},{"Start":"09:14.550 ","End":"09:19.875","Text":"a quarter x squared d x."},{"Start":"09:19.875 ","End":"09:22.220","Text":"Straightforward in integral."},{"Start":"09:22.220 ","End":"09:25.070","Text":"Let\u0027s do the integral."},{"Start":"09:25.070 ","End":"09:34.084","Text":"This equals integral of this is 3 x minus x squared over 2 or 1/2 x squared."},{"Start":"09:34.084 ","End":"09:35.960","Text":"Here I raise the power by 1,"},{"Start":"09:35.960 ","End":"09:37.940","Text":"I get 3 and divide by 3,"},{"Start":"09:37.940 ","End":"09:41.415","Text":"so it\u0027s minus 1/12th x cubed,"},{"Start":"09:41.415 ","End":"09:45.690","Text":"and all this I want between minus 6 and 2,"},{"Start":"09:45.690 ","End":"09:48.000","Text":"so I just going to subtract 2 numbers."},{"Start":"09:48.000 ","End":"09:50.220","Text":"Let\u0027s see if I plug in 2,"},{"Start":"09:50.220 ","End":"09:53.580","Text":"I get 6 times 2 is 6."},{"Start":"09:53.580 ","End":"09:59.970","Text":"The 1/2 2 squared is 2,"},{"Start":"09:59.970 ","End":"10:03.280","Text":"1/12th x cubed is 8 over 12."},{"Start":"10:03.280 ","End":"10:06.990","Text":"I\u0027ll leave it as 8 over 12 for now,"},{"Start":"10:07.090 ","End":"10:11.780","Text":"and then minus what I get when I plug in minus 6,"},{"Start":"10:11.780 ","End":"10:15.050","Text":"then that would be minus 18,"},{"Start":"10:15.050 ","End":"10:24.160","Text":"minus 1/2 times 36 is 18,"},{"Start":"10:24.160 ","End":"10:30.535","Text":"and then minus 6 cubed is,"},{"Start":"10:30.535 ","End":"10:33.440","Text":"well, maybe I\u0027ll just write it as."},{"Start":"10:33.440 ","End":"10:35.240","Text":"That\u0027s what\u0027s going to be a plus,"},{"Start":"10:35.240 ","End":"10:36.350","Text":"because we\u0027ve got a minus, minus,"},{"Start":"10:36.350 ","End":"10:38.435","Text":"minus and another minus."},{"Start":"10:38.435 ","End":"10:41.750","Text":"I\u0027ll just leave it as 6 times 6 times 6."},{"Start":"10:41.750 ","End":"10:44.825","Text":"Rather than multiplying it out to the 6 cubed,"},{"Start":"10:44.825 ","End":"10:48.240","Text":"because I know I seem to have a 1/12th and middle cancel."},{"Start":"10:49.750 ","End":"10:53.150","Text":"Let\u0027s see, I\u0027ll continue over here."},{"Start":"10:53.150 ","End":"10:55.430","Text":"Let\u0027s do the whole number parts."},{"Start":"10:55.430 ","End":"10:59.525","Text":"The whole number parts I have 6 minus 2,"},{"Start":"10:59.525 ","End":"11:07.485","Text":"and then I have plus 18 plus 18 the whole numbers,"},{"Start":"11:07.485 ","End":"11:09.590","Text":"and then the fractions,"},{"Start":"11:09.590 ","End":"11:11.510","Text":"I have a minus 8/12ths,"},{"Start":"11:11.510 ","End":"11:19.160","Text":"a is minus 2/3, and here,"},{"Start":"11:19.160 ","End":"11:21.890","Text":"well, let\u0027s see, 6 times 6 is 36,"},{"Start":"11:21.890 ","End":"11:26.015","Text":"36 over 12 is 3,"},{"Start":"11:26.015 ","End":"11:33.570","Text":"so 3 times 6 is 18."},{"Start":"11:33.570 ","End":"11:38.890","Text":"Actually I have a minus 18 is not a fraction in the end."},{"Start":"11:38.890 ","End":"11:43.395","Text":"Which means that this will cancel with this. What do I get?"},{"Start":"11:43.395 ","End":"11:47.535","Text":"18 and 6 is 24 minus 2 is 22,"},{"Start":"11:47.535 ","End":"11:55.950","Text":"22 minus 2/3 is 21 and 1/3,"},{"Start":"11:55.950 ","End":"12:01.320","Text":"and I will highlight this result and that is our answer and we are done."}],"ID":8679},{"Watched":false,"Name":"Exercise 1 part b","Duration":"14m 49s","ChapterTopicVideoID":8464,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.580","Text":"In this exercise, we\u0027re given 2 curves"},{"Start":"00:03.580 ","End":"00:07.285","Text":"and we have to find the area of the region bounded by them."},{"Start":"00:07.285 ","End":"00:12.325","Text":"The curves defined in terms of a parameter a."},{"Start":"00:12.325 ","End":"00:15.910","Text":"If you have difficulty with the parameter,"},{"Start":"00:15.910 ","End":"00:19.690","Text":"you could first try and sketch it with the specific a,"},{"Start":"00:19.690 ","End":"00:23.860","Text":"say a equals 1 or a equals 2 and see how it looks."},{"Start":"00:23.860 ","End":"00:25.180","Text":"Then to either general case,"},{"Start":"00:25.180 ","End":"00:28.330","Text":"I\u0027m going to go straight for the general case."},{"Start":"00:28.330 ","End":"00:34.000","Text":"You ought to be familiar with the hyperbola xy equals some positive"},{"Start":"00:34.000 ","End":"00:39.790","Text":"constant and know straight away to draw some axes and know that it goes here and here,"},{"Start":"00:39.790 ","End":"00:42.555","Text":"there\u0027s 2 bits, an asymptote here and here."},{"Start":"00:42.555 ","End":"00:45.215","Text":"But let\u0027s assume you forgot that."},{"Start":"00:45.215 ","End":"00:47.375","Text":"I\u0027ll start again."},{"Start":"00:47.375 ","End":"00:52.280","Text":"We could try just making a table of some values for"},{"Start":"00:52.280 ","End":"00:57.430","Text":"x and y. I\u0027m talking about this hyperbola."},{"Start":"00:57.430 ","End":"01:01.754","Text":"It\u0027s nice to note that if x and y were both a,"},{"Start":"01:01.754 ","End":"01:03.825","Text":"then that would work."},{"Start":"01:03.825 ","End":"01:06.560","Text":"X could be a, y could be a."},{"Start":"01:06.560 ","End":"01:11.255","Text":"If x was, let\u0027s say twice a,"},{"Start":"01:11.255 ","End":"01:15.230","Text":"then y would be a squared over 2a,"},{"Start":"01:15.230 ","End":"01:22.549","Text":"that would be a over 2, and it\u0027s symmetrical."},{"Start":"01:22.549 ","End":"01:24.965","Text":"So if I switch x and y,"},{"Start":"01:24.965 ","End":"01:26.960","Text":"if y is a over 2,"},{"Start":"01:26.960 ","End":"01:30.275","Text":"then x is 2a and the product is still a,"},{"Start":"01:30.275 ","End":"01:32.030","Text":"because even another symmetry,"},{"Start":"01:32.030 ","End":"01:34.115","Text":"if I make them both negative,"},{"Start":"01:34.115 ","End":"01:37.310","Text":"then it\u0027s still going to work because minus times minus."},{"Start":"01:37.310 ","End":"01:39.590","Text":"I can get 3 extra points, minus a,"},{"Start":"01:39.590 ","End":"01:43.735","Text":"minus a and minus 2a,"},{"Start":"01:43.735 ","End":"01:46.250","Text":"minus a over 2,"},{"Start":"01:46.250 ","End":"01:49.250","Text":"and minus a over 2, minus 2a."},{"Start":"01:49.250 ","End":"01:50.660","Text":"Let\u0027s see what this looks like."},{"Start":"01:50.660 ","End":"01:56.805","Text":"Let\u0027s suppose that this unit here is a and we have here a."},{"Start":"01:56.805 ","End":"02:02.074","Text":"We have a, a on the parabola."},{"Start":"02:02.074 ","End":"02:04.210","Text":"If I take 2a,"},{"Start":"02:04.210 ","End":"02:06.960","Text":"then it\u0027s only 1/2,"},{"Start":"02:06.960 ","End":"02:08.580","Text":"so it gets lower."},{"Start":"02:08.580 ","End":"02:10.970","Text":"If I take 1/2 a here,"},{"Start":"02:10.970 ","End":"02:14.150","Text":"I\u0027ve got 2a here."},{"Start":"02:14.150 ","End":"02:17.445","Text":"So I\u0027ve got this point."},{"Start":"02:17.445 ","End":"02:22.400","Text":"Like I said, you should have some familiarity with this."},{"Start":"02:22.400 ","End":"02:26.270","Text":"You can see that there are asymptotes."},{"Start":"02:26.270 ","End":"02:28.790","Text":"If you wrote it as y equals a squared over x,"},{"Start":"02:28.790 ","End":"02:32.405","Text":"when x goes to infinity, it goes to 0."},{"Start":"02:32.405 ","End":"02:36.589","Text":"It\u0027s not important to be all that accurate."},{"Start":"02:36.589 ","End":"02:40.314","Text":"This is the general shape and I\u0027ve got a 1 on the other side."},{"Start":"02:40.314 ","End":"02:48.750","Text":"You might have a minus a here and a minus 2a here and here also,"},{"Start":"02:48.750 ","End":"02:53.040","Text":"maybe a minus a, minus 2a."},{"Start":"02:53.040 ","End":"02:58.210","Text":"I\u0027ve got this point here and halfway here,"},{"Start":"02:58.210 ","End":"03:03.350","Text":"and given 2a it\u0027s 1/2 a here."},{"Start":"03:03.530 ","End":"03:05.800","Text":"Not the greatest sketches,"},{"Start":"03:05.800 ","End":"03:08.410","Text":"but that\u0027s not so important."},{"Start":"03:08.410 ","End":"03:13.165","Text":"Now that\u0027s the hyperbola with the 2 branches of the hyperbola."},{"Start":"03:13.165 ","End":"03:15.654","Text":"Now what about the straight line?"},{"Start":"03:15.654 ","End":"03:20.425","Text":"Perhaps I\u0027ll just do the intercept with the axis."},{"Start":"03:20.425 ","End":"03:25.225","Text":"Let\u0027s see if x is 0,"},{"Start":"03:25.225 ","End":"03:34.900","Text":"then y is 5 over 2a."},{"Start":"03:34.900 ","End":"03:41.400","Text":"I\u0027ll write it as 2.5a and if y is 0,"},{"Start":"03:41.400 ","End":"03:45.585","Text":"then x is also 2.5a."},{"Start":"03:45.585 ","End":"03:48.645","Text":"Let\u0027s see, this was a, this was 2a,"},{"Start":"03:48.645 ","End":"03:52.280","Text":"a bit more. I don\u0027t know."},{"Start":"03:52.280 ","End":"03:56.639","Text":"Let\u0027s say here and here again,"},{"Start":"03:57.280 ","End":"04:05.490","Text":"a, 2a, let\u0027s say 2.5a somewhere."},{"Start":"04:06.140 ","End":"04:14.320","Text":"Let\u0027s say 2.5a here and here\u0027s a straight line going through them."},{"Start":"04:14.320 ","End":"04:16.165","Text":"It\u0027s a little approximate."},{"Start":"04:16.165 ","End":"04:18.430","Text":"But we can see that the region"},{"Start":"04:18.430 ","End":"04:21.040","Text":"between them has nothing to do with this part of the hyperbola."},{"Start":"04:21.040 ","End":"04:23.710","Text":"We want this here, I\u0027ll shade it."},{"Start":"04:23.710 ","End":"04:26.710","Text":"Let\u0027s give it a name D, it\u0027s too small,"},{"Start":"04:26.710 ","End":"04:31.990","Text":"so I\u0027ll just indicate that this is my region D. The area"},{"Start":"04:31.990 ","End":"04:38.420","Text":"is the double integral over D of just the function 1."},{"Start":"04:38.420 ","End":"04:42.490","Text":"I\u0027ll write it as dA, and then we\u0027ll decide whether we want to do it as dx,"},{"Start":"04:42.490 ","End":"04:44.385","Text":"dy or dy, dx."},{"Start":"04:44.385 ","End":"04:46.335","Text":"Actually, it doesn\u0027t really matter,"},{"Start":"04:46.335 ","End":"04:49.820","Text":"it\u0027s so symmetrical in x and y. I\u0027ll tell you what."},{"Start":"04:49.820 ","End":"04:52.385","Text":"Let\u0027s just decide to do vertical slices,"},{"Start":"04:52.385 ","End":"04:55.110","Text":"which will mean dy, dx."},{"Start":"04:55.110 ","End":"04:58.910","Text":"To find the limits on x and where it goes from and to,"},{"Start":"04:58.910 ","End":"05:02.540","Text":"I need to find the intersection of these 2 curves."},{"Start":"05:02.540 ","End":"05:06.485","Text":"The equation of this hyperbola,"},{"Start":"05:06.485 ","End":"05:14.250","Text":"I can get simply by extracting y in terms of x. I could say that y equals 5 over 2,"},{"Start":"05:14.250 ","End":"05:21.440","Text":"I\u0027ll use decimal 2.5a minus x."},{"Start":"05:21.440 ","End":"05:24.020","Text":"Sorry, that\u0027s the line."},{"Start":"05:24.020 ","End":"05:27.140","Text":"Sorry, gotten the wrong way round and moved this here."},{"Start":"05:27.140 ","End":"05:28.685","Text":"The hyperbola of course,"},{"Start":"05:28.685 ","End":"05:31.175","Text":"is by extracting y from here,"},{"Start":"05:31.175 ","End":"05:38.515","Text":"is that y is equal to a squared over x."},{"Start":"05:38.515 ","End":"05:41.480","Text":"Now I want to see where they intersect,"},{"Start":"05:41.480 ","End":"05:42.770","Text":"this point and this point."},{"Start":"05:42.770 ","End":"05:44.645","Text":"I just equate."},{"Start":"05:44.645 ","End":"05:48.110","Text":"I say y equals this and y equals this."},{"Start":"05:48.110 ","End":"05:53.940","Text":"I get the equation a squared over x is equal to"},{"Start":"05:53.940 ","End":"06:03.020","Text":"2.5a minus x. I want to get rid of fractions I\u0027ll multiply both sides by x."},{"Start":"06:03.020 ","End":"06:04.940","Text":"Now multiply by 2x,"},{"Start":"06:04.940 ","End":"06:08.665","Text":"and then I can get rid of this 1/2 as well."},{"Start":"06:08.665 ","End":"06:18.450","Text":"I get 2a squared equals 5ax minus 2x squared,"},{"Start":"06:18.450 ","End":"06:19.995","Text":"everything to the left."},{"Start":"06:19.995 ","End":"06:29.030","Text":"2x squared minus 5ax,"},{"Start":"06:29.030 ","End":"06:36.285","Text":"x plus 2a squared equals 0."},{"Start":"06:36.285 ","End":"06:38.375","Text":"Now we\u0027re going to use,"},{"Start":"06:38.375 ","End":"06:41.990","Text":"well either the quadratic formula or you could factorize."},{"Start":"06:41.990 ","End":"06:44.495","Text":"Let me just tell you what the answers are."},{"Start":"06:44.495 ","End":"06:46.535","Text":"We have 2 solutions,"},{"Start":"06:46.535 ","End":"06:52.710","Text":"x equals a over 2 and x equals 2a."},{"Start":"06:55.190 ","End":"06:57.920","Text":"Well, the picture was a bit off."},{"Start":"06:57.920 ","End":"06:59.840","Text":"I mean, when x is a over 2,"},{"Start":"06:59.840 ","End":"07:02.400","Text":"we already had that y is 2a."},{"Start":"07:02.740 ","End":"07:06.830","Text":"This point should have really been more here."},{"Start":"07:06.830 ","End":"07:11.685","Text":"In any event, the x is,"},{"Start":"07:11.685 ","End":"07:15.360","Text":"some dotted lines, a over 2."},{"Start":"07:15.360 ","End":"07:21.990","Text":"Here, this intersection here really should have been this point here,"},{"Start":"07:22.220 ","End":"07:25.410","Text":"should be 2a exactly."},{"Start":"07:25.410 ","End":"07:28.010","Text":"Yeah. Well, the picture was rough, but anyway,"},{"Start":"07:28.010 ","End":"07:35.625","Text":"we\u0027ve got the limits and now we can write it as a double integral."},{"Start":"07:35.625 ","End":"07:41.410","Text":"What we wrote here can be written as an integral where x goes"},{"Start":"07:41.410 ","End":"07:48.460","Text":"from a/2 on the bottom"},{"Start":"07:48.460 ","End":"07:52.705","Text":"to 2a on the top,"},{"Start":"07:52.705 ","End":"08:01.120","Text":"this is the left and this is the right a/2-2a and that\u0027s dx."},{"Start":"08:01.120 ","End":"08:06.220","Text":"Then for each such x let say I take a typical x here,"},{"Start":"08:06.220 ","End":"08:14.800","Text":"then a vertical line through a will cut here and here."},{"Start":"08:14.800 ","End":"08:19.780","Text":"We need to go from the hyperbola to the line which means"},{"Start":"08:19.780 ","End":"08:28.090","Text":"that y goes from the hyperbola was a squared over x."},{"Start":"08:28.090 ","End":"08:34.465","Text":"The line was 2 and a 1/2 a minus x,"},{"Start":"08:34.465 ","End":"08:38.860","Text":"let\u0027s say 2.5a minus x dy."},{"Start":"08:38.860 ","End":"08:42.160","Text":"The function for the area is just 1."},{"Start":"08:42.160 ","End":"08:45.550","Text":"Let\u0027s do the inner integral first."},{"Start":"08:45.550 ","End":"08:48.500","Text":"I\u0027ll highlight it."},{"Start":"08:50.250 ","End":"08:55.190","Text":"I\u0027ll do this bit at the side, call it asterisk."},{"Start":"08:57.660 ","End":"09:06.730","Text":"The integral of 1 is y and y taken between the lower limit,"},{"Start":"09:06.730 ","End":"09:09.625","Text":"which is a squared over x,"},{"Start":"09:09.625 ","End":"09:15.460","Text":"upper limit, 2.5 a minus x."},{"Start":"09:15.460 ","End":"09:18.325","Text":"Substitute this, substitute this and subtract."},{"Start":"09:18.325 ","End":"09:21.310","Text":"It just comes out to be"},{"Start":"09:21.310 ","End":"09:31.765","Text":"2.5a minus x minus a squared over x."},{"Start":"09:31.765 ","End":"09:34.570","Text":"Now I plug it back in here."},{"Start":"09:34.570 ","End":"09:38.200","Text":"Wherever it was, asterisk I put that."},{"Start":"09:38.200 ","End":"09:41.620","Text":"Now I get the integral, this time,"},{"Start":"09:41.620 ","End":"09:49.550","Text":"the outer 1 from a/2-2a of just this."},{"Start":"09:49.710 ","End":"09:52.929","Text":"I\u0027m thinking I\u0027ll go back to fractions."},{"Start":"09:52.929 ","End":"09:54.610","Text":"I shouldn\u0027t really be mixing."},{"Start":"09:54.610 ","End":"10:04.075","Text":"This is 5/2a minus x minus a squared over x dx,"},{"Start":"10:04.075 ","End":"10:08.330","Text":"just a simple integral in a variable x."},{"Start":"10:08.370 ","End":"10:14.829","Text":"Now let\u0027s actually do the integral, equals."},{"Start":"10:14.829 ","End":"10:21.295","Text":"The integral of a constant is just that constant times x,"},{"Start":"10:21.295 ","End":"10:23.575","Text":"a is just a parameter constant."},{"Start":"10:23.575 ","End":"10:33.460","Text":"The integral of x is 1/2 x squared and the integral of a squared over x,"},{"Start":"10:33.460 ","End":"10:35.470","Text":"if it was just 1/x,"},{"Start":"10:35.470 ","End":"10:39.130","Text":"that would be natural log of x. I don\u0027t need"},{"Start":"10:39.130 ","End":"10:43.120","Text":"absolute value because x is in positive area,"},{"Start":"10:43.120 ","End":"10:47.965","Text":"but I do need the a squared and the minus,"},{"Start":"10:47.965 ","End":"10:50.800","Text":"minus a squared natural log of x."},{"Start":"10:50.800 ","End":"10:57.650","Text":"All this taken between a/2 and 2a."},{"Start":"10:58.050 ","End":"11:00.820","Text":"Let\u0027s plug in the 2a."},{"Start":"11:00.820 ","End":"11:05.515","Text":"The top part, if I get 2a x is 2a,"},{"Start":"11:05.515 ","End":"11:14.530","Text":"2a times 5/2 is 5a times a, that\u0027s 5a squared."},{"Start":"11:14.530 ","End":"11:24.410","Text":"If I plug it in here 2a all squared is 4a squared over 2 is 2a squared."},{"Start":"11:25.110 ","End":"11:32.875","Text":"Here I have a squared natural log of 2a."},{"Start":"11:32.875 ","End":"11:43.180","Text":"First part minus plug-in a/2."},{"Start":"11:43.180 ","End":"11:46.180","Text":"a/2, I still have a squared x,"},{"Start":"11:46.180 ","End":"11:53.665","Text":"but the 1/2 with the 5/2 gives me 5/4 a squared."},{"Start":"11:53.665 ","End":"11:56.620","Text":"Then if I plug in a/2 here,"},{"Start":"11:56.620 ","End":"12:02.215","Text":"I\u0027ll get a 1/4 with the 1/2 will be minus 1/8 a squared"},{"Start":"12:02.215 ","End":"12:12.170","Text":"and minus a squared natural log of a/2."},{"Start":"12:12.180 ","End":"12:17.980","Text":"This is the answer and you could stop here."},{"Start":"12:17.980 ","End":"12:20.750","Text":"I\u0027d just like to tidy it up a bit."},{"Start":"12:20.750 ","End":"12:24.090","Text":"Just maybe make it neater."},{"Start":"12:24.090 ","End":"12:29.740","Text":"Basically, this could be left as an answer. Let\u0027s see it."},{"Start":"12:29.740 ","End":"12:32.470","Text":"Let\u0027s combine the terms that contain a squared."},{"Start":"12:32.470 ","End":"12:33.550","Text":"This one, this one, this one,"},{"Start":"12:33.550 ","End":"12:37.060","Text":"and this one, 5 minus 2 is 3."},{"Start":"12:37.060 ","End":"12:44.620","Text":"5/4 minus a 1/8 comes out to be 9/8."},{"Start":"12:44.620 ","End":"12:54.040","Text":"Let\u0027s see 3 is 24/8 minus 9/8 is 15/8 and this was a squared."},{"Start":"12:54.040 ","End":"13:01.120","Text":"The other thing we can do is we also have this with this is a"},{"Start":"13:01.120 ","End":"13:10.090","Text":"squared and let me put it as minus a squared, I\u0027ll take out."},{"Start":"13:10.090 ","End":"13:14.440","Text":"What I\u0027m left with is natural log"},{"Start":"13:14.440 ","End":"13:24.505","Text":"of 2a and because I took a minus out,"},{"Start":"13:24.505 ","End":"13:26.200","Text":"then this will still be"},{"Start":"13:26.200 ","End":"13:34.990","Text":"a minus natural log of a/2."},{"Start":"13:34.990 ","End":"13:37.780","Text":"I am actually even going to go further."},{"Start":"13:37.780 ","End":"13:42.280","Text":"I\u0027m going to simplify this bit at the side,"},{"Start":"13:42.280 ","End":"13:46.075","Text":"the difference of the logs is the log of the quotient."},{"Start":"13:46.075 ","End":"13:53.095","Text":"This is natural log of 2a divided by a/2,"},{"Start":"13:53.095 ","End":"13:57.680","Text":"which is the natural log of,"},{"Start":"13:59.790 ","End":"14:07.150","Text":"let\u0027s see, 2 over a 1/2 is 4."},{"Start":"14:07.150 ","End":"14:09.625","Text":"Since 4 is 2 squared,"},{"Start":"14:09.625 ","End":"14:11.680","Text":"it\u0027s natural log of 2 squared."},{"Start":"14:11.680 ","End":"14:14.995","Text":"I can write this as 2 natural log of 2."},{"Start":"14:14.995 ","End":"14:18.865","Text":"If I put back in here 2 natural log of 2,"},{"Start":"14:18.865 ","End":"14:25.210","Text":"I can still take a squared outside the brackets and I get a squared"},{"Start":"14:25.210 ","End":"14:33.925","Text":"times 15/8 minus 2 natural log 2."},{"Start":"14:33.925 ","End":"14:35.890","Text":"That\u0027s about the simplest you can get."},{"Start":"14:35.890 ","End":"14:38.935","Text":"Another reason I bothered with this simplification"},{"Start":"14:38.935 ","End":"14:42.430","Text":"is that this is the answer as given in the exercise book,"},{"Start":"14:42.430 ","End":"14:44.755","Text":"and I wanted to get the same answer."},{"Start":"14:44.755 ","End":"14:49.520","Text":"I\u0027ll just highlight the answer and we are done."}],"ID":8680},{"Watched":false,"Name":"Exercise 1 part c","Duration":"17m 36s","ChapterTopicVideoID":8465,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we have to compute the area of"},{"Start":"00:03.060 ","End":"00:06.540","Text":"the region bounded by these 3 curves here,"},{"Start":"00:06.540 ","End":"00:08.235","Text":"1, 2, and 3."},{"Start":"00:08.235 ","End":"00:11.385","Text":"The last 2 are obviously straight lines."},{"Start":"00:11.385 ","End":"00:14.295","Text":"This 1 is actually a circle."},{"Start":"00:14.295 ","End":"00:17.655","Text":"We\u0027re going to use the familiar tricks of completing the square."},{"Start":"00:17.655 ","End":"00:24.630","Text":"Let me start by writing x squared minus 2x, now,"},{"Start":"00:24.630 ","End":"00:26.505","Text":"I\u0027ve deliberately left a gap,"},{"Start":"00:26.505 ","End":"00:30.420","Text":"plus y squared equals 0,"},{"Start":"00:30.420 ","End":"00:32.805","Text":"I brought the 2x to the left-hand side."},{"Start":"00:32.805 ","End":"00:39.745","Text":"Now, completing the square means adding something here so that this is something squared."},{"Start":"00:39.745 ","End":"00:44.435","Text":"It has to be x minus half of this squared,"},{"Start":"00:44.435 ","End":"00:48.185","Text":"and if I square it out, I get x squared minus 2x plus 1,"},{"Start":"00:48.185 ","End":"00:54.710","Text":"so I add plus 1 here but if I add 1 to the left-hand side,"},{"Start":"00:54.710 ","End":"00:57.050","Text":"I must add it also to the right-hand side,"},{"Start":"00:57.050 ","End":"01:00.275","Text":"and 1 can be written as 1 squared."},{"Start":"01:00.275 ","End":"01:03.565","Text":"At this point, you can see that it\u0027s a circle,"},{"Start":"01:03.565 ","End":"01:10.950","Text":"and the center is at the point where x is 1."},{"Start":"01:10.950 ","End":"01:13.680","Text":"This is like y minus 0 squared,"},{"Start":"01:13.680 ","End":"01:20.205","Text":"so y is 0, and the radius, which is this,"},{"Start":"01:20.205 ","End":"01:27.060","Text":"is also 1, so I can now sketch the circle,"},{"Start":"01:27.060 ","End":"01:29.190","Text":"and here it is,"},{"Start":"01:29.190 ","End":"01:32.970","Text":"the center is at 1, 0."},{"Start":"01:32.970 ","End":"01:36.390","Text":"Since the radius is 1, this is 2,"},{"Start":"01:36.390 ","End":"01:39.975","Text":"this is 0, if I need it,"},{"Start":"01:39.975 ","End":"01:42.300","Text":"then this is 1,"},{"Start":"01:42.300 ","End":"01:45.435","Text":"and this is minus 1,"},{"Start":"01:45.435 ","End":"01:50.030","Text":"and that\u0027s so much for the circle."},{"Start":"01:50.030 ","End":"01:52.055","Text":"Now, let\u0027s look at the others."},{"Start":"01:52.055 ","End":"01:57.335","Text":"Y equals 0 is simply the x axis."},{"Start":"01:57.335 ","End":"02:04.245","Text":"I\u0027ll just emphasize it a bit that this is the x-axis,"},{"Start":"02:04.245 ","End":"02:06.455","Text":"and perhaps I\u0027ll start labeling."},{"Start":"02:06.455 ","End":"02:09.229","Text":"This is the circle,"},{"Start":"02:09.229 ","End":"02:12.870","Text":"is x squared plus y squared equals 2x."},{"Start":"02:12.870 ","End":"02:16.910","Text":"This is the x-axis, y equals 0."},{"Start":"02:16.910 ","End":"02:18.835","Text":"Now, I need the last 1,"},{"Start":"02:18.835 ","End":"02:21.055","Text":"y equals x root 3."},{"Start":"02:21.055 ","End":"02:22.430","Text":"There\u0027s no constant term,"},{"Start":"02:22.430 ","End":"02:23.870","Text":"so it goes through the origin,"},{"Start":"02:23.870 ","End":"02:26.225","Text":"so I know that this is the point on here."},{"Start":"02:26.225 ","End":"02:29.405","Text":"Lets get another point, say x equals 1,"},{"Start":"02:29.405 ","End":"02:31.425","Text":"y would equal root 3,"},{"Start":"02:31.425 ","End":"02:35.640","Text":"is about 1.7, I don\u0027t know exactly, say here."},{"Start":"02:35.640 ","End":"02:37.305","Text":"Let\u0027s join these up,"},{"Start":"02:37.305 ","End":"02:38.580","Text":"and just label it,"},{"Start":"02:38.580 ","End":"02:44.000","Text":"y equals x root 3, something like this."},{"Start":"02:44.000 ","End":"02:49.205","Text":"I can already see that the area they\u0027re talking about,"},{"Start":"02:49.205 ","End":"02:54.850","Text":"the only area that has these 3 curves as borders is this bit here, I\u0027ll shade it,"},{"Start":"02:54.850 ","End":"02:58.820","Text":"and they\u0027ll label it as a D for domain,"},{"Start":"02:58.820 ","End":"03:04.370","Text":"even though its region should be R. I\u0027m looking at it and I\u0027m"},{"Start":"03:04.370 ","End":"03:10.760","Text":"trying to decide whether I should slice it vertically or horizontally,"},{"Start":"03:10.760 ","End":"03:12.950","Text":"but I\u0027m getting ahead of myself."},{"Start":"03:12.950 ","End":"03:16.175","Text":"The first thing I should say is that already I know the area"},{"Start":"03:16.175 ","End":"03:19.250","Text":"to express it as a double integral over a region."},{"Start":"03:19.250 ","End":"03:23.400","Text":"It\u0027s double integral over D of 1,"},{"Start":"03:23.400 ","End":"03:26.135","Text":"the function 1 and dA,"},{"Start":"03:26.135 ","End":"03:28.325","Text":"meaning I don\u0027t know if it\u0027s dx, dy or dy,"},{"Start":"03:28.325 ","End":"03:31.490","Text":"dx, and that\u0027s what I\u0027m trying to decide."},{"Start":"03:31.490 ","End":"03:37.375","Text":"Either way, I\u0027m going to have to split things up into 2 parts."},{"Start":"03:37.375 ","End":"03:40.685","Text":"If I break it up vertically,"},{"Start":"03:40.685 ","End":"03:46.990","Text":"I\u0027m going to have to take 1 part from here to here because then,"},{"Start":"03:46.990 ","End":"03:51.170","Text":"my slices will be from this line to this line,"},{"Start":"03:51.170 ","End":"03:54.140","Text":"the x-axis, and then from here,"},{"Start":"03:54.140 ","End":"04:01.490","Text":"it will be the upper semicircle to the x-axis but if I break it up horizontally,"},{"Start":"04:01.490 ","End":"04:02.840","Text":"I\u0027ll still have 2 cases,"},{"Start":"04:02.840 ","End":"04:08.810","Text":"I\u0027ll have to take a horizontal line here and say, okay,"},{"Start":"04:08.810 ","End":"04:13.580","Text":"between this value and this value is between"},{"Start":"04:13.580 ","End":"04:21.045","Text":"this line and this semicircle,"},{"Start":"04:21.045 ","End":"04:30.515","Text":"the right semicircle but from here to here it\u0027ll be between 2 parts of the semicircle."},{"Start":"04:30.515 ","End":"04:40.120","Text":"It seems much simpler to me to take vertical slices because on this side,"},{"Start":"04:40.120 ","End":"04:47.980","Text":"it\u0027s between the straight line and they already have y in terms of x and this x axis,"},{"Start":"04:47.980 ","End":"04:52.720","Text":"I already have y in terms of x plus y equals 0 and on the other part,"},{"Start":"04:52.720 ","End":"04:56.235","Text":"if I take my typical vertical strip,"},{"Start":"04:56.235 ","End":"05:01.730","Text":"say here, then the lower 1 is still 0,"},{"Start":"05:01.730 ","End":"05:04.455","Text":"which is convenient and the upper 1,"},{"Start":"05:04.455 ","End":"05:09.670","Text":"I just have to extract y in terms of x from this semicircle or circle."},{"Start":"05:09.670 ","End":"05:14.160","Text":"I\u0027ll take the square root and get these here."},{"Start":"05:14.160 ","End":"05:16.915","Text":"So that\u0027s what we\u0027ll go for."},{"Start":"05:16.915 ","End":"05:19.670","Text":"What I need to know though,"},{"Start":"05:19.670 ","End":"05:22.160","Text":"because if I\u0027m breaking it up into 2 separate bits,"},{"Start":"05:22.160 ","End":"05:24.484","Text":"is what is this point here?"},{"Start":"05:24.484 ","End":"05:26.225","Text":"What is the X of it?"},{"Start":"05:26.225 ","End":"05:28.085","Text":"Or if you like,"},{"Start":"05:28.085 ","End":"05:30.650","Text":"what is this point here?"},{"Start":"05:30.650 ","End":"05:37.040","Text":"Where do the sloped line and the circle intersect?"},{"Start":"05:37.040 ","End":"05:40.760","Text":"The simplest thing to do is just to take these 2,"},{"Start":"05:40.760 ","End":"05:43.250","Text":"as 2 equations and 2 unknowns,"},{"Start":"05:43.250 ","End":"05:46.940","Text":"and actually all I really care about is the x."},{"Start":"05:46.940 ","End":"05:53.750","Text":"The most obvious thing to do is to substitute y into this second equation,"},{"Start":"05:53.750 ","End":"05:58.420","Text":"and then we will get from here that x squared."},{"Start":"05:58.420 ","End":"06:05.030","Text":"Maybe I\u0027ll indicate that I\u0027m putting y into here in this equation,"},{"Start":"06:05.030 ","End":"06:09.710","Text":"now y squared is x squared root 3 squared,"},{"Start":"06:09.710 ","End":"06:12.024","Text":"it\u0027s just 3x squared,"},{"Start":"06:12.024 ","End":"06:13.740","Text":"when I square this,"},{"Start":"06:13.740 ","End":"06:17.350","Text":"equals 2x,"},{"Start":"06:17.350 ","End":"06:25.714","Text":"so I get 4x squared equals 2x or if I bring it to the other side,"},{"Start":"06:25.714 ","End":"06:30.120","Text":"I\u0027ll divide by 2 also,"},{"Start":"06:30.120 ","End":"06:35.900","Text":"4x squared is just 2x squared minus x equals 0."},{"Start":"06:35.900 ","End":"06:41.510","Text":"Okay? It was 4x squared minus 2x such divided by 2 and then that gives me,"},{"Start":"06:41.510 ","End":"06:44.060","Text":"I can factorize this because there\u0027s a missing c,"},{"Start":"06:44.060 ","End":"06:48.900","Text":"x times 2x minus 1 equals 0,"},{"Start":"06:48.900 ","End":"06:50.790","Text":"that gives me 2 solutions."},{"Start":"06:50.790 ","End":"06:55.429","Text":"Either x equals 0 or 2x minus 1 is 0,"},{"Start":"06:55.429 ","End":"06:58.685","Text":"which means that x equals 1/2."},{"Start":"06:58.685 ","End":"07:02.220","Text":"Now, we actually see it does intersect at 2 points,"},{"Start":"07:02.220 ","End":"07:05.105","Text":"but this is not the 1 we\u0027re looking for, x is 0."},{"Start":"07:05.105 ","End":"07:08.330","Text":"Well, good to know that we found it, but we want the other 1,"},{"Start":"07:08.330 ","End":"07:12.610","Text":"which means that x is equal to 1/2."},{"Start":"07:12.610 ","End":"07:19.130","Text":"Now, I can write it in here and I can split my integral up."},{"Start":"07:19.130 ","End":"07:24.005","Text":"I\u0027m doing it as a type 1 region vertical slices,"},{"Start":"07:24.005 ","End":"07:25.940","Text":"but in 2 cases,"},{"Start":"07:25.940 ","End":"07:28.580","Text":"up to 1/2, this slice,"},{"Start":"07:28.580 ","End":"07:30.335","Text":"and from 1/2 onwards,"},{"Start":"07:30.335 ","End":"07:34.730","Text":"so what we get the first bit from 0-1/2,"},{"Start":"07:34.730 ","End":"07:38.970","Text":"integral x goes from 0-1/2,"},{"Start":"07:39.820 ","End":"07:43.415","Text":"and this would be dx."},{"Start":"07:43.415 ","End":"07:48.660","Text":"Then we go on this slice"},{"Start":"07:48.660 ","End":"07:54.545","Text":"with y from x axis,"},{"Start":"07:54.545 ","End":"08:01.990","Text":"y equals 0, this curve to this 1 which is the straight line x root 3,"},{"Start":"08:01.990 ","End":"08:06.300","Text":"to x root 3, and that\u0027s dy,"},{"Start":"08:06.300 ","End":"08:09.960","Text":"and the function is just 1 for the area,"},{"Start":"08:09.960 ","End":"08:11.685","Text":"so that\u0027s the first bit."},{"Start":"08:11.685 ","End":"08:13.410","Text":"Now, the second bit."},{"Start":"08:13.410 ","End":"08:17.505","Text":"The second bit x is going to go from 1/2-2,"},{"Start":"08:17.505 ","End":"08:21.915","Text":"so x equals 1/2 up to 2,"},{"Start":"08:21.915 ","End":"08:25.845","Text":"and that\u0027s going to be dx."},{"Start":"08:25.845 ","End":"08:33.440","Text":"Then y is going to go from also the x-axis,"},{"Start":"08:33.440 ","End":"08:36.270","Text":"y equals 0 up to,"},{"Start":"08:36.270 ","End":"08:38.360","Text":"I see I\u0027m stuck here,"},{"Start":"08:38.360 ","End":"08:41.480","Text":"I haven\u0027t computed the equation of the upper semicircle."},{"Start":"08:41.480 ","End":"08:45.695","Text":"So let\u0027s just take a moment here to compute this."},{"Start":"08:45.695 ","End":"08:50.780","Text":"What I want to do is extract y in terms of x for this equation,"},{"Start":"08:50.780 ","End":"08:52.760","Text":"but I only want the upper semicircle."},{"Start":"08:52.760 ","End":"08:54.365","Text":"So let\u0027s see."},{"Start":"08:54.365 ","End":"08:57.530","Text":"If I bring this to the other side."},{"Start":"08:57.530 ","End":"09:04.095","Text":"I\u0027ve got y squared equals 2x minus x squared,"},{"Start":"09:04.095 ","End":"09:13.075","Text":"so y is plus or minus the square root of 2x minus x squared."},{"Start":"09:13.075 ","End":"09:16.690","Text":"Since you want the upper semicircle,"},{"Start":"09:16.690 ","End":"09:25.100","Text":"we want to put here just the square root of 2x minus x squared,"},{"Start":"09:25.100 ","End":"09:27.530","Text":"and this is what y travels from,"},{"Start":"09:27.530 ","End":"09:31.860","Text":"and again, the function is just 1 for the area."},{"Start":"09:32.090 ","End":"09:34.790","Text":"We\u0027ll do these from the inside out."},{"Start":"09:34.790 ","End":"09:39.660","Text":"The inner integral here is this."},{"Start":"09:39.840 ","End":"09:45.505","Text":"The inner integral here is this,"},{"Start":"09:45.505 ","End":"09:50.170","Text":"but we\u0027ve done this enough times already I can take a little shortcut."},{"Start":"09:50.170 ","End":"09:52.540","Text":"When I have the integral of 1,"},{"Start":"09:52.540 ","End":"09:56.725","Text":"the answer I get is just the top limit minus the bottom limit."},{"Start":"09:56.725 ","End":"10:02.755","Text":"Think of it, the integral of 1 dy is y and I let it be this and this and subtract."},{"Start":"10:02.755 ","End":"10:06.400","Text":"The inner integral is just this minus this,"},{"Start":"10:06.400 ","End":"10:11.320","Text":"which is x root 3 and I still wrap it"},{"Start":"10:11.320 ","End":"10:17.230","Text":"with integral from 0 to 1 1/2 dx."},{"Start":"10:17.230 ","End":"10:23.830","Text":"For the other one, let\u0027s do the out bit first."},{"Start":"10:23.830 ","End":"10:26.890","Text":"The outer bit is 1 1/2 to 2,"},{"Start":"10:26.890 ","End":"10:30.085","Text":"the inner bit same as before will have the integral of 1,"},{"Start":"10:30.085 ","End":"10:34.450","Text":"it\u0027s just the upper limit minus the lower limit which is the square root"},{"Start":"10:34.450 ","End":"10:40.370","Text":"of 2x minus x squared from 1 1/2 to 2dx."},{"Start":"10:41.130 ","End":"10:45.190","Text":"Now, we have 2 integrals to solve."},{"Start":"10:45.190 ","End":"10:49.630","Text":"I can tell you now that the first integral is very straightforward and we\u0027re"},{"Start":"10:49.630 ","End":"10:54.280","Text":"going to be spending some time doing the second integral."},{"Start":"10:54.280 ","End":"10:58.600","Text":"The first one, this is a constant times x,"},{"Start":"10:58.600 ","End":"11:01.855","Text":"so it\u0027s just, a constant, I\u0027ll put it in front,"},{"Start":"11:01.855 ","End":"11:05.290","Text":"and the integral of x is x squared over 2 so I\u0027ll"},{"Start":"11:05.290 ","End":"11:08.920","Text":"write it over 2 and I\u0027ll put the x squared here,"},{"Start":"11:08.920 ","End":"11:14.870","Text":"and this I have to evaluate between 0 and 1 1/2."},{"Start":"11:15.390 ","End":"11:18.445","Text":"This thing I\u0027ll do at the side, you know what?"},{"Start":"11:18.445 ","End":"11:24.925","Text":"I\u0027ll call this one asterisk and I\u0027ll do that later."},{"Start":"11:24.925 ","End":"11:29.365","Text":"For this substitution, when I substitute x equals 0, I\u0027m going to get nothing."},{"Start":"11:29.365 ","End":"11:32.020","Text":"All I have to do is substitute 1 1/2."},{"Start":"11:32.020 ","End":"11:34.630","Text":"1 1/2 squared is a 1/4."},{"Start":"11:34.630 ","End":"11:36.235","Text":"1/4 with the 1/2 makes it an 1/8,"},{"Start":"11:36.235 ","End":"11:43.495","Text":"so this bit is root 3 over 8 plus asterisk."},{"Start":"11:43.495 ","End":"11:47.470","Text":"Now, let\u0027s go and do the difficult integral, this one."},{"Start":"11:47.470 ","End":"11:56.050","Text":"This asterisk, what it is equal to is the integral from 1 1/2 to 2."},{"Start":"11:56.050 ","End":"11:59.785","Text":"Now, I want to rewrite what\u0027s under the square root sign."},{"Start":"11:59.785 ","End":"12:05.275","Text":"Let me just first of all write the square root of something and dx."},{"Start":"12:05.275 ","End":"12:11.500","Text":"Now, this 2x minus x squared came from the original equation from y squared."},{"Start":"12:11.500 ","End":"12:19.310","Text":"But if I had used this equation which is closer to the shape of a circle,"},{"Start":"12:19.680 ","End":"12:23.605","Text":"it represents a circle more clearly,"},{"Start":"12:23.605 ","End":"12:32.630","Text":"I would have got that y squared was equal to1 minus x minus 1 squared."},{"Start":"12:33.630 ","End":"12:42.010","Text":"The reason I would rather write it this way is that there is a substitution."},{"Start":"12:42.010 ","End":"12:44.365","Text":"There is a way of solving this,"},{"Start":"12:44.365 ","End":"12:46.840","Text":"if I let x minus 1 equals t,"},{"Start":"12:46.840 ","End":"12:50.425","Text":"the square root of 1 minus t squared is solvable."},{"Start":"12:50.425 ","End":"12:52.630","Text":"You want to make sure I didn\u0027t cheat you,"},{"Start":"12:52.630 ","End":"12:56.785","Text":"you can expand the brackets here x squared minus 2x plus 1,"},{"Start":"12:56.785 ","End":"12:58.240","Text":"if I subtract it from 1,"},{"Start":"12:58.240 ","End":"13:00.535","Text":"I get exactly 2x minus x squared,"},{"Start":"13:00.535 ","End":"13:02.230","Text":"so this is okay."},{"Start":"13:02.230 ","End":"13:05.290","Text":"As I said, we\u0027re going to do it with a substitution."},{"Start":"13:05.290 ","End":"13:11.650","Text":"The substitution will be that t is equal to x minus 1,"},{"Start":"13:11.650 ","End":"13:18.925","Text":"then dt is equal to just dx."},{"Start":"13:18.925 ","End":"13:23.935","Text":"This is 1 dt and the derivative of this is also 1 dx."},{"Start":"13:23.935 ","End":"13:33.325","Text":"The thing is, I also have to substitute the limits because when x equals the upper one 2,"},{"Start":"13:33.325 ","End":"13:38.650","Text":"that gives me that t is equal to 1."},{"Start":"13:38.650 ","End":"13:42.565","Text":"When x equals 1/2,"},{"Start":"13:42.565 ","End":"13:45.565","Text":"then t is minus 1/2."},{"Start":"13:45.565 ","End":"13:53.380","Text":"This integral comes out as the integral from minus 1/2 to"},{"Start":"13:53.380 ","End":"14:02.605","Text":"1 of the square root of 1 minus t squared dt."},{"Start":"14:02.605 ","End":"14:05.844","Text":"This is a rather tedious integral."},{"Start":"14:05.844 ","End":"14:14.095","Text":"Normally, you would do it by a substitution like t equals sine of u,"},{"Start":"14:14.095 ","End":"14:18.790","Text":"or you could just look it up in the table of integrals."},{"Start":"14:18.790 ","End":"14:22.030","Text":"I found one that has the square root of"},{"Start":"14:22.030 ","End":"14:25.975","Text":"a squared minus t squared and I just let a equals 1."},{"Start":"14:25.975 ","End":"14:27.490","Text":"In any event either way,"},{"Start":"14:27.490 ","End":"14:29.455","Text":"I\u0027m just going to quote the answer,"},{"Start":"14:29.455 ","End":"14:36.970","Text":"and it is t over 2 square root of 1 minus t squared"},{"Start":"14:36.970 ","End":"14:47.590","Text":"plus 1 1/2 arc sine of t,"},{"Start":"14:47.590 ","End":"14:50.605","Text":"and then there would be plus a constant,"},{"Start":"14:50.605 ","End":"14:55.450","Text":"but you don\u0027t need the constant because we\u0027re going to do a definite integral"},{"Start":"14:55.450 ","End":"15:03.755","Text":"and take this from minus 1/2 to 1."},{"Start":"15:03.755 ","End":"15:06.945","Text":"Let\u0027s see what we get."},{"Start":"15:06.945 ","End":"15:09.450","Text":"If we plug in 1,"},{"Start":"15:09.450 ","End":"15:12.015","Text":"1 minus 1 squared is 0,"},{"Start":"15:12.015 ","End":"15:14.370","Text":"so the first term is 0,"},{"Start":"15:14.370 ","End":"15:23.215","Text":"and if we put here 1 arc sine of 1 is Pi over 2,"},{"Start":"15:23.215 ","End":"15:28.284","Text":"we get, I\u0027ll just write it as 1/2 Pi over 2,"},{"Start":"15:28.284 ","End":"15:30.475","Text":"that\u0027s for the case of 1."},{"Start":"15:30.475 ","End":"15:33.955","Text":"Now, we have to put in the minus 1/2."},{"Start":"15:33.955 ","End":"15:38.110","Text":"If it\u0027s minus 1/2 squared,"},{"Start":"15:38.110 ","End":"15:42.250","Text":"it\u0027s a 1/4,1 minus a 1/4 is 3/4,"},{"Start":"15:42.250 ","End":"15:52.420","Text":"and the square root of 3/4 is root 3 over 2,"},{"Start":"15:52.420 ","End":"15:54.640","Text":"but we still have t over 2,"},{"Start":"15:54.640 ","End":"15:58.150","Text":"which is minus 1/2 over 2,"},{"Start":"15:58.150 ","End":"16:04.435","Text":"so it\u0027s minus 1/4,"},{"Start":"16:04.435 ","End":"16:08.560","Text":"then here we have arc 1 1/2,"},{"Start":"16:08.560 ","End":"16:18.535","Text":"arc sine of minus 1/2 it\u0027s minus 30 degrees or minus Pi over 6,"},{"Start":"16:18.535 ","End":"16:20.560","Text":"and I\u0027ll just write it as minus."},{"Start":"16:20.560 ","End":"16:24.835","Text":"You have to do it in radians minus Pi over 6."},{"Start":"16:24.835 ","End":"16:27.370","Text":"Let\u0027s see what we have."},{"Start":"16:27.370 ","End":"16:29.815","Text":"Let\u0027s go back here."},{"Start":"16:29.815 ","End":"16:33.260","Text":"This is the asterisk bit."},{"Start":"16:33.390 ","End":"16:37.765","Text":"We have root 3 over 8 from before plus,"},{"Start":"16:37.765 ","End":"16:40.210","Text":"now the stuff from here,"},{"Start":"16:40.210 ","End":"16:46.165","Text":"1/2 times Pi over 2 is Pi over 4."},{"Start":"16:46.165 ","End":"16:48.040","Text":"Then I need minus,"},{"Start":"16:48.040 ","End":"16:49.975","Text":"minus will give me plus,"},{"Start":"16:49.975 ","End":"16:53.480","Text":"that will give me plus root 3 over 8."},{"Start":"16:54.210 ","End":"16:57.910","Text":"The final term is minus, plus,"},{"Start":"16:57.910 ","End":"17:04.525","Text":"it\u0027s minus, but there\u0027s a minus here,"},{"Start":"17:04.525 ","End":"17:08.120","Text":"that\u0027s going to be plus Pi over 12."},{"Start":"17:09.210 ","End":"17:20.155","Text":"Let\u0027s see if I combine all this root 3 over 8 and root 3 over 8 is root 3 over 4,"},{"Start":"17:20.155 ","End":"17:25.460","Text":"1/4 plus 1/12 is 1/3, it\u0027s Pi over 3,"},{"Start":"17:27.180 ","End":"17:29.890","Text":"and so this will be my answer,"},{"Start":"17:29.890 ","End":"17:36.950","Text":"root 3 over 4 plus Pi over 3 and we are done."}],"ID":8681},{"Watched":false,"Name":"Exercise 1 part d","Duration":"9m 30s","ChapterTopicVideoID":8466,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this exercise, we have to compute the area of the region bounded by 2 curves."},{"Start":"00:05.250 ","End":"00:07.530","Text":"Well, this 1 is a straight line."},{"Start":"00:07.530 ","End":"00:10.230","Text":"This 1 you should be able to recognize as a parabola,"},{"Start":"00:10.230 ","End":"00:14.115","Text":"but on its side because it\u0027s y squared and not x squared."},{"Start":"00:14.115 ","End":"00:18.540","Text":"Let\u0027s see, I want to do a quick sketch."},{"Start":"00:18.540 ","End":"00:20.190","Text":"x plus y equals 3."},{"Start":"00:20.190 ","End":"00:22.410","Text":"Best to do with the intercepts, when x is 0,"},{"Start":"00:22.410 ","End":"00:25.050","Text":"y is 3, when y is 0, x is 3."},{"Start":"00:25.050 ","End":"00:34.125","Text":"If I put in here the point 3 and the point 3 and here with a line through them,"},{"Start":"00:34.125 ","End":"00:35.445","Text":"that\u0027s the first 1."},{"Start":"00:35.445 ","End":"00:38.340","Text":"As for the other 1, y squared equals 4x,"},{"Start":"00:38.340 ","End":"00:41.160","Text":"I could extract x in terms of y,"},{"Start":"00:41.160 ","End":"00:44.345","Text":"but we could just try plugging in values of x."},{"Start":"00:44.345 ","End":"00:47.120","Text":"When x is 0, y squared is 0,"},{"Start":"00:47.120 ","End":"00:49.825","Text":"so it goes through here."},{"Start":"00:49.825 ","End":"00:53.864","Text":"If x is 1, y squared is 4,"},{"Start":"00:53.864 ","End":"00:57.585","Text":"so y is plus or minus 2."},{"Start":"00:57.585 ","End":"01:00.030","Text":"We have to go up and down too."},{"Start":"01:00.030 ","End":"01:05.145","Text":"Now, it turns out that 1,2 actually lies on this curve."},{"Start":"01:05.145 ","End":"01:07.830","Text":"1 plus 2 equals 3,"},{"Start":"01:07.830 ","End":"01:09.480","Text":"but if you missed it,"},{"Start":"01:09.480 ","End":"01:11.180","Text":"then later in the calculations,"},{"Start":"01:11.180 ","End":"01:14.480","Text":"you would see that if you drew it differently,"},{"Start":"01:14.480 ","End":"01:16.045","Text":"that they\u0027re actually the same point,"},{"Start":"01:16.045 ","End":"01:20.090","Text":"and minus 2 is somewhere down here equidistant."},{"Start":"01:20.090 ","End":"01:25.805","Text":"Let\u0027s say this is 2 here and this is minus 2."},{"Start":"01:25.805 ","End":"01:28.625","Text":"Now we can draw the parabola."},{"Start":"01:28.625 ","End":"01:33.545","Text":"Sketch it. Do it in a different color,"},{"Start":"01:33.545 ","End":"01:35.585","Text":"make sure it goes through here,"},{"Start":"01:35.585 ","End":"01:39.310","Text":"and then through this point here."},{"Start":"01:39.310 ","End":"01:42.320","Text":"It looks like there\u0027s going to be another collision somewhere."},{"Start":"01:42.320 ","End":"01:47.270","Text":"I just extended the line and we\u0027re going to get another point here."},{"Start":"01:47.270 ","End":"01:51.770","Text":"The region we\u0027re talking about is here, I\u0027ll shade it."},{"Start":"01:51.860 ","End":"01:59.345","Text":"There it is in yellow and I\u0027ll give it a label D for domain or region."},{"Start":"01:59.345 ","End":"02:09.740","Text":"The area we\u0027re looking for is just the double integral over the region D of 1 dA."},{"Start":"02:09.740 ","End":"02:15.790","Text":"I\u0027m writing dA because I don\u0027t know if I\u0027m going to do it dx dy or dy dx, we shall see."},{"Start":"02:15.790 ","End":"02:19.415","Text":"Some people don\u0027t even bother to write the 1, they just write dA."},{"Start":"02:19.415 ","End":"02:24.145","Text":"We want to see whether we want to slice it horizontally or vertically."},{"Start":"02:24.145 ","End":"02:28.015","Text":"Vertically would make it a Type 1 region, horizontally Type 2."},{"Start":"02:28.015 ","End":"02:29.820","Text":"Let\u0027s see which is better."},{"Start":"02:29.820 ","End":"02:31.640","Text":"But either way we slice it,"},{"Start":"02:31.640 ","End":"02:32.920","Text":"we\u0027re still going to need this point,"},{"Start":"02:32.920 ","End":"02:35.990","Text":"so why don\u0027t we just do this computation."},{"Start":"02:35.990 ","End":"02:42.605","Text":"What I suggest is to look at this as 2 equations and 2 unknowns."},{"Start":"02:42.605 ","End":"02:45.500","Text":"Let me just maybe write them."},{"Start":"02:45.500 ","End":"02:53.790","Text":"We have that x plus y equals 3 and y squared equals 4x."},{"Start":"02:54.790 ","End":"03:01.310","Text":"If we substitute, rather isolate y from here,"},{"Start":"03:01.310 ","End":"03:06.040","Text":"I get that y is equal to 3 minus x."},{"Start":"03:06.040 ","End":"03:14.590","Text":"Then I can substitute that in here and get that 3 minus x squared equals 4x."},{"Start":"03:17.990 ","End":"03:22.085","Text":"I could think of it as x minus 3 also squared,"},{"Start":"03:22.085 ","End":"03:26.930","Text":"x squared minus 6x plus 9. x squared minus 6x,"},{"Start":"03:26.930 ","End":"03:28.160","Text":"but I\u0027ll bring this overall,"},{"Start":"03:28.160 ","End":"03:33.850","Text":"so it\u0027s minus 10x"},{"Start":"03:33.850 ","End":"03:40.465","Text":"plus 9 equals 0."},{"Start":"03:40.465 ","End":"03:42.870","Text":"I\u0027ll just give you the solutions."},{"Start":"03:42.870 ","End":"03:45.100","Text":"1 solution is x equals 1,"},{"Start":"03:45.100 ","End":"03:47.755","Text":"the other solution is x equals 9."},{"Start":"03:47.755 ","End":"03:54.060","Text":"I guess this is really not to scale because this comes out. No, that\u0027s okay."},{"Start":"03:54.060 ","End":"03:58.090","Text":"Over here it comes out 9 and the y,"},{"Start":"03:58.090 ","End":"03:59.590","Text":"let me give the full point."},{"Start":"03:59.590 ","End":"04:02.875","Text":"If x is 1 and I plug it in,"},{"Start":"04:02.875 ","End":"04:04.950","Text":"you could plug it into either,"},{"Start":"04:04.950 ","End":"04:06.675","Text":"I\u0027ll plug it into here,"},{"Start":"04:06.675 ","End":"04:09.450","Text":"then y is 3 minus 1,"},{"Start":"04:09.450 ","End":"04:14.245","Text":"is 2, which gives us this point which we already knew about."},{"Start":"04:14.245 ","End":"04:17.950","Text":"If x is 9 and we plug it in,"},{"Start":"04:17.950 ","End":"04:21.515","Text":"y is 3 minus 9, is minus 6."},{"Start":"04:21.515 ","End":"04:26.690","Text":"I guess this is minus 6 and somewhere here should be 9."},{"Start":"04:26.690 ","End":"04:28.535","Text":"It\u0027s off the scale, that doesn\u0027t matter,"},{"Start":"04:28.535 ","End":"04:32.324","Text":"but I\u0027ll label this point as 9,"},{"Start":"04:32.324 ","End":"04:36.000","Text":"-6 and here we have 1, 2."},{"Start":"04:36.000 ","End":"04:38.905","Text":"How are we going to slice this?"},{"Start":"04:38.905 ","End":"04:43.280","Text":"I say that it\u0027s best to take horizontal slices,"},{"Start":"04:43.280 ","End":"04:48.700","Text":"because if we take horizontal slices for a given y,"},{"Start":"04:48.700 ","End":"04:50.335","Text":"we\u0027ll slice it with x,"},{"Start":"04:50.335 ","End":"04:57.275","Text":"we\u0027ll always be going from the parabola to the line."},{"Start":"04:57.275 ","End":"05:01.850","Text":"It doesn\u0027t matter if we did it here or here,"},{"Start":"05:01.850 ","End":"05:04.590","Text":"it\u0027s always from the parabola to the line,"},{"Start":"05:04.590 ","End":"05:08.165","Text":"so there\u0027s no need to divide up into special cases."},{"Start":"05:08.165 ","End":"05:10.910","Text":"But if I did it vertically,"},{"Start":"05:10.910 ","End":"05:14.660","Text":"something changes around x equals 1."},{"Start":"05:14.660 ","End":"05:18.680","Text":"Here, I\u0027ll have the line minus the parabola,"},{"Start":"05:18.680 ","End":"05:21.530","Text":"but here I\u0027ll have 2 parts of the parabola."},{"Start":"05:21.530 ","End":"05:26.330","Text":"It\u0027s generally better to just take it in 1 case rather than sub-dividing."},{"Start":"05:26.330 ","End":"05:29.705","Text":"For our typical y,"},{"Start":"05:29.705 ","End":"05:34.530","Text":"it will go from minus 6 up to 2,"},{"Start":"05:34.530 ","End":"05:37.980","Text":"these are the points we found here."},{"Start":"05:37.980 ","End":"05:47.365","Text":"Let\u0027s write it. The integral y goes from minus 6-2 and for each such y,"},{"Start":"05:47.365 ","End":"05:51.025","Text":"x will go, that\u0027s dy,"},{"Start":"05:51.025 ","End":"05:54.325","Text":"then x will go from,"},{"Start":"05:54.325 ","End":"05:57.385","Text":"we\u0027ll see in a moment, dx."},{"Start":"05:57.385 ","End":"06:00.300","Text":"We need the formula for these 2 blue points,"},{"Start":"06:00.300 ","End":"06:06.190","Text":"in other words for these 2 curves as x in terms of y. No problem."},{"Start":"06:06.190 ","End":"06:09.160","Text":"The straight line, which is x plus y is 3,"},{"Start":"06:09.160 ","End":"06:15.345","Text":"I can write as x equals 3 minus y."},{"Start":"06:15.345 ","End":"06:16.880","Text":"As for the parabola,"},{"Start":"06:16.880 ","End":"06:19.490","Text":"if I just take this and divide it by 4,"},{"Start":"06:19.490 ","End":"06:27.120","Text":"I have that x is equal to y squared over 4."},{"Start":"06:27.350 ","End":"06:29.640","Text":"This is on the parabola."},{"Start":"06:29.640 ","End":"06:31.550","Text":"Lower limit is the y squared over"},{"Start":"06:31.550 ","End":"06:41.140","Text":"4 and the upper 1 or the rightmost 1 is the 3 minus y."},{"Start":"06:41.140 ","End":"06:44.245","Text":"The function for area is just 1,"},{"Start":"06:44.245 ","End":"06:47.300","Text":"which is sometimes not even written."},{"Start":"06:47.300 ","End":"06:50.885","Text":"Now, we do the inner 1 first,"},{"Start":"06:50.885 ","End":"06:55.320","Text":"that\u0027s this dx integral."},{"Start":"06:57.070 ","End":"07:00.050","Text":"We\u0027ll use our shortcut,"},{"Start":"07:00.050 ","End":"07:02.120","Text":"that when you have the integral of 1,"},{"Start":"07:02.120 ","End":"07:04.680","Text":"it\u0027s just the upper limit minus the lower limit,"},{"Start":"07:04.680 ","End":"07:10.190","Text":"it\u0027s 3 minus y minus y squared over 4."},{"Start":"07:10.190 ","End":"07:17.835","Text":"Just to remind you, the integral of this would be x,"},{"Start":"07:17.835 ","End":"07:20.280","Text":"and I\u0027d substitute x equals 3 minus y,"},{"Start":"07:20.280 ","End":"07:23.970","Text":"and I\u0027d subtract when x equals y squared over 4, I just get this."},{"Start":"07:23.970 ","End":"07:25.380","Text":"Let\u0027s put this in brackets,"},{"Start":"07:25.380 ","End":"07:28.110","Text":"that\u0027s the highlighted bit."},{"Start":"07:28.110 ","End":"07:34.120","Text":"Then there\u0027s the dy and there\u0027s the integral from minus 6-2."},{"Start":"07:35.270 ","End":"07:38.510","Text":"This is a straightforward integral."},{"Start":"07:38.510 ","End":"07:40.310","Text":"What we have here is"},{"Start":"07:40.310 ","End":"07:49.800","Text":"3y minus y squared over 2, that\u0027s 1/2y squared."},{"Start":"07:49.800 ","End":"07:52.020","Text":"Here, y cubed over 3,"},{"Start":"07:52.020 ","End":"07:55.805","Text":"I\u0027ll let the 3 and the 4 blend to make it a 12,"},{"Start":"07:55.805 ","End":"08:03.150","Text":"1/12y cubed and all this between minus 6 and 2."},{"Start":"08:03.150 ","End":"08:05.310","Text":"Let\u0027s see what we get."},{"Start":"08:05.310 ","End":"08:10.340","Text":"If we plug in 2, we get 6."},{"Start":"08:10.340 ","End":"08:13.540","Text":"2 squared over 2 is 2."},{"Start":"08:13.540 ","End":"08:15.350","Text":"2 cubed over 12,"},{"Start":"08:15.350 ","End":"08:20.925","Text":"8 over 12 is minus 2/3."},{"Start":"08:20.925 ","End":"08:23.445","Text":"Now the minus 6,"},{"Start":"08:23.445 ","End":"08:27.555","Text":"thrice minus 6 is minus 18."},{"Start":"08:27.555 ","End":"08:30.645","Text":"Minus 6 squared is 36,"},{"Start":"08:30.645 ","End":"08:31.950","Text":"minus 1/2 of it,"},{"Start":"08:31.950 ","End":"08:34.695","Text":"it\u0027s another minus 18."},{"Start":"08:34.695 ","End":"08:40.320","Text":"Here, y cubed is minus 6,"},{"Start":"08:40.320 ","End":"08:42.510","Text":"minus 6, minus 6,"},{"Start":"08:42.510 ","End":"08:46.690","Text":"before minuses, it\u0027ll be a plus."},{"Start":"08:46.720 ","End":"08:49.970","Text":"Now 6 times 6 times 6 over 12."},{"Start":"08:49.970 ","End":"08:52.970","Text":"Well, 6 times 6 over 12 is 36 over 12,"},{"Start":"08:52.970 ","End":"08:54.620","Text":"is 3 times the 6."},{"Start":"08:54.620 ","End":"08:56.675","Text":"You get another 18 here."},{"Start":"08:56.675 ","End":"08:58.715","Text":"This will go with this."},{"Start":"08:58.715 ","End":"09:01.450","Text":"Let\u0027s see what\u0027s left."},{"Start":"09:01.450 ","End":"09:07.485","Text":"Plus 18 and then 6 minus 2 is 4, that\u0027s 22."},{"Start":"09:07.485 ","End":"09:14.550","Text":"22 minus 2/3 is 21 and a 1/3."},{"Start":"09:14.550 ","End":"09:16.995","Text":"That is our answer."},{"Start":"09:16.995 ","End":"09:21.305","Text":"Unless you like improper fractions,"},{"Start":"09:21.305 ","End":"09:24.890","Text":"in which case you could say 21 times 3 is 63 plus 1 is"},{"Start":"09:24.890 ","End":"09:28.100","Text":"64 over 3 if you prefer it this way."},{"Start":"09:28.100 ","End":"09:31.170","Text":"I\u0027m leaving it like that and we\u0027re done."}],"ID":8682},{"Watched":false,"Name":"Exercise 2 part a","Duration":"11m 40s","ChapterTopicVideoID":8467,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this exercise, we have to compute the volume of the"},{"Start":"00:03.240 ","End":"00:07.005","Text":"solid that\u0027s bounded by the following surfaces."},{"Start":"00:07.005 ","End":"00:08.550","Text":"There\u0027s actually 5 of them,"},{"Start":"00:08.550 ","End":"00:09.960","Text":"1, 2, 3, 4,"},{"Start":"00:09.960 ","End":"00:13.770","Text":"5 and they found a couple of pictures on"},{"Start":"00:13.770 ","End":"00:18.270","Text":"the internet that might help explain the scenario,"},{"Start":"00:18.270 ","End":"00:19.965","Text":"what\u0027s going on here."},{"Start":"00:19.965 ","End":"00:23.700","Text":"I\u0027ll show you soon that this is one of those cases where we"},{"Start":"00:23.700 ","End":"00:27.840","Text":"have a regional domain in the x,"},{"Start":"00:27.840 ","End":"00:34.500","Text":"y plane and 2 surfaces above this region."},{"Start":"00:34.500 ","End":"00:38.250","Text":"When I say above, I mean the projection of each 1 onto the x,"},{"Start":"00:38.250 ","End":"00:43.980","Text":"y plane is the same region D. They are both colored in this green,"},{"Start":"00:43.980 ","End":"00:47.660","Text":"this is the upper surface and the lower surface and it just happens to"},{"Start":"00:47.660 ","End":"00:51.785","Text":"look like a circle or ellipse here in the x, y plane."},{"Start":"00:51.785 ","End":"01:00.045","Text":"The other picture I found showed a rectangular D. Like it says here,"},{"Start":"01:00.045 ","End":"01:03.575","Text":"the volume is given by the double integral."},{"Start":"01:03.575 ","End":"01:05.390","Text":"The D is written at the side here,"},{"Start":"01:05.390 ","End":"01:10.790","Text":"we would write double integral and the D we would write over here of"},{"Start":"01:10.790 ","End":"01:16.815","Text":"the upper minus the lower surface and this should be,"},{"Start":"01:16.815 ","End":"01:23.050","Text":"of course, dx, dy or least da."},{"Start":"01:23.050 ","End":"01:26.060","Text":"Think I\u0027ll get rid of the lower picture."},{"Start":"01:26.060 ","End":"01:29.060","Text":"Now we\u0027re not going to be drawing anything in 3D."},{"Start":"01:29.060 ","End":"01:34.219","Text":"What interests us is the region D which is the projection of these surfaces."},{"Start":"01:34.219 ","End":"01:39.740","Text":"In our case, notice that there are some equations,"},{"Start":"01:39.740 ","End":"01:41.780","Text":"this one, this one and this one,"},{"Start":"01:41.780 ","End":"01:46.040","Text":"they don\u0027t contain z at all and that means that they"},{"Start":"01:46.040 ","End":"01:51.840","Text":"are completely vertical so"},{"Start":"01:51.840 ","End":"01:57.270","Text":"that we could just draw their intersection with the x,"},{"Start":"01:57.270 ","End":"02:02.495","Text":"y plane, like y equals 0 would be,"},{"Start":"02:02.495 ","End":"02:04.745","Text":"well, it wouldn\u0027t be the x axis."},{"Start":"02:04.745 ","End":"02:13.395","Text":"It\u0027s the x, z plane but I\u0027ll draw part of it and maybe in a different color and here,"},{"Start":"02:13.395 ","End":"02:17.085","Text":"x equals 0 would be this."},{"Start":"02:17.085 ","End":"02:22.410","Text":"The projection is just the y axis and then there\u0027s x plus y equals"},{"Start":"02:22.410 ","End":"02:27.500","Text":"1 which we can quickly see that if x is 0,"},{"Start":"02:27.500 ","End":"02:29.750","Text":"y is 1, if y is 0, x is 1."},{"Start":"02:29.750 ","End":"02:40.175","Text":"If I put here 1 and here 1 then it goes through here."},{"Start":"02:40.175 ","End":"02:47.445","Text":"All together, we have this region and I highlighted it"},{"Start":"02:47.445 ","End":"02:54.970","Text":"and let\u0027s call this D. Maybe also mark the origin here."},{"Start":"02:54.970 ","End":"02:56.800","Text":"This is looking down,"},{"Start":"02:56.800 ","End":"02:58.194","Text":"we\u0027re in 3 dimensions,"},{"Start":"02:58.194 ","End":"03:01.645","Text":"we\u0027re just looking above at the x, y plane."},{"Start":"03:01.645 ","End":"03:04.100","Text":"If I take this, it looks like a triangle,"},{"Start":"03:04.100 ","End":"03:05.435","Text":"it\u0027s really a prism."},{"Start":"03:05.435 ","End":"03:10.850","Text":"Extends infinitely up and down and it\u0027s going to intersect 2 surfaces."},{"Start":"03:10.850 ","End":"03:16.670","Text":"Now the 2 surfaces given to us explicitly as here we"},{"Start":"03:16.670 ","End":"03:19.280","Text":"have z equals something and here we have"},{"Start":"03:19.280 ","End":"03:22.670","Text":"z equals something so probably 1 of them is going to be the upper,"},{"Start":"03:22.670 ","End":"03:24.790","Text":"1 of them is the lower."},{"Start":"03:24.790 ","End":"03:26.820","Text":"Now in this region,"},{"Start":"03:26.820 ","End":"03:32.070","Text":"we\u0027re in the first quadrant so x and y are non-negative."},{"Start":"03:32.070 ","End":"03:35.335","Text":"Z is already at least 1."},{"Start":"03:35.335 ","End":"03:45.429","Text":"What I\u0027m saying is that this is the upper and this 1 is the lower."},{"Start":"03:45.570 ","End":"03:48.925","Text":"Better if I wrote it down here."},{"Start":"03:48.925 ","End":"03:56.220","Text":"I\u0027ll use the notation from this picture so I have z equals the f of x,"},{"Start":"03:56.220 ","End":"04:01.289","Text":"y which is 1 plus x plus y,"},{"Start":"04:01.289 ","End":"04:11.830","Text":"that\u0027s the upper surface and the other 1 which here is called g of x, y equals 0,"},{"Start":"04:11.830 ","End":"04:18.580","Text":"that\u0027s the lower and the volume we\u0027re talking about is given simply"},{"Start":"04:18.580 ","End":"04:25.450","Text":"by the formula of the double integral of the upper minus the lower over this region."},{"Start":"04:25.450 ","End":"04:30.350","Text":"What we want is the double integral over"},{"Start":"04:30.350 ","End":"04:35.600","Text":"D of this minus this,"},{"Start":"04:35.600 ","End":"04:39.740","Text":"well, something minus 0 is just this itself so we just want"},{"Start":"04:39.740 ","End":"04:46.160","Text":"1 plus x plus y and since I haven\u0027t decided yet dx,"},{"Start":"04:46.160 ","End":"04:49.050","Text":"dy, I\u0027ll just write dA."},{"Start":"04:49.810 ","End":"04:52.475","Text":"That\u0027s an integral to compute."},{"Start":"04:52.475 ","End":"04:57.845","Text":"What we do is decide whether this is going to be a type 1 or type 2 region."},{"Start":"04:57.845 ","End":"05:00.020","Text":"Everything seems very symmetrical to me in"},{"Start":"05:00.020 ","End":"05:02.540","Text":"x and y that I don\u0027t think it makes any difference."},{"Start":"05:02.540 ","End":"05:07.675","Text":"Let\u0027s take it then as a type 1 region where we take vertical slices."},{"Start":"05:07.675 ","End":"05:14.830","Text":"Before I continue, there\u0027s something that\u0027s good practice to do is to label these."},{"Start":"05:14.830 ","End":"05:17.895","Text":"This 1 is y equals 0."},{"Start":"05:17.895 ","End":"05:21.155","Text":"This is the 1 where x is 0,"},{"Start":"05:21.155 ","End":"05:25.070","Text":"and this is x plus y equals 1."},{"Start":"05:25.070 ","End":"05:26.735","Text":"It\u0027s good to label them."},{"Start":"05:26.735 ","End":"05:29.335","Text":"Now let\u0027s proceed."},{"Start":"05:29.335 ","End":"05:35.020","Text":"This other we\u0027re going to take it as a type 1 region, meaning vertical slices."},{"Start":"05:35.300 ","End":"05:41.850","Text":"This is the typical x and what we\u0027ll get when we do"},{"Start":"05:41.850 ","End":"05:48.665","Text":"it as iterated is we\u0027ll get the outer integral will be x goes from 0-1,"},{"Start":"05:48.665 ","End":"05:54.440","Text":"that\u0027s this and for each such dx and for each such x,"},{"Start":"05:54.440 ","End":"05:56.755","Text":"I need to tell you what y goes from,"},{"Start":"05:56.755 ","End":"05:59.775","Text":"y goes from the lower to the upper."},{"Start":"05:59.775 ","End":"06:04.280","Text":"The lower 1, we\u0027re lucky we already have it is 0, the upper,"},{"Start":"06:04.280 ","End":"06:13.655","Text":"we just have to extract what y equals so I\u0027ll just rewrite this as y equals"},{"Start":"06:13.655 ","End":"06:23.730","Text":"1 minus x and so here it\u0027s 1 minus x, and that\u0027s dy."},{"Start":"06:23.730 ","End":"06:27.680","Text":"Then I need the function itself or, well, the difference,"},{"Start":"06:27.680 ","End":"06:36.075","Text":"this bit here, the 1 plus x plus y, dy, dx."},{"Start":"06:36.075 ","End":"06:40.190","Text":"We do these things from the inside out."},{"Start":"06:40.190 ","End":"06:46.260","Text":"The inside means this bit here."},{"Start":"06:46.370 ","End":"06:50.330","Text":"I\u0027ll do that at the side and label"},{"Start":"06:50.330 ","End":"06:55.640","Text":"this inner bit as asterisk and do the asterisk computation at the side."},{"Start":"06:55.640 ","End":"07:01.585","Text":"What it is equal to is the integral from 0-1 minus x,"},{"Start":"07:01.585 ","End":"07:05.680","Text":"1 plus x plus y, dy."},{"Start":"07:05.680 ","End":"07:07.550","Text":"Now it\u0027s dy, remember,"},{"Start":"07:07.550 ","End":"07:10.685","Text":"so x is a constant as far as this goes,"},{"Start":"07:10.685 ","End":"07:16.640","Text":"this is equal to the integral of 1 is y."},{"Start":"07:16.640 ","End":"07:20.735","Text":"The integral of x is a constant, so x, y,"},{"Start":"07:20.735 ","End":"07:26.160","Text":"and the integral of y is a 1/2 y squared."},{"Start":"07:26.750 ","End":"07:30.990","Text":"But this has to be taken between"},{"Start":"07:30.990 ","End":"07:37.970","Text":"the limits 0 up to 1 minus x. I Just to emphasize this,"},{"Start":"07:37.970 ","End":"07:42.305","Text":"of course, this is y equals 0 and then y equals 1 minus x."},{"Start":"07:42.305 ","End":"07:45.830","Text":"We plug in the upper plug in the lower and subtract."},{"Start":"07:45.830 ","End":"07:53.175","Text":"The upper, I put in y equals 1 minus x. I get,"},{"Start":"07:53.175 ","End":"07:56.595","Text":"put in y equals 1 minus x."},{"Start":"07:56.595 ","End":"08:01.245","Text":"Here I get 1 minus x,"},{"Start":"08:01.245 ","End":"08:05.570","Text":"and then plus x times again,"},{"Start":"08:05.570 ","End":"08:08.970","Text":"y is 1 minus x."},{"Start":"08:10.480 ","End":"08:18.155","Text":"Here I get 1/2 y is 1 minus x squared."},{"Start":"08:18.155 ","End":"08:22.280","Text":"Now, I also have to substitute y equals 0,"},{"Start":"08:22.280 ","End":"08:26.850","Text":"but look when y is 0, all of this is 0."},{"Start":"08:27.790 ","End":"08:31.610","Text":"I\u0027ll write minus 0 to show that I didn\u0027t forget."},{"Start":"08:31.610 ","End":"08:33.290","Text":"Normally I put this in brackets,"},{"Start":"08:33.290 ","End":"08:35.690","Text":"this whole thing in brackets and subtract."},{"Start":"08:35.690 ","End":"08:37.640","Text":"This is what we have."},{"Start":"08:37.640 ","End":"08:39.110","Text":"Before I substitute back,"},{"Start":"08:39.110 ","End":"08:41.180","Text":"I\u0027d like to simplify it."},{"Start":"08:41.180 ","End":"08:43.890","Text":"Let\u0027s see what we get."},{"Start":"08:44.350 ","End":"08:52.160","Text":"1 minus x plus x minus x"},{"Start":"08:52.160 ","End":"08:57.050","Text":"squared plus 1 minus x squared,"},{"Start":"08:57.050 ","End":"09:03.010","Text":"using the formula is 1 minus 2x plus x squared."},{"Start":"09:03.010 ","End":"09:13.045","Text":"With the 1/2, so I\u0027ve got plus 1/2 minus x plus 1/2x squared."},{"Start":"09:13.045 ","End":"09:19.144","Text":"Collecting terms let\u0027s see how many x squareds do I have?"},{"Start":"09:19.144 ","End":"09:23.679","Text":"This cancels with this already makes it easier."},{"Start":"09:23.679 ","End":"09:26.595","Text":"I can just do it with the constants first."},{"Start":"09:26.595 ","End":"09:30.825","Text":"1 plus 1/2 is 1 1/2."},{"Start":"09:30.825 ","End":"09:34.095","Text":"That\u0027s 3/2."},{"Start":"09:34.095 ","End":"09:39.170","Text":"The x\u0027s, I have just this 1 left,"},{"Start":"09:39.170 ","End":"09:45.200","Text":"minus x and minus x squared plus 1/2 x squared."},{"Start":"09:45.200 ","End":"09:49.070","Text":"It\u0027s minus 1/2 x squared."},{"Start":"09:49.070 ","End":"09:56.725","Text":"Now, back here, way to do another level of simplification."},{"Start":"09:56.725 ","End":"09:58.240","Text":"To get rid of the fractions,"},{"Start":"09:58.240 ","End":"10:00.250","Text":"why don\u0027t I take 1/2?"},{"Start":"10:00.250 ","End":"10:04.750","Text":"Forgot an equals here, 1/2 and just make it"},{"Start":"10:04.750 ","End":"10:11.755","Text":"as 3 minus 2x minus x squared."},{"Start":"10:11.755 ","End":"10:14.365","Text":"Because then I can put the 1/2 in front."},{"Start":"10:14.365 ","End":"10:16.330","Text":"Now when I get here,"},{"Start":"10:16.330 ","End":"10:20.335","Text":"I\u0027ve got that this equals I put the 1/2 in front of the integral."},{"Start":"10:20.335 ","End":"10:31.365","Text":"Now I\u0027m working just with x from 0-1 of 3 minus 2x minus x squared dx."},{"Start":"10:31.365 ","End":"10:36.120","Text":"Fine. This is equal to 1/2."},{"Start":"10:36.120 ","End":"10:43.925","Text":"Now, the integral 3x minus integral of 2x is x squared."},{"Start":"10:43.925 ","End":"10:49.820","Text":"Integral of x squared is 1/3x cubed."},{"Start":"10:49.820 ","End":"10:55.725","Text":"I want all this between 0 and 1."},{"Start":"10:55.725 ","End":"10:59.625","Text":"Now, when I plug in 0,"},{"Start":"10:59.625 ","End":"11:01.160","Text":"I\u0027m not going to get anything,"},{"Start":"11:01.160 ","End":"11:03.300","Text":"so only just need to plug in 1."},{"Start":"11:03.300 ","End":"11:11.850","Text":"I get 1/2 of 3 minus 1 minus 1/3."},{"Start":"11:11.850 ","End":"11:17.210","Text":"Maybe I\u0027ll write also minus 0 to show that I haven\u0027t forgotten the lower limit."},{"Start":"11:17.210 ","End":"11:20.285","Text":"Let\u0027s see what happens here."},{"Start":"11:20.285 ","End":"11:22.430","Text":"3 minus 1 is 2."},{"Start":"11:22.430 ","End":"11:31.340","Text":"2 minus 1/3 is 1 and 2/3, which is 5/3."},{"Start":"11:31.340 ","End":"11:41.650","Text":"I make this 5/3 over 2, 5/6."}],"ID":8683},{"Watched":false,"Name":"Exercise 2 part b","Duration":"11m 23s","ChapterTopicVideoID":8468,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this exercise, we have to compute the volume of the solid and"},{"Start":"00:03.300 ","End":"00:06.750","Text":"it\u0027s bounded by the following surfaces."},{"Start":"00:06.750 ","End":"00:12.150","Text":"It\u0027s one of those cases where some of the surfaces"},{"Start":"00:12.150 ","End":"00:20.220","Text":"describe the prism or cylinder that give the projection D,"},{"Start":"00:20.220 ","End":"00:22.575","Text":"and 2 of the surfaces,"},{"Start":"00:22.575 ","End":"00:26.940","Text":"usually z equals something and z equals something give us an upper"},{"Start":"00:26.940 ","End":"00:32.100","Text":"and a lower surface that bounds the solid."},{"Start":"00:32.100 ","End":"00:39.180","Text":"There\u0027s the formula that the volume is given by the double integral over this region D,"},{"Start":"00:39.180 ","End":"00:44.650","Text":"which is the projection of the upper surface minus the lower surface."},{"Start":"00:44.650 ","End":"00:49.550","Text":"I also found another picture that might give you an idea of what\u0027s going on,"},{"Start":"00:49.550 ","End":"00:53.840","Text":"upper surface, lower surface, the projection D,"},{"Start":"00:53.840 ","End":"00:57.440","Text":"and some of these equations will describe the borders"},{"Start":"00:57.440 ","End":"01:03.335","Text":"of D. We\u0027re not going to do any 3D sketching."},{"Start":"01:03.335 ","End":"01:04.955","Text":"We want to figure out what\u0027s what."},{"Start":"01:04.955 ","End":"01:07.325","Text":"What defines the sides?"},{"Start":"01:07.325 ","End":"01:08.495","Text":"What\u0027s the upper surface?"},{"Start":"01:08.495 ","End":"01:10.504","Text":"What\u0027s the lower surface?"},{"Start":"01:10.504 ","End":"01:14.120","Text":"Well, the ones without any z,"},{"Start":"01:14.120 ","End":"01:19.205","Text":"usually like y equals x squared and y equals 1,"},{"Start":"01:19.205 ","End":"01:21.680","Text":"which looked like equations in the plane,"},{"Start":"01:21.680 ","End":"01:23.030","Text":"and in some ways they are,"},{"Start":"01:23.030 ","End":"01:27.755","Text":"that will define the region D. You could also look at them in 3 dimensional,"},{"Start":"01:27.755 ","End":"01:32.425","Text":"and that way they would define the prism or cylinder that goes through D,"},{"Start":"01:32.425 ","End":"01:33.680","Text":"whichever way you want to look at it."},{"Start":"01:33.680 ","End":"01:36.095","Text":"I\u0027ll just look at it now in 2 dimensions,"},{"Start":"01:36.095 ","End":"01:45.150","Text":"y equals 1 will be a horizontal line through 1 and just see if I can do it freehand."},{"Start":"01:45.150 ","End":"01:49.905","Text":"There we go, and y equals x squared."},{"Start":"01:49.905 ","End":"01:55.460","Text":"Let\u0027s see, when x is 0, y is 0."},{"Start":"01:55.460 ","End":"01:58.340","Text":"Well, we know the x squared, it\u0027s standard parabola."},{"Start":"01:58.340 ","End":"02:01.645","Text":"Let\u0027s see, when x is 1, y is 1,"},{"Start":"02:01.645 ","End":"02:06.270","Text":"so will be here when x is 1."},{"Start":"02:06.270 ","End":"02:12.404","Text":"When x is minus 1 also y will be 1."},{"Start":"02:12.404 ","End":"02:16.535","Text":"It\u0027s important for us to know where these 2 intersect,"},{"Start":"02:16.535 ","End":"02:25.035","Text":"it\u0027s minus 1.Then the parabola would go something like this and this,"},{"Start":"02:25.035 ","End":"02:26.975","Text":"and it doesn\u0027t have to be accurate."},{"Start":"02:26.975 ","End":"02:30.095","Text":"So this is y equals 1,"},{"Start":"02:30.095 ","End":"02:32.270","Text":"this is y equals x squared."},{"Start":"02:32.270 ","End":"02:40.820","Text":"So the region we want is this bit that I\u0027ve highlighted and we\u0027ll call it D,"},{"Start":"02:40.820 ","End":"02:46.105","Text":"and that\u0027s the projection of the solid onto the x, y plane."},{"Start":"02:46.105 ","End":"02:50.475","Text":"Maybe I\u0027ll label this the origin, and this is 1."},{"Start":"02:50.475 ","End":"02:53.480","Text":"Now, what about the upper and lower?"},{"Start":"02:53.480 ","End":"02:56.450","Text":"Well, it looks like 1 of these surfaces,"},{"Start":"02:56.450 ","End":"02:59.675","Text":"it\u0027s going to be above the other."},{"Start":"02:59.675 ","End":"03:04.985","Text":"Clearly, x squared plus y squared is always bigger or equal to 0,"},{"Start":"03:04.985 ","End":"03:07.640","Text":"so this will be the,"},{"Start":"03:07.640 ","End":"03:09.395","Text":"I\u0027ll write it over here."},{"Start":"03:09.395 ","End":"03:14.845","Text":"So the upper surface"},{"Start":"03:14.845 ","End":"03:21.130","Text":"is z equals x squared plus y squared,"},{"Start":"03:21.130 ","End":"03:23.585","Text":"and that\u0027s the upper."},{"Start":"03:23.585 ","End":"03:29.190","Text":"The lower will be z equals 0."},{"Start":"03:29.190 ","End":"03:36.200","Text":"There always have to be 1 bigger or equal to the other for this formula to work."},{"Start":"03:37.170 ","End":"03:41.229","Text":"The formula is written here in small, but basically,"},{"Start":"03:41.229 ","End":"03:45.070","Text":"what it\u0027s saying is that the volume of the solid is the double"},{"Start":"03:45.070 ","End":"03:50.620","Text":"integral over our region D of the upper minus the lower,"},{"Start":"03:50.620 ","End":"03:59.220","Text":"and the upper minus the lower is this minus 0 is just x squared plus y squared da."},{"Start":"03:59.220 ","End":"04:02.030","Text":"Now, we\u0027re going to do this as an iterated integral."},{"Start":"04:02.030 ","End":"04:05.090","Text":"The question is whether we should slice it vertically or horizontally."},{"Start":"04:05.090 ","End":"04:08.095","Text":"In other words, whether it\u0027s a type 1 or type 2."},{"Start":"04:08.095 ","End":"04:15.060","Text":"It seems natural, we\u0027ve already have y in terms of x extracted."},{"Start":"04:15.370 ","End":"04:24.109","Text":"We\u0027ll just go for vertical slices so that for any given x going from minus 1 to 1,"},{"Start":"04:24.109 ","End":"04:29.630","Text":"the vertical slice will hit these 2 equations."},{"Start":"04:29.630 ","End":"04:33.215","Text":"This will be where y equals 1,"},{"Start":"04:33.215 ","End":"04:37.065","Text":"and this will be where y equals x squared."},{"Start":"04:37.065 ","End":"04:43.024","Text":"We have everything, and so I can rewrite the volume as the integral,"},{"Start":"04:43.024 ","End":"04:46.825","Text":"like I said, from minus 1 to 1."},{"Start":"04:46.825 ","End":"04:48.930","Text":"We just stumbled on this,"},{"Start":"04:48.930 ","End":"04:52.640","Text":"but really what you could have done would be to equate these"},{"Start":"04:52.640 ","End":"04:57.125","Text":"to say x squared equals 1 and get x equals plus or minus 1,"},{"Start":"04:57.125 ","End":"05:02.265","Text":"and then discover where they really intersect but we just saw that it was 1."},{"Start":"05:02.265 ","End":"05:05.234","Text":"Anyway, from minus 1 to 1,"},{"Start":"05:05.234 ","End":"05:11.260","Text":"and that\u0027s going to be dx, x runs here."},{"Start":"05:13.970 ","End":"05:17.465","Text":"Perhaps here I should say x goes from here,"},{"Start":"05:17.465 ","End":"05:20.855","Text":"and then y goes from the lower,"},{"Start":"05:20.855 ","End":"05:25.415","Text":"which is x squared to the upper,"},{"Start":"05:25.415 ","End":"05:30.025","Text":"which is y equals 1 dy."},{"Start":"05:30.025 ","End":"05:36.005","Text":"Then the function x squared plus y squared."},{"Start":"05:36.005 ","End":"05:39.560","Text":"As usual, we work from inside out,"},{"Start":"05:39.560 ","End":"05:42.545","Text":"so we do the inner integral first,"},{"Start":"05:42.545 ","End":"05:46.040","Text":"which is the integral dy."},{"Start":"05:46.040 ","End":"05:48.890","Text":"I like to do this as a side computation,"},{"Start":"05:48.890 ","End":"05:50.360","Text":"let\u0027s say I call it asterisk."},{"Start":"05:50.360 ","End":"05:53.165","Text":"Over here, I\u0027ll compute asterisk,"},{"Start":"05:53.165 ","End":"05:58.615","Text":"which is the integral from x squared to"},{"Start":"05:58.615 ","End":"06:05.415","Text":"1 of x squared plus y squared, dy."},{"Start":"06:05.415 ","End":"06:07.860","Text":"Now, when you do it dy,"},{"Start":"06:07.860 ","End":"06:10.750","Text":"then x is like a constant."},{"Start":"06:14.930 ","End":"06:17.795","Text":"We can start by doing the integral."},{"Start":"06:17.795 ","End":"06:22.175","Text":"The integral of x squared with respect to y is just x squared y,"},{"Start":"06:22.175 ","End":"06:29.135","Text":"and y squared is 1/3y cubed or y cubed over 3, whichever way."},{"Start":"06:29.135 ","End":"06:36.560","Text":"This we have to evaluate from y equals x squared to y equals 1."},{"Start":"06:36.560 ","End":"06:38.135","Text":"In other words, this is the upper limit."},{"Start":"06:38.135 ","End":"06:41.214","Text":"We substitute lower and subtract."},{"Start":"06:41.214 ","End":"06:44.580","Text":"I\u0027ll emphasize it\u0027s y equals."},{"Start":"06:44.580 ","End":"06:47.555","Text":"So what we get here is if we put in,"},{"Start":"06:47.555 ","End":"06:49.520","Text":"let\u0027s put the top one first."},{"Start":"06:49.520 ","End":"06:51.820","Text":"That means that y equals 1,"},{"Start":"06:51.820 ","End":"06:56.515","Text":"so we get x squared plus 1/3,"},{"Start":"06:56.515 ","End":"06:59.915","Text":"that\u0027s the upper minus the lower."},{"Start":"06:59.915 ","End":"07:02.030","Text":"If y is x squared,"},{"Start":"07:02.030 ","End":"07:09.605","Text":"then it\u0027s x squared times x squared is x^4 plus 1/3."},{"Start":"07:09.605 ","End":"07:17.760","Text":"Now y cubed would be x^6, so 1/3x^6."},{"Start":"07:17.760 ","End":"07:21.870","Text":"That\u0027s the answer for the inner integral."},{"Start":"07:21.870 ","End":"07:24.624","Text":"Now let\u0027s go back here."},{"Start":"07:24.624 ","End":"07:30.530","Text":"So what we have is the integral from minus"},{"Start":"07:30.530 ","End":"07:36.870","Text":"1 to 1 of this bit, the asterisk."},{"Start":"07:38.440 ","End":"07:41.135","Text":"I write it as it comes,"},{"Start":"07:41.135 ","End":"07:49.205","Text":"x squared plus 1/3 minus x^4,"},{"Start":"07:49.205 ","End":"07:59.355","Text":"minus 1/3x^6, and this will be dx."},{"Start":"07:59.355 ","End":"08:01.240","Text":"We don\u0027t need the pictures anymore,"},{"Start":"08:01.240 ","End":"08:03.310","Text":"it\u0027s just technical now."},{"Start":"08:03.310 ","End":"08:07.210","Text":"Let\u0027s see, the integral of x squared is"},{"Start":"08:07.210 ","End":"08:14.960","Text":"1/3x cubed plus 1/3x."},{"Start":"08:14.960 ","End":"08:20.770","Text":"Here, a raise by 1 is 5 and divide by 5, so minus 1/5x^5."},{"Start":"08:21.260 ","End":"08:24.990","Text":"Here I raise x^7,"},{"Start":"08:24.990 ","End":"08:29.700","Text":"and it\u0027s 1/3 and 1/7 minus 1 over 21,"},{"Start":"08:29.700 ","End":"08:32.350","Text":"21 from 3 times 7x^7."},{"Start":"08:33.460 ","End":"08:37.775","Text":"I don\u0027t need the constant because it\u0027s a definite integral,"},{"Start":"08:37.775 ","End":"08:43.395","Text":"I\u0027ll just put the limits of integration minus 1 to 1."},{"Start":"08:43.395 ","End":"08:48.585","Text":"We will get, if we plug in 1,"},{"Start":"08:48.585 ","End":"08:55.050","Text":"we get 1/3 plus 1/3."},{"Start":"08:55.050 ","End":"08:56.640","Text":"Just basically the coefficients,"},{"Start":"08:56.640 ","End":"08:57.960","Text":"we can ignore the Xs."},{"Start":"08:57.960 ","End":"09:05.445","Text":"Like minus 1/5, minus 1 over 21 minus,"},{"Start":"09:05.445 ","End":"09:08.800","Text":"and then we plug in minus 1."},{"Start":"09:09.260 ","End":"09:13.740","Text":"But all these powers are odd,"},{"Start":"09:13.740 ","End":"09:21.430","Text":"so we\u0027ll just get the same as this but with minus."},{"Start":"09:21.430 ","End":"09:24.940","Text":"I could have taken the shortcut and just put a 2 here right away,"},{"Start":"09:24.940 ","End":"09:26.785","Text":"but to show you what I mean,"},{"Start":"09:26.785 ","End":"09:28.330","Text":"we\u0027ll get exactly the same,"},{"Start":"09:28.330 ","End":"09:30.960","Text":"it will be minus 1/3,"},{"Start":"09:30.960 ","End":"09:34.860","Text":"and then it\u0027ll be minus 1/3."},{"Start":"09:34.860 ","End":"09:37.360","Text":"Wherever it\u0027s a minus it will be plus and vice versa."},{"Start":"09:37.360 ","End":"09:41.950","Text":"Plus 1/5 plus 1 over 21."},{"Start":"09:41.950 ","End":"09:44.665","Text":"Because all of these are odd powers."},{"Start":"09:44.665 ","End":"09:49.090","Text":"So what I was going to do before,"},{"Start":"09:49.090 ","End":"09:50.340","Text":"and I can still do it is saying,"},{"Start":"09:50.340 ","End":"09:51.850","Text":"\"If everything here is minus,"},{"Start":"09:51.850 ","End":"09:55.070","Text":"why don\u0027t I just put a 2 in front of this?\""},{"Start":"09:56.720 ","End":"10:02.220","Text":"Say it\u0027s 1/3 plus 1/3 minus 1/5,"},{"Start":"10:02.220 ","End":"10:07.704","Text":"minus 1 over 21 but I still have to compute this,"},{"Start":"10:07.704 ","End":"10:11.945","Text":"so let\u0027s see what would be a common denominator."},{"Start":"10:11.945 ","End":"10:14.690","Text":"I need a 3, I need a 5."},{"Start":"10:14.690 ","End":"10:16.400","Text":"21 is 3 times 7,"},{"Start":"10:16.400 ","End":"10:17.450","Text":"so I still need a 7."},{"Start":"10:17.450 ","End":"10:22.520","Text":"So 3 times 5 times 7, that\u0027s 105."},{"Start":"10:22.520 ","End":"10:27.540","Text":"So I\u0027ve got twice and then something over 105."},{"Start":"10:28.540 ","End":"10:35.715","Text":"Let\u0027s see, well, this is 2/3 together."},{"Start":"10:35.715 ","End":"10:38.910","Text":"Let\u0027s see, 3 into 105,"},{"Start":"10:38.910 ","End":"10:48.350","Text":"goes 35 times, 35 plus 35 is 70."},{"Start":"10:48.350 ","End":"10:52.345","Text":"I\u0027ll just do these together."},{"Start":"10:52.345 ","End":"10:55.770","Text":"5 into 105 goes 21 times,"},{"Start":"10:55.770 ","End":"10:59.835","Text":"times minus 1 is 21."},{"Start":"10:59.835 ","End":"11:03.030","Text":"21 into 105 goes 5 times,"},{"Start":"11:03.030 ","End":"11:05.760","Text":"so we have here a minus 5."},{"Start":"11:05.760 ","End":"11:09.815","Text":"I\u0027ll make the numerator 70 minus 26 is 44,"},{"Start":"11:09.815 ","End":"11:11.840","Text":"but times the 2 is 88."},{"Start":"11:11.840 ","End":"11:19.315","Text":"So the answer is 88 over 105."},{"Start":"11:19.315 ","End":"11:21.285","Text":"I\u0027ll just highlight that,"},{"Start":"11:21.285 ","End":"11:23.560","Text":"and we are done."}],"ID":8684},{"Watched":false,"Name":"Exercise 2 part c","Duration":"21m 23s","ChapterTopicVideoID":8469,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.070","Text":"In this exercise, we have to compute the volume of the"},{"Start":"00:03.070 ","End":"00:06.550","Text":"solid and it\u0027s bounded by the following surfaces."},{"Start":"00:06.550 ","End":"00:09.510","Text":"There\u0027s actually 5 of them,"},{"Start":"00:09.510 ","End":"00:12.430","Text":"but the surfaces that have 2 different kinds."},{"Start":"00:12.430 ","End":"00:15.260","Text":"Let me first introduce a picture."},{"Start":"00:15.630 ","End":"00:19.090","Text":"This is not the first such exercise,"},{"Start":"00:19.090 ","End":"00:22.570","Text":"so you probably know what the scenario I\u0027m talking about."},{"Start":"00:22.570 ","End":"00:31.405","Text":"That we have this solid which has sides that are part of a prism or cylinder,"},{"Start":"00:31.405 ","End":"00:35.815","Text":"that has a projection D in the x, y plane."},{"Start":"00:35.815 ","End":"00:38.155","Text":"That\u0027s the sides."},{"Start":"00:38.155 ","End":"00:43.579","Text":"The 2 extra surfaces for the upper and lower."},{"Start":"00:44.240 ","End":"00:50.620","Text":"We usually identify 2 of them like z is a function of x and y."},{"Start":"00:50.620 ","End":"00:55.270","Text":"In this case, this 1 would be 1 of the surfaces,"},{"Start":"00:55.270 ","End":"00:59.050","Text":"and this would be another 1 of those surfaces here,"},{"Start":"00:59.050 ","End":"01:00.420","Text":"f and g. Well,"},{"Start":"01:00.420 ","End":"01:03.370","Text":"the others don\u0027t contain z as you know."},{"Start":"01:03.370 ","End":"01:05.140","Text":"If I look at them in the plane,"},{"Start":"01:05.140 ","End":"01:09.785","Text":"they\u0027re just curves but in actual fact,"},{"Start":"01:09.785 ","End":"01:14.695","Text":"they are surfaces because there\u0027s also a vertical z. but"},{"Start":"01:14.695 ","End":"01:20.845","Text":"these will define the projection in the x, y plane."},{"Start":"01:20.845 ","End":"01:24.935","Text":"We\u0027re not going to be doing any 3D drawing."},{"Start":"01:24.935 ","End":"01:28.970","Text":"What I want to do is get a reasonable sketch of"},{"Start":"01:28.970 ","End":"01:33.980","Text":"the projection D of this surface onto the x, y plane."},{"Start":"01:33.980 ","End":"01:37.945","Text":"That\u0027s going to be defined by these 3 equations."},{"Start":"01:37.945 ","End":"01:44.700","Text":"The first 1, y equals 2/x or xy equals 2 is a hyperbola."},{"Start":"01:44.700 ","End":"01:48.770","Text":"Without even drawing any scale or anything,"},{"Start":"01:48.770 ","End":"01:52.730","Text":"I know that the hyperbola has 2 asymptotes."},{"Start":"01:52.730 ","End":"01:55.895","Text":"It goes to infinity and it goes to 0 here."},{"Start":"01:55.895 ","End":"02:02.900","Text":"There is another branch over here but because of the x bigger or equal to 0,"},{"Start":"02:02.900 ","End":"02:06.690","Text":"I\u0027m only going to draw this branch."},{"Start":"02:06.690 ","End":"02:10.220","Text":"There was another symmetrical branch over here."},{"Start":"02:10.220 ","End":"02:16.280","Text":"These 2 are 2 straight lines that go through the origin at different slopes;"},{"Start":"02:16.280 ","End":"02:18.065","Text":"1 of them is going to be like this,"},{"Start":"02:18.065 ","End":"02:21.320","Text":"and 1 of them\u0027s going to be like this."},{"Start":"02:21.440 ","End":"02:23.705","Text":"Let\u0027s just label them."},{"Start":"02:23.705 ","End":"02:27.440","Text":"This will be the y equals 2/x."},{"Start":"02:27.440 ","End":"02:29.900","Text":"This will be the steeper 1."},{"Start":"02:29.900 ","End":"02:32.435","Text":"The biggest slope, y equals 2x."},{"Start":"02:32.435 ","End":"02:35.790","Text":"This is y equals 1/2 x,"},{"Start":"02:37.550 ","End":"02:41.804","Text":"was 0.5x, doesn\u0027t matter."},{"Start":"02:41.804 ","End":"02:51.705","Text":"The domain that we\u0027re talking about that between these 3 would be here."},{"Start":"02:51.705 ","End":"02:54.875","Text":"Here, I\u0027ve highlighted or shaded it,"},{"Start":"02:54.875 ","End":"02:57.649","Text":"and let\u0027s give it a name D,"},{"Start":"02:57.649 ","End":"03:01.259","Text":"for domain or region."},{"Start":"03:01.640 ","End":"03:05.210","Text":"Now, we want to know which of these 2 surfaces,"},{"Start":"03:05.210 ","End":"03:08.729","Text":"the upper and which is the lower."},{"Start":"03:09.190 ","End":"03:14.165","Text":"Well, our region is in the first quadrant,"},{"Start":"03:14.165 ","End":"03:17.810","Text":"or at any rate, it\u0027s in the right half plane."},{"Start":"03:17.810 ","End":"03:20.435","Text":"What I\u0027m saying is that y is non-negative,"},{"Start":"03:20.435 ","End":"03:22.625","Text":"x squared is always non-negative."},{"Start":"03:22.625 ","End":"03:26.780","Text":"This thing, x squared plus y is bigger or equal to 0."},{"Start":"03:26.780 ","End":"03:33.050","Text":"This is going to be the upper and this is going to be the lower,"},{"Start":"03:33.050 ","End":"03:36.245","Text":"what here is called f and g, but doesn\u0027t matter."},{"Start":"03:36.245 ","End":"03:42.020","Text":"The volume of our region is going to be the double"},{"Start":"03:42.020 ","End":"03:49.340","Text":"integral over the region D of the upper minus the lower."},{"Start":"03:49.340 ","End":"03:51.980","Text":"Subtracting 0 doesn\u0027t make any difference,"},{"Start":"03:51.980 ","End":"03:57.515","Text":"so we just get the upper minus lower is just x squared plus y."},{"Start":"03:57.515 ","End":"03:59.675","Text":"That\u0027s going to be dA."},{"Start":"03:59.675 ","End":"04:04.640","Text":"I want to compute this though as an iterated integral."},{"Start":"04:04.640 ","End":"04:07.290","Text":"I have to decide whether to do it dx,"},{"Start":"04:07.290 ","End":"04:08.900","Text":"dy or dy, dx."},{"Start":"04:08.900 ","End":"04:12.365","Text":"In other words, this is going to be a type 1 or type 2 region."},{"Start":"04:12.365 ","End":"04:15.515","Text":"It looks like either way I do it,"},{"Start":"04:15.515 ","End":"04:19.330","Text":"if I slice it vertically,"},{"Start":"04:19.330 ","End":"04:21.855","Text":"I\u0027m going to get 2 cases."},{"Start":"04:21.855 ","End":"04:24.240","Text":"This divide the region into 2."},{"Start":"04:24.240 ","End":"04:26.505","Text":"I\u0027m going to have the left half and the right half."},{"Start":"04:26.505 ","End":"04:31.950","Text":"Even if I do it the other way, which is dx,"},{"Start":"04:31.950 ","End":"04:40.040","Text":"dy, I\u0027ll still have to divide it up into 2 bits according to this point here."},{"Start":"04:40.040 ","End":"04:42.710","Text":"There\u0027s no escaping."},{"Start":"04:42.710 ","End":"04:47.570","Text":"Let us rather do it vertically if only for the simple reason that I"},{"Start":"04:47.570 ","End":"04:51.590","Text":"already have y extracted in terms of x for all these limits."},{"Start":"04:51.590 ","End":"04:54.035","Text":"When y is extracted in terms of x,"},{"Start":"04:54.035 ","End":"04:57.020","Text":"then we wanna do it as a type 1 region,"},{"Start":"04:57.020 ","End":"04:59.645","Text":"which is dy, dx."},{"Start":"04:59.645 ","End":"05:03.665","Text":"We\u0027re going to need the points of intersection."},{"Start":"05:03.665 ","End":"05:06.590","Text":"Well, these 2 straight lines intersect at the origin,"},{"Start":"05:06.590 ","End":"05:12.470","Text":"so that\u0027s 1 of the points but I need to know what is this and what is this."},{"Start":"05:12.470 ","End":"05:14.540","Text":"Why don\u0027t I label them?"},{"Start":"05:14.540 ","End":"05:19.070","Text":"I\u0027ll call this 1 A and this 1, I\u0027ll call B."},{"Start":"05:19.070 ","End":"05:20.600","Text":"Let\u0027s first of all,"},{"Start":"05:20.600 ","End":"05:23.195","Text":"work on finding A."},{"Start":"05:23.195 ","End":"05:26.574","Text":"A is on the intersection between these 2 curves."},{"Start":"05:26.574 ","End":"05:29.755","Text":"So I just solve 2 equations and 2 unknowns,"},{"Start":"05:29.755 ","End":"05:37.830","Text":"y equals 2/x and y equals 2x."},{"Start":"05:37.830 ","End":"05:39.870","Text":"Y equals this and y equals that,"},{"Start":"05:39.870 ","End":"05:41.640","Text":"so I just equate these 2."},{"Start":"05:41.640 ","End":"05:45.405","Text":"So 2/x equals 2x."},{"Start":"05:45.405 ","End":"05:47.415","Text":"The 2 cancels."},{"Start":"05:47.415 ","End":"05:53.075","Text":"I get x squared equals 1 and because x is bigger or equal to 0,"},{"Start":"05:53.075 ","End":"05:55.010","Text":"I get that x equals 1."},{"Start":"05:55.010 ","End":"06:00.335","Text":"I know already that this point is the point 1."},{"Start":"06:00.335 ","End":"06:04.025","Text":"We don\u0027t need it, but if we needed the y,"},{"Start":"06:04.025 ","End":"06:07.430","Text":"we could substitute into either 1 of these and we\u0027d get that y"},{"Start":"06:07.430 ","End":"06:11.315","Text":"is equal to 2 but I don\u0027t think we\u0027ll need that."},{"Start":"06:11.315 ","End":"06:14.225","Text":"We do need the x of B."},{"Start":"06:14.225 ","End":"06:18.415","Text":"For B, we want the 2 equations."},{"Start":"06:18.415 ","End":"06:22.365","Text":"First 1 is the same, y equals 2/x."},{"Start":"06:22.365 ","End":"06:29.010","Text":"The other 1 is y equals 1/2 x or x/2."},{"Start":"06:29.010 ","End":"06:31.440","Text":"If I equate the 2 right-hand sides,"},{"Start":"06:31.440 ","End":"06:34.845","Text":"I get 2/x equals x/2."},{"Start":"06:34.845 ","End":"06:40.670","Text":"Cross-multiplying, we would get that x"},{"Start":"06:40.670 ","End":"06:46.490","Text":"squared equals 4 and x would be normally plus or minus 2 but again,"},{"Start":"06:46.490 ","End":"06:48.500","Text":"we\u0027re in the non-negative,"},{"Start":"06:48.500 ","End":"06:50.830","Text":"so x equals 2."},{"Start":"06:50.830 ","End":"06:53.700","Text":"We got the x equals 1 here,"},{"Start":"06:53.700 ","End":"06:55.480","Text":"and now, we\u0027ve got x equals 2."},{"Start":"06:55.480 ","End":"06:59.275","Text":"It will give us this point here, 2."},{"Start":"06:59.275 ","End":"07:05.900","Text":"We know that the outer integral is dx,"},{"Start":"07:05.900 ","End":"07:11.370","Text":"that will be from 0 to 1 plus from 1-2 but there will be"},{"Start":"07:11.370 ","End":"07:16.969","Text":"2 different kinds of vertical slices depending on when the first region or the second."},{"Start":"07:16.969 ","End":"07:20.070","Text":"If x was here,"},{"Start":"07:20.170 ","End":"07:27.290","Text":"then the vertical slice would go from this line to this line,"},{"Start":"07:27.290 ","End":"07:35.510","Text":"from 1/2x to 2x."},{"Start":"07:35.510 ","End":"07:40.669","Text":"If x was in the right part of the region, say here,"},{"Start":"07:40.669 ","End":"07:45.765","Text":"then the vertical cut would go from here,"},{"Start":"07:45.765 ","End":"07:48.330","Text":"which is also on the 1/2x,"},{"Start":"07:48.330 ","End":"07:52.855","Text":"but this 1 is on the hyperbola, 2/x."},{"Start":"07:52.855 ","End":"07:55.860","Text":"Let\u0027s see if we can write this out."},{"Start":"07:55.880 ","End":"08:01.595","Text":"This integral over a region becomes the iterated integral"},{"Start":"08:01.595 ","End":"08:06.890","Text":"outwardly with respect to x but as I say,"},{"Start":"08:06.890 ","End":"08:08.630","Text":"it\u0027s not going to be from 0-2,"},{"Start":"08:08.630 ","End":"08:12.425","Text":"it\u0027s going to be from 0-1 separately."},{"Start":"08:12.425 ","End":"08:14.450","Text":"Then we\u0027re going to have somewhere,"},{"Start":"08:14.450 ","End":"08:19.455","Text":"a plus integral from 1-2 separately."},{"Start":"08:19.455 ","End":"08:23.000","Text":"Now, each of these will be dx."},{"Start":"08:23.000 ","End":"08:26.405","Text":"I\u0027m going to get something here, dx."},{"Start":"08:26.405 ","End":"08:31.260","Text":"Then for this 1, for each x between 0 and 1,"},{"Start":"08:31.260 ","End":"08:34.365","Text":"y goes from, let\u0027s see,"},{"Start":"08:34.365 ","End":"08:38.695","Text":"we\u0027ll also emphasize that this is the limits for x."},{"Start":"08:38.695 ","End":"08:40.845","Text":"The inner integral is B,"},{"Start":"08:40.845 ","End":"08:43.730","Text":"y goes from this line,"},{"Start":"08:43.730 ","End":"08:46.795","Text":"which is 1/2 x or x/2."},{"Start":"08:46.795 ","End":"08:49.635","Text":"The upper 1 will be this line,"},{"Start":"08:49.635 ","End":"08:53.275","Text":"2x, and this is dy."},{"Start":"08:53.275 ","End":"09:01.925","Text":"Here, I just need the function which is x squared plus y in parentheses."},{"Start":"09:01.925 ","End":"09:05.490","Text":"For the next 1, again,"},{"Start":"09:05.490 ","End":"09:07.725","Text":"from 1 to 2 dx but here,"},{"Start":"09:07.725 ","End":"09:10.995","Text":"we go for y,"},{"Start":"09:10.995 ","End":"09:15.220","Text":"we\u0027ll need to y goes from something to something dy."},{"Start":"09:15.220 ","End":"09:20.185","Text":"The lower bit is also 1/2x or x/2."},{"Start":"09:20.185 ","End":"09:24.860","Text":"The upper 1 is the hyperbola, 2/x."},{"Start":"09:24.860 ","End":"09:27.519","Text":"The function is still the same function,"},{"Start":"09:27.519 ","End":"09:30.490","Text":"x squared plus y."},{"Start":"09:30.490 ","End":"09:35.795","Text":"This is actually the volume of the bit above part."},{"Start":"09:35.795 ","End":"09:41.835","Text":"Maybe this is part 1 and this is part 2,"},{"Start":"09:41.835 ","End":"09:46.590","Text":"this integral part 1, integral part 2."},{"Start":"09:46.590 ","End":"09:49.530","Text":"We\u0027ll do each bit separately."},{"Start":"09:49.530 ","End":"09:51.300","Text":"At this point, it\u0027s just purely technical."},{"Start":"09:51.300 ","End":"09:53.474","Text":"I don\u0027t need the pictures anymore."},{"Start":"09:53.474 ","End":"10:00.240","Text":"Let\u0027s start with the first integral and see what we get."},{"Start":"10:00.240 ","End":"10:04.050","Text":"Then we\u0027ll get to do the second, then we\u0027ll add the results."},{"Start":"10:04.050 ","End":"10:09.360","Text":"We do these integrals from the inside out."},{"Start":"10:09.360 ","End":"10:13.270","Text":"First of all, the inner bit,"},{"Start":"10:14.290 ","End":"10:17.180","Text":"I\u0027d like to compute the inner bit separately."},{"Start":"10:17.180 ","End":"10:20.525","Text":"Let me call this asterisk and I\u0027ll do this at the side."},{"Start":"10:20.525 ","End":"10:27.570","Text":"Asterisk is the integral from x"},{"Start":"10:27.570 ","End":"10:35.895","Text":"over 2 to 2x of x squared plus y dy."},{"Start":"10:35.895 ","End":"10:38.655","Text":"This is equal to,"},{"Start":"10:38.655 ","End":"10:41.280","Text":"let\u0027s see if it\u0027s dy,"},{"Start":"10:41.280 ","End":"10:42.975","Text":"x is a constant,"},{"Start":"10:42.975 ","End":"10:47.400","Text":"so this is just x squared y,"},{"Start":"10:47.400 ","End":"10:51.825","Text":"and the second bit is a half y squared, y squared over 2."},{"Start":"10:51.825 ","End":"10:53.250","Text":"I\u0027ll write it this way,"},{"Start":"10:53.250 ","End":"10:54.615","Text":"a half y squared."},{"Start":"10:54.615 ","End":"11:01.710","Text":"We need to evaluate this between x/2 and 2x,"},{"Start":"11:01.710 ","End":"11:04.695","Text":"remember, it\u0027s y that we\u0027re substituting."},{"Start":"11:04.695 ","End":"11:08.595","Text":"Let\u0027s do the upper limit, first, the 2x."},{"Start":"11:08.595 ","End":"11:13.245","Text":"What we get will be,"},{"Start":"11:13.245 ","End":"11:16.050","Text":"if we put in y equals 2x, here,"},{"Start":"11:16.050 ","End":"11:21.300","Text":"we get 2x times x squared is 2x cubed."},{"Start":"11:21.300 ","End":"11:23.280","Text":"If I put y equals 2x,"},{"Start":"11:23.280 ","End":"11:25.780","Text":"and that\u0027s 4x squared."},{"Start":"11:28.340 ","End":"11:31.660","Text":"Yeah, that\u0027s right. Wait a minute."},{"Start":"11:31.730 ","End":"11:36.750","Text":"Yeah, 4x squared over 2, 2x squared, sorry."},{"Start":"11:36.750 ","End":"11:42.300","Text":"Then x over 2 will give us x"},{"Start":"11:42.300 ","End":"11:49.120","Text":"squared times x/2 will be a half x cubed."},{"Start":"11:50.420 ","End":"11:53.880","Text":"If I put x over 2 here,"},{"Start":"11:53.880 ","End":"11:56.145","Text":"I\u0027ll get x squared over 4,"},{"Start":"11:56.145 ","End":"12:05.110","Text":"so it\u0027ll be 1/8 of x squared."},{"Start":"12:06.320 ","End":"12:11.205","Text":"I can collect together like terms,"},{"Start":"12:11.205 ","End":"12:13.439","Text":"x cubed and x cubed,"},{"Start":"12:13.439 ","End":"12:17.580","Text":"2 minus 1/2 is 1 1/2."},{"Start":"12:17.580 ","End":"12:21.810","Text":"I\u0027ll write that as 3/2, x cubed."},{"Start":"12:21.810 ","End":"12:23.835","Text":"Let\u0027s see, for the x squared,"},{"Start":"12:23.835 ","End":"12:27.045","Text":"I have 2 minus 1/8."},{"Start":"12:27.045 ","End":"12:34.560","Text":"If we put it all as 8, 16/8 minus 1/8, 15/8,"},{"Start":"12:34.560 ","End":"12:42.405","Text":"so it\u0027s plus 15 over 8 x squared."},{"Start":"12:42.405 ","End":"12:44.625","Text":"That\u0027s the asterisk bit."},{"Start":"12:44.625 ","End":"12:54.720","Text":"Now, back here, we have the integral from 0 to 1 of what I have here,"},{"Start":"12:54.720 ","End":"13:01.395","Text":"3 over 2 x cubed plus"},{"Start":"13:01.395 ","End":"13:10.470","Text":"15 over 8 x squared dx."},{"Start":"13:10.470 ","End":"13:13.725","Text":"That\u0027s what 1 equals."},{"Start":"13:13.725 ","End":"13:15.855","Text":"Let\u0027s do the integral,"},{"Start":"13:15.855 ","End":"13:17.220","Text":"raise the power by 1,"},{"Start":"13:17.220 ","End":"13:21.120","Text":"it\u0027s x to the fourth divided by the 4,"},{"Start":"13:21.120 ","End":"13:26.415","Text":"so I\u0027ve got 3/8 x to the fourth,"},{"Start":"13:26.415 ","End":"13:28.650","Text":"here, raise the power by 1,"},{"Start":"13:28.650 ","End":"13:30.795","Text":"it\u0027s x cubed divide by 3."},{"Start":"13:30.795 ","End":"13:33.735","Text":"I can just divide the numerator by 3,"},{"Start":"13:33.735 ","End":"13:42.165","Text":"that gives me 5/8 x cubed."},{"Start":"13:42.165 ","End":"13:48.195","Text":"This has to be taken between 0 and 1."},{"Start":"13:48.195 ","End":"13:50.280","Text":"If I put in 0,"},{"Start":"13:50.280 ","End":"13:53.235","Text":"everything is 0, so that doesn\u0027t matter."},{"Start":"13:53.235 ","End":"13:55.185","Text":"I just need to put in 1,"},{"Start":"13:55.185 ","End":"14:01.365","Text":"so I get 3/8 plus 5/8."},{"Start":"14:01.365 ","End":"14:03.180","Text":"That works out nicely."},{"Start":"14:03.180 ","End":"14:05.775","Text":"That is equal to 1."},{"Start":"14:05.775 ","End":"14:13.380","Text":"This is the answer to part 1."},{"Start":"14:13.380 ","End":"14:17.230","Text":"Now let\u0027s get on with part 2."},{"Start":"14:17.330 ","End":"14:26.330","Text":"In part 2, we also do it from inside out,"},{"Start":"14:26.330 ","End":"14:32.390","Text":"so do this bit first."},{"Start":"14:32.390 ","End":"14:34.600","Text":"Then we\u0027ll also call that,"},{"Start":"14:34.600 ","End":"14:38.625","Text":"maybe an asterisk, me I\u0027ll call it double asterisk."},{"Start":"14:38.625 ","End":"14:41.640","Text":"Let\u0027s see what double asterisk is."},{"Start":"14:41.640 ","End":"14:45.420","Text":"Double asterisk is the integral"},{"Start":"14:45.420 ","End":"14:52.800","Text":"from x/2 to 2/x,"},{"Start":"14:52.800 ","End":"14:56.530","Text":"of x squared plus y dy."},{"Start":"15:01.430 ","End":"15:10.035","Text":"Now, this bit, the integral is the same as before."},{"Start":"15:10.035 ","End":"15:19.665","Text":"It\u0027s x squared y plus 1/2 y squared."},{"Start":"15:19.665 ","End":"15:26.160","Text":"The difference is that the limits are not the same."},{"Start":"15:26.160 ","End":"15:29.940","Text":"The x/2 below is the same,"},{"Start":"15:29.940 ","End":"15:36.820","Text":"but above it was previously 2x, now it\u0027s 2/x."},{"Start":"15:38.810 ","End":"15:43.935","Text":"When we substitute, the second part will be the same,"},{"Start":"15:43.935 ","End":"15:45.960","Text":"the first part won\u0027t."},{"Start":"15:45.960 ","End":"15:51.520","Text":"What we get is that this is equal to."},{"Start":"15:54.800 ","End":"15:59.055","Text":"Here, I have to substitute, y is 2/x."},{"Start":"15:59.055 ","End":"16:02.970","Text":"If it\u0027s 2/x, then it\u0027s 2x squared over x."},{"Start":"16:02.970 ","End":"16:05.230","Text":"This is just 2x."},{"Start":"16:06.320 ","End":"16:11.000","Text":"Here, if I substitute 2/x,"},{"Start":"16:11.000 ","End":"16:20.160","Text":"it\u0027s 4/x squared, and it\u0027s 1/2 of that,"},{"Start":"16:20.160 ","End":"16:23.830","Text":"so it\u0027s 2/x squared."},{"Start":"16:24.260 ","End":"16:26.805","Text":"The second part with x/2,"},{"Start":"16:26.805 ","End":"16:29.190","Text":"I can just copy from here."},{"Start":"16:29.190 ","End":"16:38.590","Text":"It\u0027s 1/2 x cubed plus 1/8 x squared."},{"Start":"16:41.750 ","End":"16:47.160","Text":"This time there\u0027s no like terms to collect so I\u0027ll just straightaway go back to here."},{"Start":"16:47.160 ","End":"16:52.995","Text":"This time the limits are different,"},{"Start":"16:52.995 ","End":"16:57.510","Text":"this time integral number 2"},{"Start":"16:57.510 ","End":"17:03.210","Text":"is from 1 to 2 of the double asterisk,"},{"Start":"17:03.210 ","End":"17:11.505","Text":"which is 2x plus 2/x squared"},{"Start":"17:11.505 ","End":"17:22.900","Text":"minus 1/2 x cubed minus an 1/8 x squared."},{"Start":"17:23.900 ","End":"17:29.025","Text":"All this dx."},{"Start":"17:29.025 ","End":"17:35.460","Text":"We get the integral of 2x is x squared."},{"Start":"17:35.460 ","End":"17:42.600","Text":"The integral of 1/x squared is minus 1/x."},{"Start":"17:42.600 ","End":"17:46.305","Text":"This comes out minus 2/x."},{"Start":"17:46.305 ","End":"17:49.770","Text":"If you\u0027re not sure, you can differentiate it to see."},{"Start":"17:49.770 ","End":"17:53.505","Text":"Here, I raise the power by 1 is 4."},{"Start":"17:53.505 ","End":"17:55.875","Text":"This gives me, divide by 4,"},{"Start":"17:55.875 ","End":"17:59.430","Text":"minus 1/8 x to the fourth."},{"Start":"17:59.430 ","End":"18:03.270","Text":"Here, raise it by 1 is 3,"},{"Start":"18:03.270 ","End":"18:10.230","Text":"if I divide by 3, minus 1 over 24x cubed."},{"Start":"18:10.230 ","End":"18:19.350","Text":"All this has to go from 1 to 2. Let\u0027s see."},{"Start":"18:19.350 ","End":"18:21.270","Text":"I\u0027ll scroll a bit more."},{"Start":"18:21.270 ","End":"18:23.430","Text":"All I have to remember is at the end,"},{"Start":"18:23.430 ","End":"18:25.830","Text":"is add part 1 and part 2."},{"Start":"18:25.830 ","End":"18:30.450","Text":"Let\u0027s see if I substitute 2 what do I get?"},{"Start":"18:30.450 ","End":"18:37.760","Text":"2 squared is 4, 2/2 is 1."},{"Start":"18:37.760 ","End":"18:47.030","Text":"2 to the fourth is 16/8 is 2."},{"Start":"18:47.030 ","End":"18:51.200","Text":"Let\u0027s see, 2 cubed is 8."},{"Start":"18:51.200 ","End":"18:58.065","Text":"8/24 is 1/3, and this is just for the 2."},{"Start":"18:58.065 ","End":"19:00.195","Text":"Now, I have to do the same thing for 1."},{"Start":"19:00.195 ","End":"19:04.845","Text":"For 1, I\u0027ve got 1 minus 2"},{"Start":"19:04.845 ","End":"19:12.220","Text":"minus an 1/8 minus 1/24."},{"Start":"19:14.930 ","End":"19:18.380","Text":"I don\u0027t want to scroll anymore."},{"Start":"19:18.380 ","End":"19:21.290","Text":"Let me just continue over here through the first 1."},{"Start":"19:21.290 ","End":"19:25.790","Text":"4 minus 1 minus 2 is just 1."},{"Start":"19:25.790 ","End":"19:29.165","Text":"1 minus 1/3 is 2/3,"},{"Start":"19:29.165 ","End":"19:30.680","Text":"for the first bit."},{"Start":"19:30.680 ","End":"19:33.480","Text":"Let\u0027s see the second bit."},{"Start":"19:37.260 ","End":"19:41.965","Text":"What I can do is 1/8 plus 1/24."},{"Start":"19:41.965 ","End":"19:44.229","Text":"If I do that,"},{"Start":"19:44.229 ","End":"19:46.150","Text":"that might make things easier."},{"Start":"19:46.150 ","End":"19:48.505","Text":"If I put it all over 24,"},{"Start":"19:48.505 ","End":"19:50.890","Text":"I\u0027m adding them because they\u0027re both minus."},{"Start":"19:50.890 ","End":"19:55.280","Text":"I\u0027ve got 3/24 and 1/24 is 4/24."},{"Start":"19:55.280 ","End":"19:57.100","Text":"It\u0027s 1/6."},{"Start":"19:57.100 ","End":"20:03.750","Text":"So I\u0027ve got 1 minus 2 is minus 1,"},{"Start":"20:03.750 ","End":"20:06.600","Text":"and this is 1/6, minus 1/6."},{"Start":"20:06.600 ","End":"20:12.010","Text":"I\u0027ve got minus 1 and 1/6,"},{"Start":"20:13.670 ","End":"20:20.100","Text":"which means 2/3 plus 1 and 1/6,"},{"Start":"20:20.100 ","End":"20:27.820","Text":"that is 1 and 5/6 I make it."},{"Start":"20:28.160 ","End":"20:32.460","Text":"Yeah, because it\u0027s 1 plus 2/3 plus 1/6 and 2/3 is,"},{"Start":"20:32.460 ","End":"20:34.155","Text":"of course, 1 and 5/6."},{"Start":"20:34.155 ","End":"20:37.060","Text":"I\u0027ll circle this,"},{"Start":"20:37.550 ","End":"20:48.360","Text":"this 1 and 5/6."},{"Start":"20:48.360 ","End":"20:51.765","Text":"This is part 1, this is the answer to part 2."},{"Start":"20:51.765 ","End":"20:53.700","Text":"Now I\u0027ve got some room here."},{"Start":"20:53.700 ","End":"20:57.885","Text":"1 plus 2 is equal to,"},{"Start":"20:57.885 ","End":"20:59.940","Text":"let\u0027s do it in sixth."},{"Start":"20:59.940 ","End":"21:04.920","Text":"1 is 6/6, and 1 and 5/6,"},{"Start":"21:04.920 ","End":"21:08.700","Text":"and 6 plus 5, is 11/6."},{"Start":"21:08.700 ","End":"21:15.360","Text":"Altogether, 6 plus 11 is 17/6."},{"Start":"21:15.360 ","End":"21:20.495","Text":"This is the final answer to the question,"},{"Start":"21:20.495 ","End":"21:23.850","Text":"and we are done."}],"ID":8685},{"Watched":false,"Name":"Exercise 2 part d","Duration":"20m 6s","ChapterTopicVideoID":8470,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this exercise, we have to compute the volume of the"},{"Start":"00:03.210 ","End":"00:07.410","Text":"solid and it\u0027s bounded by these surfaces,"},{"Start":"00:07.410 ","End":"00:09.675","Text":"1, 2, 3 of them."},{"Start":"00:09.675 ","End":"00:17.145","Text":"I\u0027ve inserted the usual picture that I found which explains the general concept."},{"Start":"00:17.145 ","End":"00:23.970","Text":"We have the volume as a projection of region D on"},{"Start":"00:23.970 ","End":"00:30.960","Text":"the xy-plane and there\u0027s an upper part of the boundary of the volume,"},{"Start":"00:30.960 ","End":"00:34.110","Text":"there\u0027s an upper surface and the lower surface."},{"Start":"00:34.110 ","End":"00:41.190","Text":"The formula is the double integral of the upper minus the lower over this region,"},{"Start":"00:41.190 ","End":"00:45.430","Text":"which is the projection of the solid."},{"Start":"00:45.430 ","End":"00:49.625","Text":"Up till now, the exercises have mostly been,"},{"Start":"00:49.625 ","End":"00:58.700","Text":"we have been given the borders of this region D by functions of x and y."},{"Start":"00:58.700 ","End":"01:01.370","Text":"But this isn\u0027t always the case."},{"Start":"01:01.370 ","End":"01:04.700","Text":"Sometimes the borders of"},{"Start":"01:04.700 ","End":"01:10.520","Text":"this region D are obtained from the intersection of these 2 surfaces."},{"Start":"01:10.520 ","End":"01:14.345","Text":"In this picture, they don\u0027t intersect, but in principle,"},{"Start":"01:14.345 ","End":"01:20.435","Text":"these surfaces might continue and they might have some intersection,"},{"Start":"01:20.435 ","End":"01:22.880","Text":"and it\u0027s hard to draw."},{"Start":"01:22.880 ","End":"01:27.860","Text":"But you can also imagine a hybrid situation where they intersect and we\u0027re still"},{"Start":"01:27.860 ","End":"01:33.005","Text":"given part of the region in terms of an equation and part of it,"},{"Start":"01:33.005 ","End":"01:37.040","Text":"we determine by intersecting the 2 surfaces and this is"},{"Start":"01:37.040 ","End":"01:41.554","Text":"the case here where the middle equation is going to do double duty."},{"Start":"01:41.554 ","End":"01:47.225","Text":"It\u0027s going to partially define D and it\u0027s partly also going to be 1 of the surfaces."},{"Start":"01:47.225 ","End":"01:49.625","Text":"But this is not a convenient form."},{"Start":"01:49.625 ","End":"01:53.795","Text":"I want to extract z in terms of x and y,"},{"Start":"01:53.795 ","End":"01:55.670","Text":"and that\u0027s what I\u0027ll do first."},{"Start":"01:55.670 ","End":"01:58.369","Text":"If I multiply both sides by 4,"},{"Start":"01:58.369 ","End":"02:08.340","Text":"I\u0027ve got x because of the 4 cancels and I\u0027ll get plus 2y plus z is equal to 4."},{"Start":"02:08.340 ","End":"02:14.980","Text":"Then I extract Z so I get that z is equal to"},{"Start":"02:15.680 ","End":"02:20.130","Text":"4 minus x minus"},{"Start":"02:20.130 ","End":"02:27.945","Text":"2y and the other surface will be Z equals 0."},{"Start":"02:27.945 ","End":"02:29.520","Text":"Now I\u0027ve got 2 surfaces,"},{"Start":"02:29.520 ","End":"02:34.430","Text":"z in terms of x and y. I\u0027m going to intersect them and see,"},{"Start":"02:34.430 ","End":"02:39.050","Text":"I\u0027ll get an equation an x and y and this will give me part of the boundary."},{"Start":"02:39.050 ","End":"02:42.125","Text":"The other part I\u0027ll get from this equation."},{"Start":"02:42.125 ","End":"02:44.270","Text":"But let\u0027s, first of all, do this intersection."},{"Start":"02:44.270 ","End":"02:49.370","Text":"When they intersect, what I get is just compare the right-hand sides and I"},{"Start":"02:49.370 ","End":"02:56.120","Text":"get 4 minus x minus 2y equals 0."},{"Start":"02:56.120 ","End":"03:01.820","Text":"I think I\u0027d rather have this equation down here and note that this is just an x and y,"},{"Start":"03:01.820 ","End":"03:04.175","Text":"so it\u0027s going to define parts of the boundary,"},{"Start":"03:04.175 ","End":"03:06.605","Text":"and the other bit is what was given to us."},{"Start":"03:06.605 ","End":"03:08.300","Text":"I\u0027ll just copy this over here."},{"Start":"03:08.300 ","End":"03:13.890","Text":"We have 2y squared equals x."},{"Start":"03:14.180 ","End":"03:20.570","Text":"These 2 are going to actually define the boundary D,"},{"Start":"03:20.570 ","End":"03:24.695","Text":"which is the projection of the solid body."},{"Start":"03:24.695 ","End":"03:28.325","Text":"We use these 2 equations."},{"Start":"03:28.325 ","End":"03:30.980","Text":"The first 1 is"},{"Start":"03:30.980 ","End":"03:37.140","Text":"a straight line equation and maybe I\u0027ll compute the intersection with the axis."},{"Start":"03:37.140 ","End":"03:40.920","Text":"If x is 0, then I get 2y is 4,"},{"Start":"03:40.920 ","End":"03:43.125","Text":"so y is 2."},{"Start":"03:43.125 ","End":"03:47.204","Text":"So would go through here and if y is 0,"},{"Start":"03:47.204 ","End":"03:52.530","Text":"then I get 4 equals x,"},{"Start":"03:52.530 ","End":"03:56.610","Text":"that might be point 4,"},{"Start":"03:56.610 ","End":"03:59.370","Text":"this doesn\u0027t have to be precise."},{"Start":"03:59.370 ","End":"04:03.675","Text":"I\u0027ll just label this is 4 and this is 2."},{"Start":"04:03.675 ","End":"04:10.145","Text":"The other 1 is a sideways parabola it\u0027s x equals 2y squared."},{"Start":"04:10.145 ","End":"04:13.355","Text":"If it was y equals something x squared, it would be this way."},{"Start":"04:13.355 ","End":"04:18.020","Text":"X equals something y squared is going to be some kind of parabola like this."},{"Start":"04:18.020 ","End":"04:22.700","Text":"Maybe I should just draw a general shape of"},{"Start":"04:22.700 ","End":"04:28.860","Text":"a parabola and then let\u0027s compute the intersections."},{"Start":"04:29.540 ","End":"04:35.535","Text":"Let\u0027s intersect these 2 and see what we get."},{"Start":"04:35.535 ","End":"04:39.560","Text":"The easiest thing to do would be to let x"},{"Start":"04:39.560 ","End":"04:42.890","Text":"equal 2y squared in this equation and then I would"},{"Start":"04:42.890 ","End":"04:52.565","Text":"get 4 minus 2y squared minus 2y equals 0."},{"Start":"04:52.565 ","End":"04:54.905","Text":"If I divide by 2,"},{"Start":"04:54.905 ","End":"04:59.830","Text":"now let\u0027s divide by minus 2 and then just rearrange."},{"Start":"04:59.830 ","End":"05:04.860","Text":"I get y squared divide this by minus 2,"},{"Start":"05:04.860 ","End":"05:09.815","Text":"that leaves me plus y minus 2 equals 0."},{"Start":"05:09.815 ","End":"05:14.075","Text":"I solve for y, we get 2 solutions for y."},{"Start":"05:14.075 ","End":"05:22.730","Text":"We get y equals 1 and we get y equals minus 2 and for each 1,"},{"Start":"05:22.730 ","End":"05:24.860","Text":"I\u0027ll compute its x,"},{"Start":"05:24.860 ","End":"05:27.830","Text":"which I can get from 2y squared."},{"Start":"05:27.830 ","End":"05:31.710","Text":"Here I get x equals 2,"},{"Start":"05:33.050 ","End":"05:40.870","Text":"and if y is minus 2 then x is 2y squared, that would be 8."},{"Start":"05:41.160 ","End":"05:44.230","Text":"Maybe I\u0027ll just label it that it\u0027s the point,"},{"Start":"05:44.230 ","End":"05:47.725","Text":"minus 2, 8 it\u0027s off the chart."},{"Start":"05:47.725 ","End":"05:50.185","Text":"But I know that here,"},{"Start":"05:50.185 ","End":"05:54.150","Text":"that this is minus 2 and then we wait."},{"Start":"05:54.150 ","End":"05:58.540","Text":"It\u0027s now clear what the region D is in the plane."},{"Start":"05:58.540 ","End":"06:01.810","Text":"It\u0027s this bit here and why don\u0027t I shade it?"},{"Start":"06:01.810 ","End":"06:04.950","Text":"There we are and then just label it,"},{"Start":"06:04.950 ","End":"06:09.430","Text":"we\u0027ll call it D. Now I\u0027m going to use"},{"Start":"06:09.430 ","End":"06:16.310","Text":"the formula here that the volume is equal to the double integral."},{"Start":"06:16.410 ","End":"06:21.340","Text":"That\u0027s the region D and it\u0027s going to be the upper"},{"Start":"06:21.340 ","End":"06:26.364","Text":"minus the lower and this is where I come to the first minor snag."},{"Start":"06:26.364 ","End":"06:31.410","Text":"I don\u0027t know which is the upper and which is the lower, dA."},{"Start":"06:31.410 ","End":"06:37.375","Text":"There\u0027s 2 surfaces and it\u0027s actually quite easy to tell which is which."},{"Start":"06:37.375 ","End":"06:43.855","Text":"Because the place where they\u0027re both equal is the intersection,"},{"Start":"06:43.855 ","End":"06:50.595","Text":"and we found that this intersection is just this line here,"},{"Start":"06:50.595 ","End":"06:53.920","Text":"which reminds me that I really should have labeled these lines."},{"Start":"06:53.920 ","End":"06:58.340","Text":"This is the line where x equals 2y squared."},{"Start":"06:58.340 ","End":"07:06.560","Text":"This 1, 4 minus x minus 2y equals 0."},{"Start":"07:06.560 ","End":"07:11.150","Text":"Now, this line, because it\u0027s the intersection of the 2 surfaces,"},{"Start":"07:11.150 ","End":"07:13.090","Text":"on each side of the line,"},{"Start":"07:13.090 ","End":"07:17.885","Text":"1 surface is going to be always bigger or always smaller than the other."},{"Start":"07:17.885 ","End":"07:20.150","Text":"Now, we want this side of the line,"},{"Start":"07:20.150 ","End":"07:22.565","Text":"so all I have to do is take any point"},{"Start":"07:22.565 ","End":"07:26.825","Text":"that\u0027s inside the region"},{"Start":"07:26.825 ","End":"07:30.035","Text":"because it\u0027s going to be on this side of the line and see which is bigger."},{"Start":"07:30.035 ","End":"07:32.120","Text":"Actually, I could take the origin,"},{"Start":"07:32.120 ","End":"07:33.605","Text":"that would be the simplest to me."},{"Start":"07:33.605 ","End":"07:35.405","Text":"If I plug in the origin,"},{"Start":"07:35.405 ","End":"07:37.895","Text":"I get something positive z is 4."},{"Start":"07:37.895 ","End":"07:40.770","Text":"If you had chosen the point,"},{"Start":"07:40.970 ","End":"07:43.605","Text":"it could be inside 2,"},{"Start":"07:43.605 ","End":"07:51.965","Text":"0, then I would get 4 minus 2 minus 0 positive."},{"Start":"07:51.965 ","End":"07:54.230","Text":"Everywhere in this D,"},{"Start":"07:54.230 ","End":"07:57.640","Text":"this function will be positive."},{"Start":"07:57.640 ","End":"08:01.350","Text":"It\u0027ll be 0 on the line itself,"},{"Start":"08:01.350 ","End":"08:05.120","Text":"I\u0027m saying that this is always bigger or equal to 0,"},{"Start":"08:05.120 ","End":"08:10.175","Text":"so that this is the upper, and just divide that."},{"Start":"08:10.175 ","End":"08:12.870","Text":"That\u0027s going to be the upper,"},{"Start":"08:13.010 ","End":"08:16.990","Text":"and this 1 is going to be the lower."},{"Start":"08:16.990 ","End":"08:21.260","Text":"The other matter I have to decide on is whether I\u0027m going to use"},{"Start":"08:21.260 ","End":"08:28.080","Text":"this region as a Type 1 dy dx,"},{"Start":"08:28.080 ","End":"08:30.435","Text":"or a Type 2, dx dy,"},{"Start":"08:30.435 ","End":"08:36.080","Text":"do I want y extracted in terms of x or x extracted in terms of y?"},{"Start":"08:36.080 ","End":"08:40.730","Text":"Now, I say it\u0027s better to slice this horizontally."},{"Start":"08:40.730 ","End":"08:43.625","Text":"If I take a typical y,"},{"Start":"08:43.625 ","End":"08:52.470","Text":"say here and then I take a horizontal line through this y."},{"Start":"08:52.590 ","End":"08:59.545","Text":"Then it enters here on the parabola and exits here on the straight line."},{"Start":"08:59.545 ","End":"09:03.220","Text":"This will be true wherever I put y between."},{"Start":"09:03.220 ","End":"09:04.900","Text":"Well, what you\u0027re going to be between x?"},{"Start":"09:04.900 ","End":"09:08.140","Text":"It\u0027s going to be between 1 and minus 2 or minus 2 and"},{"Start":"09:08.140 ","End":"09:13.585","Text":"1 but why is the other way not good?"},{"Start":"09:13.585 ","End":"09:16.120","Text":"If I sliced it vertically,"},{"Start":"09:16.120 ","End":"09:19.160","Text":"then if I took x here,"},{"Start":"09:19.440 ","End":"09:28.395","Text":"then my vertical slice would hit the parabola twice but if I took x here,"},{"Start":"09:28.395 ","End":"09:32.070","Text":"then the vertical slice through the region would"},{"Start":"09:32.070 ","End":"09:35.850","Text":"hit the line above and the parabola below."},{"Start":"09:35.850 ","End":"09:37.620","Text":"I\u0027d have to divide into 2 cases,"},{"Start":"09:37.620 ","End":"09:44.290","Text":"would have to take from 0-2 and then from 2-4."},{"Start":"09:44.700 ","End":"09:49.070","Text":"It\u0027s easier when you have just 1 case."},{"Start":"09:49.230 ","End":"09:55.750","Text":"We can now write the integral as an iterated integral."},{"Start":"09:55.750 ","End":"09:58.180","Text":"It\u0027s going to be dxdy."},{"Start":"09:58.180 ","End":"10:01.165","Text":"We get that our volume equals."},{"Start":"10:01.165 ","End":"10:03.205","Text":"Now, notice that y goes from,"},{"Start":"10:03.205 ","End":"10:05.635","Text":"as I said, from minus 2 to 1."},{"Start":"10:05.635 ","End":"10:08.755","Text":"I\u0027ll write y from minus 2 to 1."},{"Start":"10:08.755 ","End":"10:12.100","Text":"Then that will be dy."},{"Start":"10:12.100 ","End":"10:13.720","Text":"Then for each such y,"},{"Start":"10:13.720 ","End":"10:19.750","Text":"x will go from parabola to the line."},{"Start":"10:19.750 ","End":"10:27.910","Text":"It\u0027s the integral at x going from the parabola,"},{"Start":"10:27.910 ","End":"10:31.750","Text":"which is 2y squared up to,"},{"Start":"10:31.750 ","End":"10:37.675","Text":"that\u0027s just 1 small piece of work we have to do is to isolate x in terms of y."},{"Start":"10:37.675 ","End":"10:41.515","Text":"Allow me to replace this equation with,"},{"Start":"10:41.515 ","End":"10:43.360","Text":"if I just put x on the other side,"},{"Start":"10:43.360 ","End":"10:50.410","Text":"I get x equals 4 minus 2y."},{"Start":"10:50.410 ","End":"10:53.005","Text":"That\u0027s what I write as the upper limit,"},{"Start":"10:53.005 ","End":"10:59.230","Text":"4 minus 2y and that\u0027s dx traveling from here to here."},{"Start":"10:59.230 ","End":"11:05.320","Text":"Then I need to put the upper minus lower, well, the lower is 0,"},{"Start":"11:05.320 ","End":"11:09.160","Text":"so I just have the upper really so what I have here is just"},{"Start":"11:09.160 ","End":"11:15.030","Text":"the 4 minus x minus 2y minus 0 theoretically,"},{"Start":"11:15.030 ","End":"11:16.650","Text":"but no need for that."},{"Start":"11:16.650 ","End":"11:20.300","Text":"Now, it\u0027s just purely technical to do the integral."},{"Start":"11:20.300 ","End":"11:23.980","Text":"As usual, we work from the inside out."},{"Start":"11:23.980 ","End":"11:31.735","Text":"First of all, do this integral with respect to x. I\u0027ll do it at the side."},{"Start":"11:31.735 ","End":"11:37.390","Text":"Maybe I\u0027ll give this a name asterisk and then the asterisk I\u0027ll compute at the side."},{"Start":"11:37.390 ","End":"11:41.620","Text":"That\u0027s going to equal. I could copy it."},{"Start":"11:41.620 ","End":"11:44.230","Text":"We can straight away jump into the integral."},{"Start":"11:44.230 ","End":"11:49.520","Text":"The integral with respect to x of 4 is 4x."},{"Start":"11:50.340 ","End":"11:53.230","Text":"Remember, the integration is with respect to x."},{"Start":"11:53.230 ","End":"11:58.615","Text":"The integral of minus x is minus 1/2x squared."},{"Start":"11:58.615 ","End":"12:05.050","Text":"The integral of minus 2y is minus 2yx because the"},{"Start":"12:05.050 ","End":"12:11.680","Text":"integral is dx but all this has to be evaluated between the upper and lower limits."},{"Start":"12:11.680 ","End":"12:20.320","Text":"We have to plug in from 2y squared up to 4 minus 2y."},{"Start":"12:20.320 ","End":"12:23.260","Text":"I want just to remind you, it\u0027s x that\u0027s being substituted,"},{"Start":"12:23.260 ","End":"12:25.525","Text":"of course, not y."},{"Start":"12:25.525 ","End":"12:29.990","Text":"Let\u0027s see what we get from here."},{"Start":"12:30.030 ","End":"12:35.560","Text":"If I substitute x equals 4 minus 2y,"},{"Start":"12:35.560 ","End":"12:41.080","Text":"I get 4 times 4 minus 2,"},{"Start":"12:41.080 ","End":"12:45.970","Text":"y minus 1/2, 4 minus"},{"Start":"12:45.970 ","End":"12:56.980","Text":"2y squared minus 2y"},{"Start":"12:56.980 ","End":"13:01.910","Text":"times 4 minus 2y,"},{"Start":"13:02.460 ","End":"13:07.930","Text":"let\u0027s just say, this is the upper bit minus,"},{"Start":"13:07.930 ","End":"13:09.985","Text":"now here I\u0027ll write the lower bit,"},{"Start":"13:09.985 ","End":"13:12.160","Text":"where x is 2y squared,"},{"Start":"13:12.160 ","End":"13:21.220","Text":"we get 4 times 2y squared minus 1/2 2y squared"},{"Start":"13:21.220 ","End":"13:29.260","Text":"squared minus 2y times"},{"Start":"13:29.260 ","End":"13:33.145","Text":"x is 2y squared."},{"Start":"13:33.145 ","End":"13:36.115","Text":"Upper minus lower."},{"Start":"13:36.115 ","End":"13:39.160","Text":"This is just a tedious calculation,"},{"Start":"13:39.160 ","End":"13:40.885","Text":"let\u0027s see if I can simplify it."},{"Start":"13:40.885 ","End":"13:43.990","Text":"Yeah, I could take 4 minus 2y out of"},{"Start":"13:43.990 ","End":"13:49.060","Text":"the first bracket and I\u0027ll stop further over and maybe I\u0027ll fit it into 1 line."},{"Start":"13:49.060 ","End":"13:52.270","Text":"We have 4 minus 2y."},{"Start":"13:52.270 ","End":"13:55.060","Text":"I\u0027m still working on the top bracket."},{"Start":"13:55.060 ","End":"13:59.290","Text":"This will be 4 minus,"},{"Start":"13:59.290 ","End":"14:02.140","Text":"I\u0027m left with another 4 minus 2y,"},{"Start":"14:02.140 ","End":"14:06.685","Text":"but there\u0027s also 1/2 so I can write that as 2 minus y."},{"Start":"14:06.685 ","End":"14:10.735","Text":"If it\u0027s 2 minus y and there\u0027s a minus in front of it,"},{"Start":"14:10.735 ","End":"14:13.045","Text":"it\u0027s minus 2 plus y."},{"Start":"14:13.045 ","End":"14:14.500","Text":"Did several steps in 1,"},{"Start":"14:14.500 ","End":"14:16.970","Text":"but I think this is fine."},{"Start":"14:19.290 ","End":"14:27.205","Text":"The last bit, the 4 minus 2y has come out front and I\u0027m just left with minus 2y."},{"Start":"14:27.205 ","End":"14:31.360","Text":"That\u0027s the top bit and now there\u0027s the minus in the lower bit."},{"Start":"14:31.360 ","End":"14:36.890","Text":"I have a y squared I can certainly take out of everything."},{"Start":"14:38.220 ","End":"14:42.880","Text":"I\u0027ll write the y squared and then let\u0027s see what\u0027s left."},{"Start":"14:42.880 ","End":"14:46.160","Text":"4 times 2 is 8."},{"Start":"14:46.230 ","End":"14:49.840","Text":"Now, if I multiply this out,"},{"Start":"14:49.840 ","End":"14:58.910","Text":"I get y^4 and the coefficient will be 2 squared over 2, which is 2."},{"Start":"14:59.070 ","End":"15:02.815","Text":"This is minus 2y^4,"},{"Start":"15:02.815 ","End":"15:09.400","Text":"but I just write minus 2y squared because the y squared has been taken out."},{"Start":"15:09.400 ","End":"15:11.620","Text":"Here, when I take the y squared out,"},{"Start":"15:11.620 ","End":"15:13.585","Text":"I\u0027ve just got 2y2,"},{"Start":"15:13.585 ","End":"15:17.420","Text":"which is minus 4y."},{"Start":"15:24.900 ","End":"15:29.560","Text":"This square bracket, 4 minus 2 is 2,"},{"Start":"15:29.560 ","End":"15:31.660","Text":"y minus 2y is minus y."},{"Start":"15:31.660 ","End":"15:37.240","Text":"Instead of this, I can say this is 2 minus y."},{"Start":"15:37.240 ","End":"15:45.290","Text":"The other bracket, not much I can collect together."},{"Start":"15:46.980 ","End":"15:52.280","Text":"Now, multiplying out this with this."},{"Start":"15:52.530 ","End":"15:54.940","Text":"Each 1 of these, with each 1 of these,"},{"Start":"15:54.940 ","End":"15:57.715","Text":"4 times 2 is 8,"},{"Start":"15:57.715 ","End":"16:01.900","Text":"4 times minus y is minus 4y,"},{"Start":"16:01.900 ","End":"16:05.860","Text":"minus 2y times 2 is minus 4y,"},{"Start":"16:05.860 ","End":"16:11.515","Text":"minus 2y minus y plus 2y squared."},{"Start":"16:11.515 ","End":"16:17.440","Text":"Here, we have minus 8y squared plus"},{"Start":"16:17.440 ","End":"16:27.830","Text":"2y^4 plus 4y cubed."},{"Start":"16:28.530 ","End":"16:33.820","Text":"Change these 2 as minus 8y."},{"Start":"16:33.820 ","End":"16:36.130","Text":"This is my asterisk."},{"Start":"16:36.130 ","End":"16:40.240","Text":"Then also changed the order of these 2 and then we will be having an ascending order."},{"Start":"16:40.240 ","End":"16:46.375","Text":"What I get is that the volume that I want is the"},{"Start":"16:46.375 ","End":"16:52.300","Text":"integral from minus 2 to 1."},{"Start":"16:52.300 ","End":"16:56.350","Text":"This is an integral dy of all this."},{"Start":"16:56.350 ","End":"16:57.865","Text":"I\u0027ll just copy it."},{"Start":"16:57.865 ","End":"17:01.580","Text":"Eight minus 8y."},{"Start":"17:02.010 ","End":"17:06.670","Text":"Silly me. These 2 can also be combined and that will give"},{"Start":"17:06.670 ","End":"17:11.065","Text":"me minus 6y squared, so back here,"},{"Start":"17:11.065 ","End":"17:16.660","Text":"minus 6y squared, and then we\u0027ll take the y cubed"},{"Start":"17:16.660 ","End":"17:22.480","Text":"for them and then plus 2y^4,"},{"Start":"17:22.480 ","End":"17:25.610","Text":"and all this dy."},{"Start":"17:26.550 ","End":"17:34.630","Text":"Continuing, the integral of this will be"},{"Start":"17:34.630 ","End":"17:42.880","Text":"8y minus y squared over 2 times a is 4y squared."},{"Start":"17:42.880 ","End":"17:44.290","Text":"Here, I raised the power,"},{"Start":"17:44.290 ","End":"17:48.505","Text":"that\u0027s a 3 divided by 3 minus 2y cubed."},{"Start":"17:48.505 ","End":"17:58.060","Text":"Here, y^4 cancels, so plus y^4."},{"Start":"17:58.060 ","End":"18:01.810","Text":"Here, y^5 divided by 5,"},{"Start":"18:01.810 ","End":"18:07.450","Text":"so 2/5 y^5, and all"},{"Start":"18:07.450 ","End":"18:14.510","Text":"this between minus 2 and 1 for y."},{"Start":"18:14.510 ","End":"18:17.410","Text":"If I plug in 1, that\u0027s the easiest."},{"Start":"18:17.410 ","End":"18:26.825","Text":"I just get 8 minus 4 minus 2 plus 1 plus 2/5."},{"Start":"18:26.825 ","End":"18:29.620","Text":"That\u0027s for the 1."},{"Start":"18:29.620 ","End":"18:32.349","Text":"Then for the minus 2,"},{"Start":"18:32.349 ","End":"18:42.165","Text":"I\u0027ll get here minus 16 minus 2 squared is 4 times 4,"},{"Start":"18:42.165 ","End":"18:44.790","Text":"is another minus 16,"},{"Start":"18:44.790 ","End":"18:49.940","Text":"minus 2 cubed is minus 8."},{"Start":"18:49.940 ","End":"18:53.750","Text":"This makes this plus 16."},{"Start":"18:53.750 ","End":"19:00.860","Text":"Minus 2^4 is plus 16."},{"Start":"19:00.860 ","End":"19:07.605","Text":"Minus 2^5 is minus 32."},{"Start":"19:07.605 ","End":"19:10.150","Text":"I\u0027ve got minus 32,"},{"Start":"19:10.150 ","End":"19:19.375","Text":"so minus 64/5."},{"Start":"19:19.375 ","End":"19:23.935","Text":"8 minus 4 minus 2 plus 1 is 3 plus 2/5."},{"Start":"19:23.935 ","End":"19:29.530","Text":"This first bit is 3 and 2/5,"},{"Start":"19:29.530 ","End":"19:31.705","Text":"we have a lot of cancellation,"},{"Start":"19:31.705 ","End":"19:33.370","Text":"minus 16 plus 16,"},{"Start":"19:33.370 ","End":"19:38.390","Text":"minus 16 plus 16, minus and minus."},{"Start":"19:38.400 ","End":"19:47.470","Text":"That makes it plus 64/5."},{"Start":"19:47.470 ","End":"19:50.350","Text":"3 and 2/5, 3 times 5 plus 2 is 17."},{"Start":"19:50.350 ","End":"19:52.700","Text":"This is 17/5."},{"Start":"19:52.700 ","End":"20:02.995","Text":"17/5 plus 64/5, 17 and 64 is 81/5."},{"Start":"20:02.995 ","End":"20:07.940","Text":"That is the answer. Finally done."}],"ID":8686},{"Watched":false,"Name":"Exercise 2 part e","Duration":"11m 28s","ChapterTopicVideoID":8471,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.820","Text":"In this exercise, we have the familiar setup of a volume of a solid bounded by surfaces."},{"Start":"00:05.820 ","End":"00:08.010","Text":"It\u0027s set up so that the solid has"},{"Start":"00:08.010 ","End":"00:14.250","Text":"a projection of a certain region or domain D in the x, y plane,"},{"Start":"00:14.250 ","End":"00:19.800","Text":"and it\u0027s bounded above and below by 2 functions where 1 is bigger or equal to the other,"},{"Start":"00:19.800 ","End":"00:25.890","Text":"and there might also be some sides which are just"},{"Start":"00:25.890 ","End":"00:33.315","Text":"above the boundary of D. Familiar situation."},{"Start":"00:33.315 ","End":"00:39.080","Text":"In this case, we know that the volume of the solid is given by the formula of the double"},{"Start":"00:39.080 ","End":"00:44.930","Text":"integral over the projection D of the upper surface minus the lower surface."},{"Start":"00:44.930 ","End":"00:47.360","Text":"We have to figure out what\u0027s going on here."},{"Start":"00:47.360 ","End":"00:49.220","Text":"What is the D? What is the upper surface?"},{"Start":"00:49.220 ","End":"00:50.705","Text":"What is the lower surface?"},{"Start":"00:50.705 ","End":"00:53.575","Text":"Sometimes these surfaces intersect,"},{"Start":"00:53.575 ","End":"00:56.940","Text":"let\u0027s see what\u0027s happening in our case."},{"Start":"00:56.940 ","End":"01:00.150","Text":"There almost seem to be too few equations here,"},{"Start":"01:00.150 ","End":"01:02.640","Text":"but if you look at this 1,"},{"Start":"01:02.640 ","End":"01:10.415","Text":"I\u0027ll just copy it, x squared plus y squared over 4 equals 1."},{"Start":"01:10.415 ","End":"01:13.445","Text":"Then this is actually a closed ellipse,"},{"Start":"01:13.445 ","End":"01:15.740","Text":"this will completely determine"},{"Start":"01:15.740 ","End":"01:20.210","Text":"D. It\u0027s not exactly an ellipse and if we\u0027re looking at it in 3D,"},{"Start":"01:20.210 ","End":"01:25.370","Text":"then it\u0027s elliptical cylinder because it extends also up and down,"},{"Start":"01:25.370 ","End":"01:27.440","Text":"but the projection onto the x,"},{"Start":"01:27.440 ","End":"01:29.825","Text":"y plane is just an ellipse."},{"Start":"01:29.825 ","End":"01:35.910","Text":"If I introduce a sketch and here\u0027s my ellipse."},{"Start":"01:35.910 ","End":"01:39.035","Text":"Actually the theory is that if I have x squared over"},{"Start":"01:39.035 ","End":"01:43.955","Text":"a squared plus y squared over b squared equals 1,"},{"Start":"01:43.955 ","End":"01:47.075","Text":"that these 2 points are plus or minus a,"},{"Start":"01:47.075 ","End":"01:49.820","Text":"and these 2 points are plus or minus b."},{"Start":"01:49.820 ","End":"01:53.090","Text":"In our case, we see that the b squared is 4,"},{"Start":"01:53.090 ","End":"01:55.090","Text":"so b is plus or minus."},{"Start":"01:55.090 ","End":"01:57.540","Text":"B is 2, but this is 2,"},{"Start":"01:57.540 ","End":"01:59.125","Text":"this is minus 2,"},{"Start":"01:59.125 ","End":"02:01.370","Text":"and this is like x squared over 1 squared,"},{"Start":"02:01.370 ","End":"02:04.710","Text":"so a is 1 and minus 1."},{"Start":"02:04.710 ","End":"02:06.455","Text":"That\u0027s the ellipse."},{"Start":"02:06.455 ","End":"02:11.255","Text":"But this isn\u0027t exactly our D,"},{"Start":"02:11.255 ","End":"02:15.830","Text":"because we notice that there\u0027s an extra condition,"},{"Start":"02:15.830 ","End":"02:18.964","Text":"z bigger or equal to 0."},{"Start":"02:18.964 ","End":"02:24.710","Text":"Now, z bigger or equal to 0 is actually going to tell me 2 things."},{"Start":"02:24.710 ","End":"02:33.620","Text":"First of all, it tells me that the lower surface is the x-y plane,"},{"Start":"02:33.620 ","End":"02:41.950","Text":"because z equals 0 is the border of this which is the x-y plane."},{"Start":"02:42.710 ","End":"02:46.660","Text":"Also the surface z equals 0."},{"Start":"02:48.420 ","End":"02:57.479","Text":"I\u0027ll write it somewhere that z equals 0 it\u0027s actually the lower surface."},{"Start":"02:57.479 ","End":"03:00.850","Text":"We\u0027ll see, it\u0027s 1 of the 2 surfaces."},{"Start":"03:00.850 ","End":"03:05.390","Text":"It also means that because z equals y,"},{"Start":"03:05.390 ","End":"03:09.529","Text":"that y is bigger or equal to 0."},{"Start":"03:09.529 ","End":"03:11.540","Text":"If y is bigger or equal to 0,"},{"Start":"03:11.540 ","End":"03:13.640","Text":"then we don\u0027t have the whole ellipse,"},{"Start":"03:13.640 ","End":"03:17.270","Text":"we only have the upper half of the ellipse and I\u0027ll label"},{"Start":"03:17.270 ","End":"03:21.665","Text":"this region D. This is our projection."},{"Start":"03:21.665 ","End":"03:25.790","Text":"Now, just to summarize what we said before,"},{"Start":"03:25.790 ","End":"03:27.875","Text":"because z is bigger or equal to 0,"},{"Start":"03:27.875 ","End":"03:29.375","Text":"this is 1 of the surfaces,"},{"Start":"03:29.375 ","End":"03:33.390","Text":"so we actually have 2 surfaces."},{"Start":"03:33.950 ","End":"03:41.915","Text":"We have z equals y as 1 surface and z equals 0 as the other surface,"},{"Start":"03:41.915 ","End":"03:45.725","Text":"and because z is bigger or equal to 0,"},{"Start":"03:45.725 ","End":"03:53.160","Text":"this exactly tells us that this is the upper and this is the lower."},{"Start":"03:54.400 ","End":"03:56.885","Text":"I\u0027m going to put in an extra sketch,"},{"Start":"03:56.885 ","End":"03:59.450","Text":"not that we need it to show you what\u0027s happening."},{"Start":"03:59.450 ","End":"04:01.565","Text":"I\u0027ll take a side view."},{"Start":"04:01.565 ","End":"04:03.170","Text":"For the side view,"},{"Start":"04:03.170 ","End":"04:08.450","Text":"I\u0027m taking z into the picture and I\u0027m looking at it from the side."},{"Start":"04:08.450 ","End":"04:10.160","Text":"This is the y-axis,"},{"Start":"04:10.160 ","End":"04:13.190","Text":"and y goes from minus 2 to 2,"},{"Start":"04:13.190 ","End":"04:15.155","Text":"at least initially it does."},{"Start":"04:15.155 ","End":"04:17.870","Text":"That\u0027s the ellipse sideways,"},{"Start":"04:17.870 ","End":"04:23.285","Text":"but we only want the y bigger or equal to 0 part of the ellipse,"},{"Start":"04:23.285 ","End":"04:27.120","Text":"so this is like the side of the domain."},{"Start":"04:27.270 ","End":"04:32.480","Text":"The line z equals y from the side"},{"Start":"04:32.480 ","End":"04:37.774","Text":"just looks like a 45 degree line through the origin z equals y,"},{"Start":"04:37.774 ","End":"04:42.050","Text":"and it goes up to just the part above the domain,"},{"Start":"04:42.050 ","End":"04:44.420","Text":"which is this half of the ellipse."},{"Start":"04:44.420 ","End":"04:48.095","Text":"The volume we\u0027re talking about is here,"},{"Start":"04:48.095 ","End":"04:52.085","Text":"v and below this part here is D,"},{"Start":"04:52.085 ","End":"04:55.770","Text":"if you throw in the x-axis also."},{"Start":"04:55.770 ","End":"04:59.405","Text":"This just gives you an idea of what\u0027s going on."},{"Start":"04:59.405 ","End":"05:02.630","Text":"The lower plane, z equals 0,"},{"Start":"05:02.630 ","End":"05:05.045","Text":"the upper plane z equals y,"},{"Start":"05:05.045 ","End":"05:10.715","Text":"and the volume between them above the semi ellipse."},{"Start":"05:10.715 ","End":"05:13.820","Text":"I could have omitted all this extra sketch,"},{"Start":"05:13.820 ","End":"05:17.770","Text":"but I just thought it might give you some idea of what\u0027s going on."},{"Start":"05:17.770 ","End":"05:20.220","Text":"If it doesn\u0027t help then just throw it out."},{"Start":"05:20.220 ","End":"05:23.915","Text":"What we have now according to this theorem,"},{"Start":"05:23.915 ","End":"05:30.880","Text":"is that our volume that we want is the double integral over the region D,"},{"Start":"05:30.880 ","End":"05:33.704","Text":"that\u0027s the upper half ellipse,"},{"Start":"05:33.704 ","End":"05:37.410","Text":"of the upper minus the lower."},{"Start":"05:37.410 ","End":"05:43.390","Text":"Upper minus lower is just y and dA."},{"Start":"05:43.390 ","End":"05:47.330","Text":"We have to decide what kind of a region D is going to be,"},{"Start":"05:47.330 ","End":"05:50.845","Text":"if we\u0027re going to slice it horizontally or vertically,"},{"Start":"05:50.845 ","End":"05:53.640","Text":"Type 2 or Type 1."},{"Start":"05:53.640 ","End":"05:55.620","Text":"It doesn\u0027t really matter,"},{"Start":"05:55.620 ","End":"05:56.790","Text":"you could do it both ways."},{"Start":"05:56.790 ","End":"06:00.525","Text":"I\u0027d like to slice it vertically, I want it to be a dy,"},{"Start":"06:00.525 ","End":"06:07.000","Text":"dx, and then this will become the integral."},{"Start":"06:07.000 ","End":"06:10.940","Text":"X will go from minus 1 to 1,"},{"Start":"06:10.940 ","End":"06:14.500","Text":"so that\u0027s part dx."},{"Start":"06:15.020 ","End":"06:18.970","Text":"I\u0027ll even write that x goes from minus 1 to 1."},{"Start":"06:18.970 ","End":"06:26.260","Text":"Then we have the integral y goes from the lower part."},{"Start":"06:26.260 ","End":"06:28.930","Text":"Well, let us put a little sketch here."},{"Start":"06:28.930 ","End":"06:37.125","Text":"We take a typical x and we take the vertical slice through this x."},{"Start":"06:37.125 ","End":"06:43.659","Text":"We go with the y from the lower to the upper,"},{"Start":"06:43.659 ","End":"06:46.120","Text":"where we enter the region where we exited,"},{"Start":"06:46.120 ","End":"06:48.265","Text":"and we have this function."},{"Start":"06:48.265 ","End":"06:51.440","Text":"This function is y equals 0."},{"Start":"06:51.980 ","End":"06:55.235","Text":"If you like I\u0027ll label it also here,"},{"Start":"06:55.235 ","End":"06:58.120","Text":"the x-axis is y equals 0."},{"Start":"06:58.120 ","End":"07:04.630","Text":"But what I\u0027m missing is the equation of the upper semi-ellipse."},{"Start":"07:04.630 ","End":"07:07.570","Text":"All I have to do to get that,"},{"Start":"07:07.570 ","End":"07:14.965","Text":"I just need to extract y from the equation of the ellipse."},{"Start":"07:14.965 ","End":"07:18.630","Text":"Let me do that at the side. Let\u0027s see."},{"Start":"07:18.630 ","End":"07:20.260","Text":"If I just copy that first,"},{"Start":"07:20.260 ","End":"07:24.295","Text":"x squared plus y squared over 4 equals 1."},{"Start":"07:24.295 ","End":"07:28.555","Text":"Multiply both sides by 4 and bring x squared over,"},{"Start":"07:28.555 ","End":"07:39.080","Text":"so I\u0027ve got y squared equals 4 minus 4x squared."},{"Start":"07:39.080 ","End":"07:46.035","Text":"Or perhaps I could have taken the 4 out,"},{"Start":"07:46.035 ","End":"07:48.165","Text":"1 minus x squared,"},{"Start":"07:48.165 ","End":"07:53.425","Text":"and then y would normally be plus or minus."},{"Start":"07:53.425 ","End":"07:56.515","Text":"But because y is bigger or equal to 0,"},{"Start":"07:56.515 ","End":"07:57.790","Text":"I\u0027m just taking the positive,"},{"Start":"07:57.790 ","End":"07:59.980","Text":"but I\u0027ll write the plus just to emphasize."},{"Start":"07:59.980 ","End":"08:02.740","Text":"I know what am doing, I\u0027m taking only the plus."},{"Start":"08:02.740 ","End":"08:09.300","Text":"Square root of 4 is 2 and then square root of 1 minus x"},{"Start":"08:09.300 ","End":"08:16.685","Text":"squared and that is the equation of the upper half of the boundary of the ellipse."},{"Start":"08:16.685 ","End":"08:19.060","Text":"Now we can write it here,"},{"Start":"08:19.060 ","End":"08:22.020","Text":"or maybe for completeness I\u0027ll write it here also,"},{"Start":"08:22.020 ","End":"08:25.290","Text":"2 square root of 1 minus x squared,"},{"Start":"08:25.290 ","End":"08:31.665","Text":"and now here, square root of 1 minus x squared."},{"Start":"08:31.665 ","End":"08:35.205","Text":"That\u0027s dy, y goes from here to here."},{"Start":"08:35.205 ","End":"08:37.040","Text":"Then we just need the function,"},{"Start":"08:37.040 ","End":"08:38.920","Text":"the difference of the upper minus the lower."},{"Start":"08:38.920 ","End":"08:42.445","Text":"We can just copy it from here, that\u0027s just y."},{"Start":"08:42.445 ","End":"08:45.330","Text":"From here on it\u0027s just purely technical,"},{"Start":"08:45.330 ","End":"08:47.405","Text":"no need for any pictures or anything."},{"Start":"08:47.405 ","End":"08:50.335","Text":"As usual we work from the inside out."},{"Start":"08:50.335 ","End":"08:54.280","Text":"Silly me, I forgot the 2 here, didn\u0027t I?"},{"Start":"08:54.470 ","End":"09:00.720","Text":"The inner integral is an integral dy."},{"Start":"09:00.720 ","End":"09:02.910","Text":"Let me do this on the side."},{"Start":"09:02.910 ","End":"09:05.410","Text":"I\u0027ll call it asterisk."},{"Start":"09:08.600 ","End":"09:18.380","Text":"The integral of y is 1/2 y squared and we need to evaluate this below."},{"Start":"09:18.380 ","End":"09:25.090","Text":"We let y equals 0 and above y equals 2 root 1 minus x squared,"},{"Start":"09:25.090 ","End":"09:28.600","Text":"so this equals, plug in the upper limit,"},{"Start":"09:28.600 ","End":"09:31.195","Text":"and we have 1/2,"},{"Start":"09:31.195 ","End":"09:36.825","Text":"y squared will just be this 2 squared is 4,"},{"Start":"09:36.825 ","End":"09:39.740","Text":"the root squared is just the thing itself,"},{"Start":"09:39.740 ","End":"09:41.630","Text":"1 minus x squared,"},{"Start":"09:41.630 ","End":"09:43.160","Text":"that\u0027s the upper limit."},{"Start":"09:43.160 ","End":"09:47.245","Text":"Lower limit, plug in y equals 0 is just 0."},{"Start":"09:47.245 ","End":"09:51.210","Text":"What we are left with is 1/2 times 4 is 2."},{"Start":"09:51.210 ","End":"09:54.255","Text":"2 times 1 minus x squared,"},{"Start":"09:54.255 ","End":"09:55.935","Text":"that\u0027s asterisk."},{"Start":"09:55.935 ","End":"09:58.120","Text":"Back to here."},{"Start":"09:58.120 ","End":"10:02.600","Text":"I now get the integral from minus 1 to 1,"},{"Start":"10:02.600 ","End":"10:04.550","Text":"everything is now dx."},{"Start":"10:04.550 ","End":"10:15.260","Text":"2 I can pull up front of 1 minus x squared dx. Let\u0027s see."},{"Start":"10:19.260 ","End":"10:22.160","Text":"I\u0027ll leave the 2 here."},{"Start":"10:22.160 ","End":"10:24.665","Text":"The integral of 1 is x,"},{"Start":"10:24.665 ","End":"10:32.670","Text":"the integral of x squared is x cubed over 3 or 1/3 x cubed,"},{"Start":"10:32.670 ","End":"10:38.060","Text":"and this has to be taken between minus 1 and 1."},{"Start":"10:38.060 ","End":"10:40.610","Text":"Let\u0027s see what we get."},{"Start":"10:40.610 ","End":"10:42.965","Text":"We get twice."},{"Start":"10:42.965 ","End":"10:51.870","Text":"Plug in 1 and we have 1 minus 1/3."},{"Start":"10:51.870 ","End":"10:54.975","Text":"Less, plug in minus 1,"},{"Start":"10:54.975 ","End":"11:01.110","Text":"and we get minus 1 and it comes out plus 1/3 if you check the signs."},{"Start":"11:01.110 ","End":"11:07.380","Text":"Basically, this is 2/3 minus minus 2/3."},{"Start":"11:07.380 ","End":"11:17.110","Text":"What we have is twice and then 2/3 minus minus 2/3 is just twice 2/3."},{"Start":"11:18.680 ","End":"11:23.115","Text":"I make it 8/3,"},{"Start":"11:23.115 ","End":"11:25.485","Text":"and I\u0027ll highlight that."},{"Start":"11:25.485 ","End":"11:28.960","Text":"That\u0027s the answer and we\u0027re done."}],"ID":8687},{"Watched":false,"Name":"Exercise 2 part f","Duration":"11m 42s","ChapterTopicVideoID":8472,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, you have to compute the volume of the solid"},{"Start":"00:03.720 ","End":"00:06.720","Text":"bounded by the following surfaces,"},{"Start":"00:06.720 ","End":"00:09.640","Text":"let\u0027s say 1, 2, 3, 4 of them."},{"Start":"00:09.920 ","End":"00:13.620","Text":"We\u0027re going to use the standard theorem that we\u0027ve been"},{"Start":"00:13.620 ","End":"00:16.500","Text":"using is that when we have a solid which has"},{"Start":"00:16.500 ","End":"00:19.035","Text":"an upper surface and a lower surface and"},{"Start":"00:19.035 ","End":"00:23.805","Text":"the projection of the whole thing is a region D in the plane,"},{"Start":"00:23.805 ","End":"00:28.650","Text":"then the volume is just given by the double integral over"},{"Start":"00:28.650 ","End":"00:34.710","Text":"the region D of the upper surface minus the lower surface."},{"Start":"00:34.710 ","End":"00:38.445","Text":"Now, usually we compute the boundary"},{"Start":"00:38.445 ","End":"00:42.560","Text":"of D by looking at the equations that don\u0027t contain Z,"},{"Start":"00:42.560 ","End":"00:45.800","Text":"but sometimes this is not enough and that\u0027s what happens here,"},{"Start":"00:45.800 ","End":"00:51.530","Text":"and then we also have to intersect the 2 surfaces."},{"Start":"00:51.530 ","End":"00:53.420","Text":"Here there\u0027s an upper and a lower,"},{"Start":"00:53.420 ","End":"00:55.700","Text":"but sometimes they actually intersect."},{"Start":"00:55.700 ","End":"01:00.065","Text":"You could imagine continuing the lines and having them intersect."},{"Start":"01:00.065 ","End":"01:03.335","Text":"Let me start with computing the intersection."},{"Start":"01:03.335 ","End":"01:05.030","Text":"I have 2 surfaces."},{"Start":"01:05.030 ","End":"01:08.815","Text":"I have z equals 6,"},{"Start":"01:08.815 ","End":"01:11.740","Text":"1 of them, and the other 1,"},{"Start":"01:11.740 ","End":"01:14.540","Text":"z equals x plus"},{"Start":"01:14.540 ","End":"01:16.760","Text":"y. I don\u0027t know which is the"},{"Start":"01:16.760 ","End":"01:20.190","Text":"upper and which is the lower that\u0027s what I\u0027m going to discover."},{"Start":"01:28.430 ","End":"01:32.940","Text":"To solve where these 2 meet,"},{"Start":"01:32.940 ","End":"01:36.360","Text":"we just equate the right-hand sides,"},{"Start":"01:36.360 ","End":"01:39.065","Text":"so we get an equation in x and y."},{"Start":"01:39.065 ","End":"01:44.664","Text":"Basically, we get x plus y equals 6"},{"Start":"01:44.664 ","End":"01:51.440","Text":"and this will be where they intersect or at least the projection onto the x y plane."},{"Start":"01:51.440 ","End":"01:54.500","Text":"x plus y equals 6 is just a straight line."},{"Start":"01:54.500 ","End":"01:57.755","Text":"When x is 0, y is"},{"Start":"01:57.755 ","End":"02:06.035","Text":"6 and when y is 0,"},{"Start":"02:06.035 ","End":"02:10.480","Text":"x is 6 and so here\u0027s the straight line,"},{"Start":"02:10.480 ","End":"02:13.675","Text":"x plus y equals 6."},{"Start":"02:13.675 ","End":"02:16.030","Text":"It\u0027s not actually the intersection,"},{"Start":"02:16.030 ","End":"02:17.470","Text":"it\u0027s just the projection."},{"Start":"02:17.470 ","End":"02:20.860","Text":"It\u0027s just the xy and actually intersect somewhere in"},{"Start":"02:20.860 ","End":"02:25.420","Text":"mid-air where at the height 6 above this but doesn\u0027t matter,"},{"Start":"02:25.420 ","End":"02:29.545","Text":"we just want the projection onto the xy plane."},{"Start":"02:29.545 ","End":"02:36.110","Text":"Now the other 2, y equals 0 is the x-axis."},{"Start":"02:38.400 ","End":"02:41.805","Text":"Let\u0027s try and highlight it here."},{"Start":"02:41.805 ","End":"02:45.540","Text":"That\u0027s where y equals 0,"},{"Start":"02:45.540 ","End":"02:49.190","Text":"the x-axis and then we have x equals 0,"},{"Start":"02:49.190 ","End":"02:52.280","Text":"which is going to be this bit here,"},{"Start":"02:52.280 ","End":"02:58.085","Text":"which is the y-axis and I\u0027ll also label that, x equals 0."},{"Start":"02:58.085 ","End":"03:04.185","Text":"Now we really do have a triangle in the projection onto the xy plane."},{"Start":"03:04.185 ","End":"03:08.000","Text":"We have our D and here it is,"},{"Start":"03:08.000 ","End":"03:16.220","Text":"and let me label it D. Now we have to decide which is the upper and which is the lower."},{"Start":"03:16.220 ","End":"03:19.820","Text":"Now whenever you intersect 2 surfaces, z equals something,"},{"Start":"03:19.820 ","End":"03:23.110","Text":"z equals something and you get the intersection,"},{"Start":"03:23.110 ","End":"03:26.445","Text":"always on 1 side of the line, it could be a curve,"},{"Start":"03:26.445 ","End":"03:29.540","Text":"on 1 side, 1 will be bigger and on the other side,"},{"Start":"03:29.540 ","End":"03:33.050","Text":"the other will be bigger because the only place"},{"Start":"03:33.050 ","End":"03:37.090","Text":"they\u0027re equal to is here and because of continuity on 1 side,"},{"Start":"03:37.090 ","End":"03:39.920","Text":"1 will be bigger and the other side the other will be bigger."},{"Start":"03:39.920 ","End":"03:42.050","Text":"Now we know which side of the line we want."},{"Start":"03:42.050 ","End":"03:44.705","Text":"We want the side of the line where our D is on."},{"Start":"03:44.705 ","End":"03:51.260","Text":"We\u0027ll pick any point on the lower left side of the line or in our region,"},{"Start":"03:51.260 ","End":"03:53.349","Text":"I\u0027ll go for the origin."},{"Start":"03:53.349 ","End":"03:57.590","Text":"If I substitute the origin, which is x equals 0,"},{"Start":"03:57.590 ","End":"04:01.370","Text":"y equals 0, this is always 6."},{"Start":"04:01.370 ","End":"04:04.880","Text":"This will come out to be 0 plus 0."},{"Start":"04:04.880 ","End":"04:10.130","Text":"This will be the upper, I\u0027ll write that;"},{"Start":"04:10.130 ","End":"04:17.260","Text":"upper, and this 1 will be the lower."},{"Start":"04:17.440 ","End":"04:23.555","Text":"I mean, anywhere on the lower left side of this line if you substitute,"},{"Start":"04:23.555 ","End":"04:27.575","Text":"you will get that this is bigger and on the other side,"},{"Start":"04:27.575 ","End":"04:29.030","Text":"this 1 will be the bigger 1,"},{"Start":"04:29.030 ","End":"04:31.715","Text":"but our D is on this side of the line."},{"Start":"04:31.715 ","End":"04:35.125","Text":"So any point here,"},{"Start":"04:35.125 ","End":"04:37.050","Text":"this will be higher than this."},{"Start":"04:37.050 ","End":"04:38.685","Text":"Now we have all we need,"},{"Start":"04:38.685 ","End":"04:41.700","Text":"almost, to compute this."},{"Start":"04:41.700 ","End":"04:45.725","Text":"What we want is according to this formula,"},{"Start":"04:45.725 ","End":"04:48.650","Text":"the double integral over D,"},{"Start":"04:48.650 ","End":"04:50.765","Text":"that\u0027s what our volume will equal,"},{"Start":"04:50.765 ","End":"04:54.780","Text":"of upper minus lower."},{"Start":"04:54.780 ","End":"05:00.170","Text":"Upper minus lower will be 6 minus x minus y."},{"Start":"05:00.170 ","End":"05:04.040","Text":"Just to practice minus this and for the moment it\u0027s DA,"},{"Start":"05:04.040 ","End":"05:07.075","Text":"we haven\u0027t decided yet which way to slice it."},{"Start":"05:07.075 ","End":"05:13.730","Text":"Both ways are equally good and I think we\u0027ll go for a vertical slicing,"},{"Start":"05:13.730 ","End":"05:18.515","Text":"which will give us dydx, y extracted in terms of x."},{"Start":"05:18.515 ","End":"05:20.390","Text":"The limit on x, of course,"},{"Start":"05:20.390 ","End":"05:22.490","Text":"will be from 0-6."},{"Start":"05:22.490 ","End":"05:24.940","Text":"Let\u0027s really start."},{"Start":"05:24.940 ","End":"05:30.470","Text":"I can say that this is going to be the integral from x equals"},{"Start":"05:30.470 ","End":"05:36.335","Text":"0 to x equals 6."},{"Start":"05:36.335 ","End":"05:39.724","Text":"I don\u0027t usually write the x twice."},{"Start":"05:39.724 ","End":"05:43.760","Text":"This will do and that will be dx,"},{"Start":"05:43.760 ","End":"05:46.715","Text":"that\u0027s the outer integral and for each x,"},{"Start":"05:46.715 ","End":"05:50.930","Text":"well, let\u0027s say this is a typical x here."},{"Start":"05:50.930 ","End":"05:57.845","Text":"What we have to do is figure the limits on y by doing a vertical slice."},{"Start":"05:57.845 ","End":"06:04.095","Text":"Here is my vertical slice and for this vertical slice,"},{"Start":"06:04.095 ","End":"06:09.740","Text":"y will travel from this point to this point."},{"Start":"06:09.740 ","End":"06:13.340","Text":"Now this point we already have the equation for y equals 0,"},{"Start":"06:13.340 ","End":"06:18.665","Text":"we\u0027re just missing the equation of this and explicit form y in terms of x."},{"Start":"06:18.665 ","End":"06:22.550","Text":"Well, it\u0027s easy to see that this will be y equals,"},{"Start":"06:22.550 ","End":"06:29.930","Text":"I\u0027ll just extract y from here so I get y equals 6 minus x,"},{"Start":"06:29.930 ","End":"06:32.135","Text":"and now I don\u0027t need this 1."},{"Start":"06:32.135 ","End":"06:34.670","Text":"Writing this mathematically over here,"},{"Start":"06:34.670 ","End":"06:42.750","Text":"the inner integral is when y goes for this given x, which travels from 0-6,"},{"Start":"06:42.750 ","End":"06:50.460","Text":"the y goes from 0-6 minus x dy,"},{"Start":"06:50.460 ","End":"06:53.450","Text":"and then I just have this function to copy,"},{"Start":"06:53.450 ","End":"06:57.475","Text":"6 minus x minus y."},{"Start":"06:57.475 ","End":"07:00.200","Text":"At this point it\u0027s totally technical."},{"Start":"07:00.200 ","End":"07:02.660","Text":"We don\u0027t need any of the pictures or anything,"},{"Start":"07:02.660 ","End":"07:04.370","Text":"and we just get to work."},{"Start":"07:04.370 ","End":"07:08.020","Text":"We always do these things from the inside out."},{"Start":"07:08.020 ","End":"07:16.580","Text":"The inner integral is the 1 that\u0027s dy and I like to do this separately at the side."},{"Start":"07:16.580 ","End":"07:21.950","Text":"Let me just call it say asterisk and I\u0027ll do the asterisk over here."},{"Start":"07:21.950 ","End":"07:25.255","Text":"What I have is the integral."},{"Start":"07:25.255 ","End":"07:32.610","Text":"I\u0027ll just copy it from 0-6 minus x of 6 minus x minus"},{"Start":"07:32.610 ","End":"07:40.700","Text":"y dy and then this is equal to the integral of 6 is 6y,"},{"Start":"07:40.700 ","End":"07:43.085","Text":"the integral of x is xy."},{"Start":"07:43.085 ","End":"07:44.915","Text":"Remember that x is a constant."},{"Start":"07:44.915 ","End":"07:49.720","Text":"The integral of y is minus a half y squared."},{"Start":"07:49.720 ","End":"07:57.155","Text":"All this is taken between y equals 0 and y equals 6 minus x."},{"Start":"07:57.155 ","End":"08:00.095","Text":"Let\u0027s substitute the upper 1 first."},{"Start":"08:00.095 ","End":"08:05.119","Text":"We get 6 times 6 minus x"},{"Start":"08:05.119 ","End":"08:11.850","Text":"minus x times y is 6 minus x minus 1/2."},{"Start":"08:11.850 ","End":"08:15.165","Text":"Y squared is 6 minus x squared."},{"Start":"08:15.165 ","End":"08:18.410","Text":"If I plug in y equals 0 everywhere here,"},{"Start":"08:18.410 ","End":"08:20.780","Text":"I\u0027ll get 0 for everything."},{"Start":"08:20.780 ","End":"08:26.580","Text":"I\u0027ll just write minus 0 to say I\u0027ve forgotten to plug in the lower limit."},{"Start":"08:26.580 ","End":"08:30.070","Text":"Let\u0027s see what this comes out to."},{"Start":"08:35.360 ","End":"08:41.720","Text":"I won\u0027t even write the minus 0 because I might need the space here."},{"Start":"08:41.720 ","End":"08:48.690","Text":"6 times 6 is 36 minus 6x minus x times"},{"Start":"08:48.690 ","End":"08:56.215","Text":"6 is 6x plus x squared and I\u0027m going to use the formula."},{"Start":"08:56.215 ","End":"09:05.665","Text":"I\u0027ll put it at the side that a minus b squared is a squared minus 2ab plus b squared."},{"Start":"09:05.665 ","End":"09:10.805","Text":"Here we get minus 1/2 of"},{"Start":"09:10.805 ","End":"09:17.400","Text":"36 minus 12x plus x squared."},{"Start":"09:17.400 ","End":"09:22.365","Text":"Let\u0027s see if I simplify this what have we got."},{"Start":"09:22.365 ","End":"09:24.600","Text":"I\u0027ll take the constants first,"},{"Start":"09:24.600 ","End":"09:28.740","Text":"I have 36 and I have minus 18."},{"Start":"09:28.740 ","End":"09:30.750","Text":"That is 18."},{"Start":"09:30.750 ","End":"09:34.065","Text":"Now xs, minus 6x minus 6x,"},{"Start":"09:34.065 ","End":"09:41.460","Text":"that\u0027s minus 12x, and minus 1/2 times minus 12 is plus 6."},{"Start":"09:41.460 ","End":"09:46.800","Text":"I end up with minus 6x and then for x squared,"},{"Start":"09:46.800 ","End":"09:51.555","Text":"I have x squared minus 1/2x squared,"},{"Start":"09:51.555 ","End":"09:56.200","Text":"so that\u0027s 1/2x squared."},{"Start":"09:56.200 ","End":"09:59.819","Text":"Now I\u0027m going to go back here."},{"Start":"09:59.819 ","End":"10:02.700","Text":"This was the asterisks."},{"Start":"10:02.700 ","End":"10:05.310","Text":"When I substitute it back here,"},{"Start":"10:05.310 ","End":"10:14.265","Text":"then what we will get will be the integral from 0-6 of this."},{"Start":"10:14.265 ","End":"10:15.810","Text":"I\u0027ll change the order,"},{"Start":"10:15.810 ","End":"10:17.310","Text":"I don\u0027t know why."},{"Start":"10:17.310 ","End":"10:20.940","Text":"1/2x squared minus 6x plus"},{"Start":"10:20.940 ","End":"10:29.860","Text":"18 dx and then this will equal,"},{"Start":"10:30.170 ","End":"10:34.425","Text":"let\u0027s say you raise the power by 1x cubed divided by 3."},{"Start":"10:34.425 ","End":"10:39.945","Text":"So 1/6x cubed, x squared over 2."},{"Start":"10:39.945 ","End":"10:50.290","Text":"It\u0027s minus 3x squared plus 18x taken between 0 and 6."},{"Start":"10:50.450 ","End":"10:53.900","Text":"I see already that when I put in x equals 0,"},{"Start":"10:53.900 ","End":"10:56.975","Text":"I get nothing, so I only have to substitute the 6."},{"Start":"10:56.975 ","End":"11:07.815","Text":"I get 1/6 times 6 cubed minus 3 times 6 squared plus 18 times 6."},{"Start":"11:07.815 ","End":"11:09.495","Text":"Let\u0027s see."},{"Start":"11:09.495 ","End":"11:12.480","Text":"One of the 6 is canceled so it\u0027s just 6 squared."},{"Start":"11:12.480 ","End":"11:16.515","Text":"This is 36 minus,"},{"Start":"11:16.515 ","End":"11:20.085","Text":"this is 3 times 36,"},{"Start":"11:20.085 ","End":"11:27.570","Text":"which is 108, and this is 18 times 6,"},{"Start":"11:27.570 ","End":"11:32.640","Text":"which is also 108."},{"Start":"11:32.640 ","End":"11:34.320","Text":"These 2 cancel."},{"Start":"11:34.320 ","End":"11:40.190","Text":"The final answer comes out to be 36 and this is our volume."},{"Start":"11:40.190 ","End":"11:43.440","Text":"That\u0027s the final answer and we are done."}],"ID":8688},{"Watched":false,"Name":"Exercise 3 part 1","Duration":"8m 38s","ChapterTopicVideoID":8473,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.315","Text":"This is an exercise from physics,"},{"Start":"00:03.315 ","End":"00:06.270","Text":"but you don\u0027t need to know too much physics for it."},{"Start":"00:06.270 ","End":"00:10.050","Text":"We\u0027re given a flat triangular board."},{"Start":"00:10.050 ","End":"00:13.305","Text":"We say flat, it\u0027s as if it was 2D."},{"Start":"00:13.305 ","End":"00:15.225","Text":"It\u0027s in the plane."},{"Start":"00:15.225 ","End":"00:17.760","Text":"It\u0027s a triangle, 1 at the origin,"},{"Start":"00:17.760 ","End":"00:21.465","Text":"1 vertex at 1, 0, and 0, 1."},{"Start":"00:21.465 ","End":"00:26.475","Text":"Now, we\u0027re given its density function is given by delta,"},{"Start":"00:26.475 ","End":"00:29.250","Text":"Greek letter delta for density of x,"},{"Start":"00:29.250 ","End":"00:30.775","Text":"y is x times y."},{"Start":"00:30.775 ","End":"00:33.070","Text":"We have to compute the mass of the board."},{"Start":"00:33.070 ","End":"00:35.660","Text":"I\u0027m going to pull out a formula in a minute that"},{"Start":"00:35.660 ","End":"00:38.240","Text":"tells us how to compute the mass and the density function."},{"Start":"00:38.240 ","End":"00:45.815","Text":"But before that, I\u0027d like to just sketch this board in the plane."},{"Start":"00:45.815 ","End":"00:49.040","Text":"Here we have an x, y plane."},{"Start":"00:49.040 ","End":"00:51.905","Text":"Let\u0027s say this is the origin."},{"Start":"00:51.905 ","End":"00:55.685","Text":"1, 0 might be here,"},{"Start":"00:55.685 ","End":"00:58.885","Text":"and 0, 1 might be here."},{"Start":"00:58.885 ","End":"01:00.710","Text":"I\u0027ll just mark this is 1,"},{"Start":"01:00.710 ","End":"01:02.960","Text":"this is 1, this is the origin."},{"Start":"01:02.960 ","End":"01:09.365","Text":"Now, I\u0027ll draw the straight line through this to complete the triangle."},{"Start":"01:09.365 ","End":"01:14.105","Text":"Let me shade this triangle. Here we are."},{"Start":"01:14.105 ","End":"01:15.620","Text":"Even let\u0027s give it a name,"},{"Start":"01:15.620 ","End":"01:19.430","Text":"a letter to let\u0027s call it D for domain or region,"},{"Start":"01:19.430 ","End":"01:23.040","Text":"is letter D. Now,"},{"Start":"01:23.040 ","End":"01:26.540","Text":"it\u0027s time for me to give you that formula from physics,"},{"Start":"01:26.540 ","End":"01:36.525","Text":"which basically says that if we have a region D,"},{"Start":"01:36.525 ","End":"01:40.520","Text":"the region D is the shape of the board."},{"Start":"01:40.520 ","End":"01:44.630","Text":"The board, when you place it on the x, y plane,"},{"Start":"01:44.630 ","End":"01:49.570","Text":"that gives a region D. We have a density function,"},{"Start":"01:49.570 ","End":"01:53.925","Text":"delta of x, y in general."},{"Start":"01:53.925 ","End":"01:59.960","Text":"We also assume that this density function is continuous and that\u0027s important."},{"Start":"01:59.960 ","End":"02:03.970","Text":"I\u0027ll write that it has to be continuous."},{"Start":"02:03.970 ","End":"02:08.715","Text":"In our case, x times y is certainly a continuous function."},{"Start":"02:08.715 ","End":"02:11.445","Text":"If all this holds true,"},{"Start":"02:11.445 ","End":"02:19.130","Text":"then the mass of the board is equal to the"},{"Start":"02:19.130 ","End":"02:28.330","Text":"double integral over the region D of the density function dA."},{"Start":"02:28.330 ","End":"02:31.470","Text":"Now, this isn\u0027t a symmetrical form."},{"Start":"02:31.470 ","End":"02:36.470","Text":"We have to decide whether we want to go with vertical slices or horizontal slices,"},{"Start":"02:36.470 ","End":"02:38.785","Text":"type 1 or type 2,"},{"Start":"02:38.785 ","End":"02:41.145","Text":"dy, dx or dx, dy."},{"Start":"02:41.145 ","End":"02:43.325","Text":"It doesn\u0027t really matter."},{"Start":"02:43.325 ","End":"02:49.860","Text":"But I would like to write the equations of these lines."},{"Start":"02:50.020 ","End":"02:52.520","Text":"Well, these 2 are clear."},{"Start":"02:52.520 ","End":"02:55.010","Text":"What I want is the equation of this line."},{"Start":"02:55.010 ","End":"02:57.440","Text":"Whenever you have the same number here and here,"},{"Start":"02:57.440 ","End":"02:59.465","Text":"if this is a and this is a,"},{"Start":"02:59.465 ","End":"03:02.195","Text":"then the equation is x plus y equals a."},{"Start":"03:02.195 ","End":"03:04.550","Text":"In this case, x plus y equals 1."},{"Start":"03:04.550 ","End":"03:06.140","Text":"You can check 1,"},{"Start":"03:06.140 ","End":"03:08.555","Text":"0 satisfies x plus y equals 1,"},{"Start":"03:08.555 ","End":"03:11.000","Text":"and 0, 1 satisfies it."},{"Start":"03:11.000 ","End":"03:12.740","Text":"We have that, of course."},{"Start":"03:12.740 ","End":"03:14.665","Text":"We have this equation,"},{"Start":"03:14.665 ","End":"03:17.690","Text":"the x-axis is given by y equals 0."},{"Start":"03:17.690 ","End":"03:19.880","Text":"I actually don\u0027t care about the third side"},{"Start":"03:19.880 ","End":"03:22.309","Text":"because I\u0027m going to be taking vertical slices."},{"Start":"03:22.309 ","End":"03:24.695","Text":"It would work just as well with horizontal."},{"Start":"03:24.695 ","End":"03:29.100","Text":"Let\u0027s do this as a type 1 region."},{"Start":"03:29.100 ","End":"03:34.640","Text":"We get that this mass is equal to the integral."},{"Start":"03:34.640 ","End":"03:38.390","Text":"Now, we\u0027re going to take x from 0 to 1."},{"Start":"03:38.390 ","End":"03:40.159","Text":"That\u0027s clear from the picture,"},{"Start":"03:40.159 ","End":"03:42.300","Text":"from 0 to 1."},{"Start":"03:42.300 ","End":"03:45.505","Text":"That\u0027s going to be dx."},{"Start":"03:45.505 ","End":"03:51.270","Text":"For each x, let\u0027s say we take a typical x here."},{"Start":"03:51.270 ","End":"03:53.755","Text":"That 1 is our x."},{"Start":"03:53.755 ","End":"03:59.540","Text":"Then we take a vertical slice through the region."},{"Start":"03:59.540 ","End":"04:04.120","Text":"For this x, our y goes from here to here,"},{"Start":"04:04.120 ","End":"04:06.880","Text":"is where it enters, where it leaves."},{"Start":"04:07.790 ","End":"04:18.710","Text":"I can write that y goes from 0 to dy."},{"Start":"04:18.710 ","End":"04:24.980","Text":"What we\u0027re missing is we want y in terms of x for the upper line."},{"Start":"04:24.980 ","End":"04:34.610","Text":"I\u0027ll just rewrite this as y equals 1 minus x, same thing."},{"Start":"04:34.610 ","End":"04:41.710","Text":"Then I can say exactly that this upper limit is 1 minus x."},{"Start":"04:41.710 ","End":"04:51.175","Text":"Then all I need is the function in our particular case, xy dy dx."},{"Start":"04:51.175 ","End":"04:52.365","Text":"This is going to be the mass."},{"Start":"04:52.365 ","End":"04:54.185","Text":"Now, all we have to do is compute it."},{"Start":"04:54.185 ","End":"04:57.020","Text":"As usual, we go from inside out."},{"Start":"04:57.020 ","End":"05:04.460","Text":"The inside is the integral dy, this bit here."},{"Start":"05:04.840 ","End":"05:09.450","Text":"I\u0027ll do this bit at the side and then I\u0027ll plug it back in."},{"Start":"05:09.450 ","End":"05:14.510","Text":"What we want is the integral."},{"Start":"05:14.900 ","End":"05:17.160","Text":"I will write it again first,"},{"Start":"05:17.160 ","End":"05:24.385","Text":"integral from 0 to 1 minus x of xy, dy."},{"Start":"05:24.385 ","End":"05:27.770","Text":"In fact, since x is a constant,"},{"Start":"05:27.770 ","End":"05:30.759","Text":"I can pull the x in front."},{"Start":"05:30.759 ","End":"05:33.560","Text":"I rewrote it, just pull the x out."},{"Start":"05:33.560 ","End":"05:35.945","Text":"It\u0027s a constant as far as y goes."},{"Start":"05:35.945 ","End":"05:43.810","Text":"This is equal to x times the integral of y is just 1/2 y squared."},{"Start":"05:43.810 ","End":"05:49.195","Text":"I need to evaluate this from 0 to 1 minus x."},{"Start":"05:49.195 ","End":"05:51.569","Text":"Clearly, when y is 0,"},{"Start":"05:51.569 ","End":"05:53.280","Text":"we just get 0."},{"Start":"05:53.280 ","End":"05:54.350","Text":"There\u0027s nothing to subtract to."},{"Start":"05:54.350 ","End":"05:57.455","Text":"I just have to substitute 1 minus x."},{"Start":"05:57.455 ","End":"05:59.390","Text":"Let me write it in the following order."},{"Start":"05:59.390 ","End":"06:02.729","Text":"It\u0027s 1/2 times the x,"},{"Start":"06:02.729 ","End":"06:05.220","Text":"and then y equals 1 minus x,"},{"Start":"06:05.220 ","End":"06:07.785","Text":"so this thing squared."},{"Start":"06:07.785 ","End":"06:15.105","Text":"Let\u0027s just simplify it or multiply out, we get 1/2x."},{"Start":"06:15.105 ","End":"06:21.400","Text":"This thing from the formula is 1 minus 2x plus x squared."},{"Start":"06:22.700 ","End":"06:31.470","Text":"Now, we could just multiply 1/2x minus 1/2x times 2x is x squared,"},{"Start":"06:31.470 ","End":"06:35.350","Text":"and here plus 1/2x cubed."},{"Start":"06:36.500 ","End":"06:43.410","Text":"At this point, I\u0027m going to now substitute back up here."},{"Start":"06:43.690 ","End":"06:53.600","Text":"We get the integral from 0 to 1 and it\u0027s just with respect to x now of what we have here,"},{"Start":"06:53.600 ","End":"07:02.210","Text":"1/2 x minus x squared plus 1/2x cubed dx."},{"Start":"07:02.210 ","End":"07:11.130","Text":"This then is equal to 1/2x."},{"Start":"07:11.130 ","End":"07:15.225","Text":"I raise the power by 1 and divide by 2,"},{"Start":"07:15.225 ","End":"07:18.805","Text":"I get 1/4x squared."},{"Start":"07:18.805 ","End":"07:25.775","Text":"Here I raised by 1 is 3 divided by 3, 1/3x cubed."},{"Start":"07:25.775 ","End":"07:34.790","Text":"Here raise it, that\u0027s x^4 divide by 4, so 1/8x^4."},{"Start":"07:34.790 ","End":"07:40.380","Text":"All this between 0 and 1."},{"Start":"07:40.610 ","End":"07:42.990","Text":"Now, just have to substitute."},{"Start":"07:42.990 ","End":"07:45.020","Text":"Once again, if I plug in x equals 0,"},{"Start":"07:45.020 ","End":"07:46.760","Text":"I\u0027m not going to get anything."},{"Start":"07:46.760 ","End":"07:49.205","Text":"I just need to substitute the 1,"},{"Start":"07:49.205 ","End":"07:56.580","Text":"so I get 1/4 minus 1/3 plus an 1/8."},{"Start":"07:56.820 ","End":"08:03.205","Text":"Exercise in fractions, common denominator looks like 24 will do it."},{"Start":"08:03.205 ","End":"08:06.415","Text":"4 into 24, 6 times."},{"Start":"08:06.415 ","End":"08:09.280","Text":"3 into 24, 8 times."},{"Start":"08:09.280 ","End":"08:13.320","Text":"8 into 24, 3 times."},{"Start":"08:13.320 ","End":"08:19.605","Text":"6 plus 3 minus 8 is 9 minus 8 is 1."},{"Start":"08:19.605 ","End":"08:23.310","Text":"I make it 1 over 24."},{"Start":"08:23.310 ","End":"08:25.185","Text":"I\u0027ll highlight it."},{"Start":"08:25.185 ","End":"08:27.865","Text":"Normally, in physics you would say the units of mass,"},{"Start":"08:27.865 ","End":"08:30.820","Text":"but here we weren\u0027t told if was kilograms,"},{"Start":"08:30.820 ","End":"08:32.545","Text":"or pounds, or whatever."},{"Start":"08:32.545 ","End":"08:34.045","Text":"We just leave it abstract."},{"Start":"08:34.045 ","End":"08:38.470","Text":"1 over 24 is the mass of the board. We\u0027re done."}],"ID":8689},{"Watched":false,"Name":"Exercise 3 part 2","Duration":"12m 35s","ChapterTopicVideoID":8474,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:06.870","Text":"This exercise is actually a continuation of part 2 of the previous exercise,"},{"Start":"00:06.870 ","End":"00:11.115","Text":"so I won\u0027t repeat everything."},{"Start":"00:11.115 ","End":"00:15.180","Text":"I\u0027ll just say that the previous exercise asked us"},{"Start":"00:15.180 ","End":"00:20.020","Text":"to find the mass of the board with this description."},{"Start":"00:20.480 ","End":"00:23.760","Text":"This time we\u0027re not asked to find the mass,"},{"Start":"00:23.760 ","End":"00:26.250","Text":"but the center of mass,"},{"Start":"00:26.250 ","End":"00:34.520","Text":"which is a concept in physics and just going to use the equations."},{"Start":"00:34.520 ","End":"00:38.705","Text":"You don\u0027t have to really understand the concepts which is going to use the formulas."},{"Start":"00:38.705 ","End":"00:43.900","Text":"First, let me copy part of the previous exercise."},{"Start":"00:43.900 ","End":"00:48.500","Text":"Here\u0027s the stuff I copied from the previous part"},{"Start":"00:48.500 ","End":"00:52.670","Text":"1 of the exercise where we found the mass and it turned out that the answer,"},{"Start":"00:52.670 ","End":"00:54.715","Text":"if we call the mass M,"},{"Start":"00:54.715 ","End":"01:02.255","Text":"turned out to be 1 over 24 and this was the formula we used."},{"Start":"01:02.255 ","End":"01:06.560","Text":"Now we\u0027re going to give the formula for center of mass."},{"Start":"01:06.560 ","End":"01:09.965","Text":"The center of mass is a point,"},{"Start":"01:09.965 ","End":"01:12.110","Text":"usually inside the region,"},{"Start":"01:12.110 ","End":"01:18.750","Text":"but not necessarily, let\u0027s call it some special x,"},{"Start":"01:18.750 ","End":"01:24.150","Text":"y, let\u0027s put it as x with a bar over it and y bar."},{"Start":"01:24.150 ","End":"01:25.875","Text":"Then we have 2 formulas,"},{"Start":"01:25.875 ","End":"01:31.740","Text":"1 for the x of the center of mass and 1 for y of"},{"Start":"01:31.740 ","End":"01:39.910","Text":"the center of mass and in each case it starts out with 1 over m,"},{"Start":"01:40.190 ","End":"01:44.660","Text":"where m is what we found in the previous exercise in our case."},{"Start":"01:44.660 ","End":"01:49.105","Text":"In each case we have a double integral"},{"Start":"01:49.105 ","End":"01:57.195","Text":"over D and the difference is that here,"},{"Start":"01:57.195 ","End":"02:03.660","Text":"it\u0027s x times the density function dA."},{"Start":"02:03.660 ","End":"02:07.980","Text":"The mass itself was just 1 times the density function."},{"Start":"02:07.980 ","End":"02:14.485","Text":"This is x and this is y times the density function dA."},{"Start":"02:14.485 ","End":"02:19.720","Text":"We have to compute these 2 double integrals over D and,"},{"Start":"02:19.720 ","End":"02:24.695","Text":"just like we did in the previous part,"},{"Start":"02:24.695 ","End":"02:35.310","Text":"we\u0027ll do vertical slices and make it into an integral that is dy dx or a type 1 region."},{"Start":"02:35.310 ","End":"02:40.505","Text":"Here I copied the integral we had in the previous section,"},{"Start":"02:40.505 ","End":"02:46.770","Text":"where this xy is just the Delta of x, y."},{"Start":"02:46.770 ","End":"02:50.960","Text":"We\u0027ll get some very similar thing here."},{"Start":"02:50.960 ","End":"02:56.515","Text":"Just that we\u0027ll get an x and a y added on."},{"Start":"02:56.515 ","End":"03:01.340","Text":"Let\u0027s start with the x coordinate of the center of mass."},{"Start":"03:01.340 ","End":"03:06.530","Text":"So we get that this x is equal to,"},{"Start":"03:06.530 ","End":"03:13.890","Text":"now if m is 1 over 24 then 1 over m is 24 because 1 over,"},{"Start":"03:13.890 ","End":"03:15.480","Text":"1 over and all that,"},{"Start":"03:15.480 ","End":"03:17.965","Text":"and then we\u0027ll get the double integral."},{"Start":"03:17.965 ","End":"03:20.345","Text":"It\u0027ll be the same as here."},{"Start":"03:20.345 ","End":"03:25.725","Text":"I\u0027m just copying x goes from 0-1,"},{"Start":"03:25.725 ","End":"03:31.585","Text":"and y goes from 0-1 minus x."},{"Start":"03:31.585 ","End":"03:34.420","Text":"That\u0027s basically that x is going from 0-1 and y is"},{"Start":"03:34.420 ","End":"03:38.140","Text":"going from this lower line to the upper line."},{"Start":"03:38.140 ","End":"03:39.850","Text":"These are the formulas."},{"Start":"03:39.850 ","End":"03:45.270","Text":"Then here instead of xy, we choose Delta."},{"Start":"03:45.270 ","End":"03:48.435","Text":"I need an extra x so it\u0027ll be x times xy,"},{"Start":"03:48.435 ","End":"03:58.020","Text":"so it\u0027ll be x squared y dy, dx."},{"Start":"03:58.020 ","End":"04:03.020","Text":"Then later similarly we\u0027ll have xy squared."},{"Start":"04:03.020 ","End":"04:04.535","Text":"Well, okay, I\u0027m jumping the gun."},{"Start":"04:04.535 ","End":"04:07.795","Text":"Let\u0027s just stick to the x part first."},{"Start":"04:07.795 ","End":"04:11.120","Text":"What I want to do is when we have a double integral,"},{"Start":"04:11.120 ","End":"04:13.745","Text":"we do it from the inside out,"},{"Start":"04:13.745 ","End":"04:17.105","Text":"so first we\u0027re going to do this bit,"},{"Start":"04:17.105 ","End":"04:22.579","Text":"the dy bit and I\u0027ll do that at the side,"},{"Start":"04:22.579 ","End":"04:25.715","Text":"so this part I\u0027ll do over here."},{"Start":"04:25.715 ","End":"04:31.235","Text":"Well, first of all, let me take the x squared outside so I\u0027ve got x squared"},{"Start":"04:31.235 ","End":"04:38.575","Text":"times the integral from 0-1 minus x of y,"},{"Start":"04:38.575 ","End":"04:48.960","Text":"dy and this is equal to x squared and the integral of this is just"},{"Start":"04:48.960 ","End":"04:55.240","Text":"1/2y squared from 0-1 minus x"},{"Start":"04:55.240 ","End":"05:03.765","Text":"and so what I get is 1/2x squared,"},{"Start":"05:03.765 ","End":"05:07.925","Text":"y squared, when I put in 0,"},{"Start":"05:07.925 ","End":"05:11.690","Text":"is not going to give me anything so I just need to put in the 1 minus x"},{"Start":"05:11.690 ","End":"05:16.100","Text":"so here it\u0027s 1 minus x squared minus 0,"},{"Start":"05:16.100 ","End":"05:17.930","Text":"which I\u0027m not going to write."},{"Start":"05:17.930 ","End":"05:20.605","Text":"If I expand it out,"},{"Start":"05:20.605 ","End":"05:27.560","Text":"this will be 1 minus x squared is 1 minus 2x plus x squared."},{"Start":"05:27.560 ","End":"05:30.530","Text":"If I multiply it by 1/2 x squared,"},{"Start":"05:30.530 ","End":"05:34.910","Text":"let\u0027s see we\u0027ll get in the end 1 will give me 1/2x squared,"},{"Start":"05:34.910 ","End":"05:39.765","Text":"the minus 2x with this will give me"},{"Start":"05:39.765 ","End":"05:45.660","Text":"minus x cubed and the plus x"},{"Start":"05:45.660 ","End":"05:52.210","Text":"squared will give me 1/2x to the 4th."},{"Start":"05:52.370 ","End":"06:01.730","Text":"At this point, I now plug it back over there and"},{"Start":"06:01.730 ","End":"06:10.595","Text":"we get 24 times the integral from 0 to 1 of whatever it is I wrote here,"},{"Start":"06:10.595 ","End":"06:16.700","Text":"1/2x squared minus x cubed plus what is it?"},{"Start":"06:16.700 ","End":"06:26.520","Text":"1/2x to the 4 dx and then this is equal to 24 times,"},{"Start":"06:26.520 ","End":"06:29.015","Text":"see the integral of this,"},{"Start":"06:29.015 ","End":"06:36.200","Text":"because this is going to be 1/6 x cubed minus 1/4x to the 4th,"},{"Start":"06:36.200 ","End":"06:44.335","Text":"plus 1/10x to the 5th,"},{"Start":"06:44.335 ","End":"06:48.160","Text":"this between 0 and 1."},{"Start":"06:48.160 ","End":"06:57.290","Text":"When I plug in 0, not get anything when I plug-in 1 it\u0027s like the x is just not there."},{"Start":"06:57.290 ","End":"07:05.695","Text":"What I get is 24 times 1/6 minus 1/4 plus 1/10."},{"Start":"07:05.695 ","End":"07:09.335","Text":"As a simple fraction exercise,"},{"Start":"07:09.335 ","End":"07:14.660","Text":"common denominator for these 3 would be 60 so it\u0027s"},{"Start":"07:14.660 ","End":"07:23.790","Text":"24 over 60 and when I put it over 60,"},{"Start":"07:23.790 ","End":"07:26.820","Text":"6 into 60 is 10,"},{"Start":"07:26.820 ","End":"07:29.295","Text":"times 4 into 60,"},{"Start":"07:29.295 ","End":"07:32.595","Text":"15 times 10 into 60,"},{"Start":"07:32.595 ","End":"07:35.700","Text":"6 times 10 plus 6,"},{"Start":"07:35.700 ","End":"07:41.010","Text":"16 minus 15 is 1 so it\u0027s 24 over 60,"},{"Start":"07:41.010 ","End":"07:48.930","Text":"and I can cancel by 12 so that makes it 2/5."},{"Start":"07:48.930 ","End":"07:53.150","Text":"That\u0027s the x bar part."},{"Start":"07:53.150 ","End":"07:55.895","Text":"Now, the y bar will be very similar,"},{"Start":"07:55.895 ","End":"07:59.520","Text":"except that I\u0027ll have xy squared here."},{"Start":"08:10.460 ","End":"08:13.870","Text":"Pretty much the same thing as here, like I said,"},{"Start":"08:13.870 ","End":"08:23.770","Text":"is 24 integral from 0 to 1 and then integral y goes from 0 to 1 minus x."},{"Start":"08:23.770 ","End":"08:26.110","Text":"But instead of x squared y,"},{"Start":"08:26.110 ","End":"08:30.590","Text":"we said we\u0027re going to have xy squared dy,"},{"Start":"08:31.760 ","End":"08:35.725","Text":"dx, same procedure as before."},{"Start":"08:35.725 ","End":"08:40.385","Text":"We first compute the inner integral dy."},{"Start":"08:40.385 ","End":"08:43.635","Text":"I\u0027ll do the inner bit to the side,"},{"Start":"08:43.635 ","End":"08:45.920","Text":"and I can take the x out."},{"Start":"08:45.920 ","End":"08:51.380","Text":"I have x times the integral from"},{"Start":"08:51.380 ","End":"08:59.210","Text":"0-1 minus x of y squared dy,"},{"Start":"08:59.210 ","End":"09:06.270","Text":"which equals x times,"},{"Start":"09:06.280 ","End":"09:14.225","Text":"this is 1/3y cubed"},{"Start":"09:14.225 ","End":"09:18.185","Text":"from 0-1 minus x,"},{"Start":"09:18.185 ","End":"09:25.510","Text":"which equals, let\u0027s see the 1/3 first,"},{"Start":"09:25.510 ","End":"09:28.730","Text":"then the x, then y cubed."},{"Start":"09:28.730 ","End":"09:33.245","Text":"I only have to substitute the top limit because 0 doesn\u0027t give me anything."},{"Start":"09:33.245 ","End":"09:43.230","Text":"So I have 1 minus x cubed and there is a formula for 1 minus x cubed and"},{"Start":"09:43.230 ","End":"09:48.915","Text":"it is 1 minus 3x plus 3x squared"},{"Start":"09:48.915 ","End":"09:57.495","Text":"minus x cubed and so now I can put this back."},{"Start":"09:57.495 ","End":"10:00.545","Text":"This I got from this bit."},{"Start":"10:00.545 ","End":"10:05.900","Text":"Now I\u0027m going to plug it back in here and we\u0027ll"},{"Start":"10:05.900 ","End":"10:12.680","Text":"get still the integral from 0-1."},{"Start":"10:12.680 ","End":"10:17.510","Text":"The 1/3 will go with the 24,"},{"Start":"10:17.510 ","End":"10:23.370","Text":"making it 8 and the inner bit,"},{"Start":"10:23.370 ","End":"10:27.760","Text":"I better expand the 1/3 of taking care of the x. I\u0027ll multiply"},{"Start":"10:27.760 ","End":"10:33.320","Text":"out so it\u0027s x minus 3x squared"},{"Start":"10:33.320 ","End":"10:38.040","Text":"plus 3x cubed minus x to"},{"Start":"10:38.040 ","End":"10:44.100","Text":"the 4th dx and now let\u0027s see,"},{"Start":"10:44.100 ","End":"10:52.080","Text":"we\u0027ll integrate it, it\u0027s 8 times and 1/2x squared"},{"Start":"10:52.080 ","End":"11:02.595","Text":"minus x cubed and plus 3/4x to the 4th,"},{"Start":"11:02.595 ","End":"11:07.050","Text":"minus 1/5x to the 5th,"},{"Start":"11:07.050 ","End":"11:11.175","Text":"this between 0 and 1."},{"Start":"11:11.175 ","End":"11:14.550","Text":"As before, when we put in 0, we\u0027re not going to get anything."},{"Start":"11:14.550 ","End":"11:20.490","Text":"We just put in 1, so you\u0027ve got 8 times 1/2 minus"},{"Start":"11:20.490 ","End":"11:26.510","Text":"1 plus 3/4 minus 1/5."},{"Start":"11:26.510 ","End":"11:32.150","Text":"Let\u0027s see what\u0027s our common denominator going to be, 20 looks good."},{"Start":"11:32.150 ","End":"11:36.410","Text":"This will be 8 over 20,"},{"Start":"11:36.410 ","End":"11:39.395","Text":"1/2 is 10 over 20,"},{"Start":"11:39.395 ","End":"11:43.295","Text":"1 is 20 over 20,"},{"Start":"11:43.295 ","End":"11:46.910","Text":"3/4 is 15 over 20,"},{"Start":"11:46.910 ","End":"11:52.480","Text":"and 1/5 is 4 over 20."},{"Start":"11:52.480 ","End":"11:54.245","Text":"Let\u0027s compute this."},{"Start":"11:54.245 ","End":"11:57.950","Text":"10 and 15 is 25. That\u0027s the pluses."},{"Start":"11:57.950 ","End":"12:00.500","Text":"Minus 20, minus 4 is minus 24,"},{"Start":"12:00.500 ","End":"12:02.555","Text":"so this is just 1,"},{"Start":"12:02.555 ","End":"12:08.895","Text":"so the answer is 8 over 20 and 8 over 20 if I divide top and bottom by 4,"},{"Start":"12:08.895 ","End":"12:13.370","Text":"gives me 2/5, same as this."},{"Start":"12:13.370 ","End":"12:15.710","Text":"So the center of mass,"},{"Start":"12:15.710 ","End":"12:19.165","Text":"the x bar, y bar,"},{"Start":"12:19.165 ","End":"12:24.840","Text":"is equal to 2/5,"},{"Start":"12:24.840 ","End":"12:30.935","Text":"2/5 and that\u0027s the coordinates of the center of mass."},{"Start":"12:30.935 ","End":"12:35.400","Text":"I\u0027ll just highlight it and declare that we are done."}],"ID":8690},{"Watched":false,"Name":"Exercise 4","Duration":"14m 4s","ChapterTopicVideoID":8475,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.189","Text":"This exercise is from physics,"},{"Start":"00:03.189 ","End":"00:05.120","Text":"but you don\u0027t have to know physics,"},{"Start":"00:05.120 ","End":"00:07.345","Text":"we\u0027ll have all the formulas that we need."},{"Start":"00:07.345 ","End":"00:11.350","Text":"We have a flat board and it has a rectangular shape and"},{"Start":"00:11.350 ","End":"00:16.600","Text":"the rectangle is described as all the xs between this and this,"},{"Start":"00:16.600 ","End":"00:20.020","Text":"and all the ys between this and this, it\u0027s a rectangle."},{"Start":"00:20.020 ","End":"00:23.499","Text":"We\u0027re given also that it has a constant density function."},{"Start":"00:23.499 ","End":"00:25.030","Text":"It\u0027s a homogeneous board."},{"Start":"00:25.030 ","End":"00:30.190","Text":"It\u0027s the same mass distribution thickness everywhere."},{"Start":"00:30.190 ","End":"00:33.430","Text":"Now we have to compute 2 things."},{"Start":"00:33.430 ","End":"00:38.620","Text":"First of all, we\u0027ll compute the mass and we\u0027ll use that to answer the other question,"},{"Start":"00:38.620 ","End":"00:44.330","Text":"which is really the moment of inertia of the board about the z-axis,"},{"Start":"00:44.330 ","End":"00:46.790","Text":"but expressed in terms of the mass."},{"Start":"00:46.790 ","End":"00:50.880","Text":"A diagram will be very helpful."},{"Start":"00:51.020 ","End":"00:53.540","Text":"Here is the x, y plane."},{"Start":"00:53.540 ","End":"00:57.185","Text":"Now let\u0027s take care of x between a/2."},{"Start":"00:57.185 ","End":"01:03.115","Text":"Let\u0027s say that\u0027s here and minus a/2, maybe here."},{"Start":"01:03.115 ","End":"01:08.650","Text":"Then for y, we need to go from b/2,"},{"Start":"01:08.650 ","End":"01:16.185","Text":"maybe here and minus b/2 on the other side, minus b/2."},{"Start":"01:16.185 ","End":"01:18.705","Text":"Now we want to draw the rectangle."},{"Start":"01:18.705 ","End":"01:20.660","Text":"Here\u0027s our rectangle."},{"Start":"01:20.660 ","End":"01:23.315","Text":"This is the board because it includes the interior."},{"Start":"01:23.315 ","End":"01:25.580","Text":"Why don\u0027t I just shade it?"},{"Start":"01:25.580 ","End":"01:31.575","Text":"I give it a label R for rectangle or region."},{"Start":"01:31.575 ","End":"01:33.320","Text":"Now I want to relate to"},{"Start":"01:33.320 ","End":"01:40.835","Text":"this constant density concept to say that it\u0027s homogeneous or uniform."},{"Start":"01:40.835 ","End":"01:47.340","Text":"It means that the density function which we call Delta."},{"Start":"01:47.340 ","End":"01:50.525","Text":"Delta of x, y is some constant."},{"Start":"01:50.525 ","End":"01:52.490","Text":"Let\u0027s give the constant a name,"},{"Start":"01:52.490 ","End":"01:56.810","Text":"I could use C or K. But I\u0027ll call it Delta naught,"},{"Start":"01:56.810 ","End":"02:00.470","Text":"just to remind us that it is a density function and it\u0027s uniform"},{"Start":"02:00.470 ","End":"02:04.145","Text":"for all x and y in the rectangle."},{"Start":"02:04.145 ","End":"02:07.130","Text":"Now I\u0027d like to bring in the formulas that we\u0027re going to use."},{"Start":"02:07.130 ","End":"02:14.465","Text":"These are the general formulas that the mass of such a flat board in general,"},{"Start":"02:14.465 ","End":"02:19.610","Text":"when the shape is a region or domain D is a"},{"Start":"02:19.610 ","End":"02:25.610","Text":"double integral of the density function Delta of x,"},{"Start":"02:25.610 ","End":"02:28.970","Text":"y, dA , that\u0027s for the mass,"},{"Start":"02:28.970 ","End":"02:31.330","Text":"and the moment of inertia,"},{"Start":"02:31.330 ","End":"02:34.850","Text":"I for Inertia about the z-axis,"},{"Start":"02:34.850 ","End":"02:38.570","Text":"which is like a vertical line through the center,"},{"Start":"02:38.570 ","End":"02:43.250","Text":"is given by the double"},{"Start":"02:43.250 ","End":"02:49.130","Text":"integral also over the region D,"},{"Start":"02:49.130 ","End":"02:51.330","Text":"which is the shape of the board."},{"Start":"02:53.380 ","End":"03:03.620","Text":"Similar to this but there\u0027s an x-squared plus y-squared proceeding the Delta of x, y dA."},{"Start":"03:03.620 ","End":"03:08.075","Text":"These are all the formulas we\u0027re going to need and you don\u0027t need to know physics."},{"Start":"03:08.075 ","End":"03:10.715","Text":"Let\u0027s see what happens in our case."},{"Start":"03:10.715 ","End":"03:12.410","Text":"We\u0027ll start with the mass."},{"Start":"03:12.410 ","End":"03:17.300","Text":"In our case we have the double integral."},{"Start":"03:17.300 ","End":"03:20.120","Text":"Now, our D is called R here,"},{"Start":"03:20.120 ","End":"03:22.645","Text":"so it\u0027s over the rectangle."},{"Start":"03:22.645 ","End":"03:26.625","Text":"Delta is a constant,"},{"Start":"03:26.625 ","End":"03:30.760","Text":"Delta naught so we can take that."},{"Start":"03:30.760 ","End":"03:32.750","Text":"Well, I\u0027ll write it first of all,"},{"Start":"03:32.750 ","End":"03:38.040","Text":"and then we\u0027ll take it outside the integral of Delta naught dA."},{"Start":"03:38.330 ","End":"03:45.780","Text":"But this is just equal to the double integral."},{"Start":"03:45.780 ","End":"03:49.269","Text":"I can take the Delta naught in front,"},{"Start":"03:49.269 ","End":"03:53.480","Text":"the double integral over R. But this time I only have 1."},{"Start":"03:53.480 ","End":"03:56.615","Text":"I can write just dA, but I like to leave the 1 in."},{"Start":"03:56.615 ","End":"04:05.060","Text":"Now, it is well-known that the integral of a region of 1 is just the area of the region."},{"Start":"04:05.060 ","End":"04:09.635","Text":"This is equal to Delta naught and I\u0027ll just write it,"},{"Start":"04:09.635 ","End":"04:18.700","Text":"the area of R."},{"Start":"04:19.130 ","End":"04:22.940","Text":"Now let\u0027s look at this R. R is a rectangle."},{"Start":"04:22.940 ","End":"04:29.895","Text":"The rectangle is length times width is the area."},{"Start":"04:29.895 ","End":"04:34.310","Text":"From here to here is a and this is b,"},{"Start":"04:34.310 ","End":"04:36.140","Text":"so it\u0027s just a b."},{"Start":"04:36.140 ","End":"04:45.190","Text":"We end up with the mass being Delta naught times a times b."},{"Start":"04:45.680 ","End":"04:49.560","Text":"That\u0027s the mass. Let\u0027s leave that at the side."},{"Start":"04:49.560 ","End":"04:52.740","Text":"Now let\u0027s tackle the moment of inertia."},{"Start":"04:52.750 ","End":"04:57.230","Text":"The moment of inertia is going to be,"},{"Start":"04:57.230 ","End":"05:00.950","Text":"once again, Delta is a constant,"},{"Start":"05:00.950 ","End":"05:02.465","Text":"so I\u0027ll bring it out front,"},{"Start":"05:02.465 ","End":"05:05.435","Text":"in front of the integral sign Delta naught,"},{"Start":"05:05.435 ","End":"05:09.955","Text":"times the double integral over,"},{"Start":"05:09.955 ","End":"05:19.715","Text":"this time we have the rectangle R of x squared plus y squared dA."},{"Start":"05:19.715 ","End":"05:24.995","Text":"I\u0027d like to write this as an iterative integral either dx dy or dy dx."},{"Start":"05:24.995 ","End":"05:26.480","Text":"Luckily it doesn\u0027t matter."},{"Start":"05:26.480 ","End":"05:33.750","Text":"Let\u0027s take the outer loop as x and the inner loop as y like a vertical slices."},{"Start":"05:33.750 ","End":"05:39.490","Text":"It\u0027s easy to see that this is equal to Delta naught."},{"Start":"05:39.490 ","End":"05:49.280","Text":"It\u0027s very simple region so we can just say x goes from minus a/2 to a/2,"},{"Start":"05:49.280 ","End":"05:55.760","Text":"that\u0027s dx and for each such x,"},{"Start":"05:55.760 ","End":"06:00.560","Text":"we get the same vertical slice from minus"},{"Start":"06:00.560 ","End":"06:08.970","Text":"b/2 up to b/2 dy."},{"Start":"06:08.970 ","End":"06:12.939","Text":"Then the x squared plus y squared, I just copy."},{"Start":"06:14.560 ","End":"06:17.870","Text":"At this point I\u0027m going to take a shortcut using"},{"Start":"06:17.870 ","End":"06:20.585","Text":"one of the standard tricks that are used."},{"Start":"06:20.585 ","End":"06:23.840","Text":"Notice that this function, well,"},{"Start":"06:23.840 ","End":"06:29.375","Text":"even say the inner function is an even function in y."},{"Start":"06:29.375 ","End":"06:32.090","Text":"If I replace y by minus y,"},{"Start":"06:32.090 ","End":"06:34.985","Text":"I get the same thing and when that happens,"},{"Start":"06:34.985 ","End":"06:40.805","Text":"we can just take the integral from 0 to b/2 and double it."},{"Start":"06:40.805 ","End":"06:44.570","Text":"Then similarly, when we get out to the x integral,"},{"Start":"06:44.570 ","End":"06:48.515","Text":"it\u0027s also replacing x by minus x is the same."},{"Start":"06:48.515 ","End":"06:53.615","Text":"So the left and the right will be equal so we just have to take one side and double it."},{"Start":"06:53.615 ","End":"06:59.070","Text":"In short, what I\u0027m saying is we can settle for 1/4 of the rectangle."},{"Start":"06:59.070 ","End":"07:00.805","Text":"I\u0027ve marked it here,"},{"Start":"07:00.805 ","End":"07:06.655","Text":"and then just do the integral over this quarter rectangle and multiply the answer by 4."},{"Start":"07:06.655 ","End":"07:15.565","Text":"We get that this is equal to Delta naught times 4,"},{"Start":"07:15.565 ","End":"07:18.280","Text":"and now the integral becomes simpler."},{"Start":"07:18.280 ","End":"07:25.150","Text":"We just go from 0-a over 2 in the x direction,"},{"Start":"07:25.150 ","End":"07:31.525","Text":"and from 0-b over 2 in the y direction,"},{"Start":"07:31.525 ","End":"07:33.400","Text":"and the rest of it is the same,"},{"Start":"07:33.400 ","End":"07:38.635","Text":"x squared plus y squared dy, dx."},{"Start":"07:38.635 ","End":"07:41.125","Text":"Let\u0027s start computing this integral."},{"Start":"07:41.125 ","End":"07:45.130","Text":"We don\u0027t need the diagram anymore,"},{"Start":"07:45.130 ","End":"07:47.395","Text":"so I can scroll, at least we all can see it."},{"Start":"07:47.395 ","End":"07:50.860","Text":"I\u0027ll just reiterate that this was M that we found,"},{"Start":"07:50.860 ","End":"07:55.630","Text":"and this is the moment of inertia about the z-axis,"},{"Start":"07:55.630 ","End":"07:58.820","Text":"and I\u0027m going to continue developing this."},{"Start":"07:58.830 ","End":"08:02.290","Text":"As usual we do the inner integral first,"},{"Start":"08:02.290 ","End":"08:05.240","Text":"that\u0027s going to be the dy integral,"},{"Start":"08:05.280 ","End":"08:08.830","Text":"and I\u0027d like to compute this separately at the side,"},{"Start":"08:08.830 ","End":"08:10.570","Text":"let me call it asterisk."},{"Start":"08:10.570 ","End":"08:13.435","Text":"At the side I\u0027ll compute asterisk,"},{"Start":"08:13.435 ","End":"08:16.135","Text":"which is, let\u0027s see,"},{"Start":"08:16.135 ","End":"08:23.305","Text":"we have the integral from 0-b over 2."},{"Start":"08:23.305 ","End":"08:28.270","Text":"I\u0027m just copying it, x squared plus y squared dy,"},{"Start":"08:28.270 ","End":"08:30.535","Text":"and this is equal 2."},{"Start":"08:30.535 ","End":"08:33.520","Text":"Integral of x squared is x squared,"},{"Start":"08:33.520 ","End":"08:39.220","Text":"y integral of y squared is 1/3, y cubed."},{"Start":"08:39.220 ","End":"08:46.810","Text":"This, we shall take between 0 and b over 2,"},{"Start":"08:46.810 ","End":"08:50.860","Text":"maybe I\u0027ll emphasize that it\u0027s y that goes from here to here."},{"Start":"08:50.860 ","End":"08:54.520","Text":"First of all, let\u0027s plug in b over 2."},{"Start":"08:54.520 ","End":"09:02.350","Text":"We get x squared times b over 2 plus,"},{"Start":"09:02.350 ","End":"09:05.155","Text":"and if I put y equals b over 2,"},{"Start":"09:05.155 ","End":"09:06.910","Text":"I get, let\u0027s see,"},{"Start":"09:06.910 ","End":"09:09.820","Text":"it\u0027s going to be b cubed over 8."},{"Start":"09:09.820 ","End":"09:13.915","Text":"But the 8 is going to combine with 3 to give 24,"},{"Start":"09:13.915 ","End":"09:23.720","Text":"so it\u0027s going to be b cubed over 24."},{"Start":"09:24.210 ","End":"09:27.730","Text":"That\u0027s just the asterisk part,"},{"Start":"09:27.730 ","End":"09:30.530","Text":"so now I\u0027m going to go back here."},{"Start":"09:31.770 ","End":"09:37.705","Text":"Continuing, I\u0027d like to write the 4 before the Delta,"},{"Start":"09:37.705 ","End":"09:45.625","Text":"and then I have the integral from 0-a over 2."},{"Start":"09:45.625 ","End":"09:49.540","Text":"Now, this bit I copy from here, so what do I have?"},{"Start":"09:49.540 ","End":"09:56.005","Text":"I have b over 2x squared plus"},{"Start":"09:56.005 ","End":"10:05.060","Text":"b squared over 24 in brackets, just dx."},{"Start":"10:05.070 ","End":"10:09.100","Text":"Now I integrate this with respect to x,"},{"Start":"10:09.100 ","End":"10:12.085","Text":"4 delta naught this."},{"Start":"10:12.085 ","End":"10:17.860","Text":"Then we have, raise the power by 1 is x cubed divide by 3,"},{"Start":"10:17.860 ","End":"10:22.675","Text":"so I\u0027ve got b over 6x cubed."},{"Start":"10:22.675 ","End":"10:24.430","Text":"Here it\u0027s a constant,"},{"Start":"10:24.430 ","End":"10:27.680","Text":"just b squared over 24x."},{"Start":"10:29.280 ","End":"10:38.755","Text":"This has to be evaluated between 0 and a over 2."},{"Start":"10:38.755 ","End":"10:41.740","Text":"Something that I didn\u0027t say before,"},{"Start":"10:41.740 ","End":"10:44.275","Text":"it\u0027s obvious but I should have said it,"},{"Start":"10:44.275 ","End":"10:46.360","Text":"I only substituted the upper limit."},{"Start":"10:46.360 ","End":"10:51.400","Text":"I should have mentioned that when I plug in y equals 0,"},{"Start":"10:51.400 ","End":"10:54.940","Text":"I get 0 so I should have really written minus 0 here."},{"Start":"10:54.940 ","End":"10:58.000","Text":"The same thing is going to happen here."},{"Start":"10:58.000 ","End":"11:00.880","Text":"When we plug in 0 for x,"},{"Start":"11:00.880 ","End":"11:04.465","Text":"we\u0027re not going to get anything so we just have to plug in the a over 2,"},{"Start":"11:04.465 ","End":"11:09.865","Text":"so we get 4 Delta naught."},{"Start":"11:09.865 ","End":"11:13.975","Text":"I\u0027ll put a big brackets here and I\u0027ll substitute a over 2,"},{"Start":"11:13.975 ","End":"11:18.945","Text":"so I\u0027ve got b over 6 times"},{"Start":"11:18.945 ","End":"11:25.650","Text":"a over 2 cubed plus b"},{"Start":"11:25.650 ","End":"11:32.410","Text":"squared over 24 and x,"},{"Start":"11:32.410 ","End":"11:35.450","Text":"I just substitute a over 2."},{"Start":"11:36.360 ","End":"11:39.490","Text":"Then minus 0, again,"},{"Start":"11:39.490 ","End":"11:42.740","Text":"I\u0027ll just mention it, I had forgotten it."},{"Start":"11:44.340 ","End":"11:47.575","Text":"I just noticed that I mis-copied something."},{"Start":"11:47.575 ","End":"11:50.080","Text":"This 3 I wrote badly,"},{"Start":"11:50.080 ","End":"11:51.460","Text":"it looks like a 2."},{"Start":"11:51.460 ","End":"11:55.000","Text":"This is actually a 3, so let me correct."},{"Start":"11:55.000 ","End":"11:59.350","Text":"I\u0027ll erase the 2s here, here, and here,"},{"Start":"11:59.350 ","End":"12:01.555","Text":"and write b cubed,"},{"Start":"12:01.555 ","End":"12:04.030","Text":"b cubed, b cubed."},{"Start":"12:04.030 ","End":"12:05.620","Text":"Sorry about that."},{"Start":"12:05.620 ","End":"12:09.890","Text":"I guess the lesson is it\u0027s important to write clearly."},{"Start":"12:10.320 ","End":"12:13.570","Text":"Let\u0027s continue."},{"Start":"12:13.570 ","End":"12:15.865","Text":"I need more space."},{"Start":"12:15.865 ","End":"12:22.720","Text":"Now, notice that if I do the denominators,"},{"Start":"12:22.720 ","End":"12:28.705","Text":"6 times 2 cubed is 6 times 8 is 48."},{"Start":"12:28.705 ","End":"12:31.615","Text":"Here 24 times 2 is 48."},{"Start":"12:31.615 ","End":"12:34.495","Text":"I can bring the 48 up front."},{"Start":"12:34.495 ","End":"12:39.440","Text":"I have 4 Delta naught over 48."},{"Start":"12:39.660 ","End":"12:42.475","Text":"Let\u0027s see what I\u0027m left with."},{"Start":"12:42.475 ","End":"12:47.380","Text":"I\u0027ve got ba cubed"},{"Start":"12:47.380 ","End":"12:54.910","Text":"plus b cubed a. I can keep simplifying,"},{"Start":"12:54.910 ","End":"12:57.530","Text":"let\u0027s go down more."},{"Start":"12:57.570 ","End":"13:03.055","Text":"What I can do is I can take ab outside the brackets."},{"Start":"13:03.055 ","End":"13:07.075","Text":"Also 4 with 48 cancels,"},{"Start":"13:07.075 ","End":"13:09.490","Text":"this goes into this 12 times."},{"Start":"13:09.490 ","End":"13:13.809","Text":"What I have is Delta naught,"},{"Start":"13:13.809 ","End":"13:16.420","Text":"and then I\u0027m taking ab out,"},{"Start":"13:16.420 ","End":"13:19.570","Text":"and then I\u0027m writing this over 12."},{"Start":"13:19.570 ","End":"13:26.630","Text":"What I\u0027m left with after I take ab is a squared plus b squared."},{"Start":"13:26.850 ","End":"13:35.425","Text":"Now, notice I still have the result that M is Delta naught ab,"},{"Start":"13:35.425 ","End":"13:41.680","Text":"so this whole bit is M. What I get in the end is"},{"Start":"13:41.680 ","End":"13:49.660","Text":"1/12 M times a squared plus b squared."},{"Start":"13:49.660 ","End":"13:51.550","Text":"This is the final answer."},{"Start":"13:51.550 ","End":"13:56.815","Text":"We have indeed expressed the moment of inertia in terms of M,"},{"Start":"13:56.815 ","End":"14:01.055","Text":"and this is the expression that it\u0027s equal to."},{"Start":"14:01.055 ","End":"14:04.050","Text":"We are done."}],"ID":8691},{"Watched":false,"Name":"Exercise 5","Duration":"12m 20s","ChapterTopicVideoID":8476,"CourseChapterTopicPlaylistID":4967,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.330","Text":"In this exercise, we have to find the surface area of part of the cylinder."},{"Start":"00:06.330 ","End":"00:08.685","Text":"This is the equation of a cylinder,"},{"Start":"00:08.685 ","End":"00:10.920","Text":"but we\u0027re not going to sketch it,"},{"Start":"00:10.920 ","End":"00:17.160","Text":"which lies above the rectangle so and so in the xy plane."},{"Start":"00:17.160 ","End":"00:19.785","Text":"Now, this I\u0027m going to sketch."},{"Start":"00:19.785 ","End":"00:22.845","Text":"Here\u0027s the xy plane,"},{"Start":"00:22.845 ","End":"00:26.715","Text":"and this definition in set theory style,"},{"Start":"00:26.715 ","End":"00:30.090","Text":"just says that the set of all pairs xy,"},{"Start":"00:30.090 ","End":"00:37.050","Text":"it should say in the plane such that x is between 0 and 1 inclusive."},{"Start":"00:37.050 ","End":"00:41.730","Text":"I\u0027m going to mark here 0 and let\u0027s say that this is 1,"},{"Start":"00:41.730 ","End":"00:45.920","Text":"and y is between 0 and 4 inclusive."},{"Start":"00:45.920 ","End":"00:48.870","Text":"Let\u0027s say that this is 4."},{"Start":"00:49.520 ","End":"00:52.370","Text":"Here\u0027s the rectangle. Of course,"},{"Start":"00:52.370 ","End":"00:55.610","Text":"it includes the interior, maybe I\u0027ll shade it."},{"Start":"00:55.610 ","End":"01:03.129","Text":"I will also label it R for rectangle or region."},{"Start":"01:03.260 ","End":"01:07.260","Text":"What we\u0027re missing now is a formula."},{"Start":"01:07.260 ","End":"01:09.670","Text":"But I forgot the formula."},{"Start":"01:09.670 ","End":"01:14.960","Text":"I went online to a search site and"},{"Start":"01:14.960 ","End":"01:22.020","Text":"typed in surface area double integral and I quickly found this."},{"Start":"01:22.160 ","End":"01:27.784","Text":"This tells me all I need is even sketch here that the sum region,"},{"Start":"01:27.784 ","End":"01:31.135","Text":"they also happen to call it R so much the better."},{"Start":"01:31.135 ","End":"01:38.345","Text":"The thing is though that this will lead to z being a function of x and y for the surface."},{"Start":"01:38.345 ","End":"01:43.850","Text":"But our cylinder is not in the form where z is a function of x and y."},{"Start":"01:43.850 ","End":"01:50.620","Text":"So we\u0027ll work on it in a moment and get it into the form z equals f of xy."},{"Start":"01:51.410 ","End":"01:54.685","Text":"After we\u0027ve done that,"},{"Start":"01:54.685 ","End":"01:58.990","Text":"then we\u0027ll be able to use the formula that\u0027s down here."},{"Start":"01:58.990 ","End":"02:04.185","Text":"First things first, when you get those z equals a function of x and y."},{"Start":"02:04.185 ","End":"02:06.555","Text":"We\u0027ll start with this,"},{"Start":"02:06.555 ","End":"02:07.790","Text":"and then from there,"},{"Start":"02:07.790 ","End":"02:11.840","Text":"we can get that z squared equals"},{"Start":"02:11.840 ","End":"02:18.945","Text":"4 minus x squared and then we know that z is,"},{"Start":"02:18.945 ","End":"02:21.560","Text":"and here we\u0027re in a dilemma."},{"Start":"02:21.560 ","End":"02:28.205","Text":"Do we take plus or minus the square root of 4 minus x squared?"},{"Start":"02:28.205 ","End":"02:31.495","Text":"Then I noticed the word above."},{"Start":"02:31.495 ","End":"02:34.470","Text":"Above would mean that we take the plus,"},{"Start":"02:34.470 ","End":"02:36.890","Text":"then I\u0027ll put a plus here just to emphasize that"},{"Start":"02:36.890 ","End":"02:39.455","Text":"I didn\u0027t forget that there is also a minus,"},{"Start":"02:39.455 ","End":"02:41.940","Text":"but that would be below."},{"Start":"02:43.070 ","End":"02:54.140","Text":"Now I can apply the formula and maybe I\u0027ll give the surface area a name S. In our case,"},{"Start":"02:54.140 ","End":"03:03.370","Text":"we get that S is equal to double integral and I put the R down here and print."},{"Start":"03:03.370 ","End":"03:06.400","Text":"I noticed they often put it at the side."},{"Start":"03:06.400 ","End":"03:09.550","Text":"Yeah, I should have said that this now is equal to"},{"Start":"03:09.550 ","End":"03:14.110","Text":"my f of xy that I was looking for, that\u0027s here,"},{"Start":"03:14.110 ","End":"03:19.195","Text":"and so I\u0027m just copying from here, but abbreviating,"},{"Start":"03:19.195 ","End":"03:26.170","Text":"we have the square root of 1 plus f with respect to"},{"Start":"03:26.170 ","End":"03:33.120","Text":"x squared plus f"},{"Start":"03:33.120 ","End":"03:38.410","Text":"with respect to y squared dA."},{"Start":"03:38.710 ","End":"03:43.280","Text":"I just mentioned that I\u0027ve seen the written slightly differently in"},{"Start":"03:43.280 ","End":"03:47.599","Text":"some places they would write it as follows with zx,"},{"Start":"03:47.599 ","End":"03:51.450","Text":"zy set of fx, fy."},{"Start":"03:51.650 ","End":"03:55.160","Text":"Next, I\u0027d like to compute this expression,"},{"Start":"03:55.160 ","End":"03:57.740","Text":"the integrant, as it\u0027s sometimes called,"},{"Start":"03:57.740 ","End":"04:00.170","Text":"that thing is going to be integrated."},{"Start":"04:00.170 ","End":"04:06.410","Text":"I\u0027ll use the f rather than the z. I\u0027ll say that f with respect to x,"},{"Start":"04:06.410 ","End":"04:11.970","Text":"it doesn\u0027t really matter, or z with respect to x is the derivative of this."},{"Start":"04:11.980 ","End":"04:19.805","Text":"The derivative of square root is 1 over twice the square root of that same thing."},{"Start":"04:19.805 ","End":"04:23.750","Text":"But because it\u0027s a function of x,"},{"Start":"04:23.750 ","End":"04:24.830","Text":"we need the chain rule,"},{"Start":"04:24.830 ","End":"04:26.735","Text":"we need the inner derivative,"},{"Start":"04:26.735 ","End":"04:29.760","Text":"which is minus 2x."},{"Start":"04:31.630 ","End":"04:35.370","Text":"The 2 cancels with the 2."},{"Start":"04:35.570 ","End":"04:40.095","Text":"As for f with respect to y, well,"},{"Start":"04:40.095 ","End":"04:43.550","Text":"maybe you notice that there was something funny about this function of x and y"},{"Start":"04:43.550 ","End":"04:47.135","Text":"because y doesn\u0027t appear explicitly, but that\u0027s okay."},{"Start":"04:47.135 ","End":"04:48.980","Text":"As far as y is concerned,"},{"Start":"04:48.980 ","End":"04:50.575","Text":"this is a constant."},{"Start":"04:50.575 ","End":"04:56.740","Text":"F with respect to y or z with respect to y, is just 0."},{"Start":"04:56.740 ","End":"05:04.700","Text":"Now I take this and this and plug it in here and so we get that this is equal to the"},{"Start":"05:04.700 ","End":"05:13.800","Text":"double integral over region or rectangle R of the square root of 1 plus."},{"Start":"05:13.800 ","End":"05:19.045","Text":"This squared, well, I can ignore the minus then and just square the top,"},{"Start":"05:19.045 ","End":"05:23.340","Text":"and that\u0027s x squared, square the bottom,"},{"Start":"05:23.340 ","End":"05:25.880","Text":"and just the square root just drops out,"},{"Start":"05:25.880 ","End":"05:28.960","Text":"so it\u0027s 4 minus x squared."},{"Start":"05:28.960 ","End":"05:32.210","Text":"Then plus fy squared,"},{"Start":"05:32.210 ","End":"05:39.240","Text":"dA and this can"},{"Start":"05:39.240 ","End":"05:43.205","Text":"be simplified double integral."},{"Start":"05:43.205 ","End":"05:45.880","Text":"If I put a common denominator,"},{"Start":"05:45.880 ","End":"05:49.675","Text":"4 minus x squared,"},{"Start":"05:49.675 ","End":"05:55.900","Text":"then what I get is 4 minus x squared plus x squared,"},{"Start":"05:55.900 ","End":"05:58.790","Text":"which is just 4dA."},{"Start":"06:00.050 ","End":"06:03.790","Text":"Finally, I\u0027ll just write it over here."},{"Start":"06:03.790 ","End":"06:09.055","Text":"This is equal, the square root of 4 is 2."},{"Start":"06:09.055 ","End":"06:11.465","Text":"I can pull the 2 out front."},{"Start":"06:11.465 ","End":"06:20.170","Text":"It\u0027s twice the double integral of 1 over the square root of"},{"Start":"06:20.170 ","End":"06:29.090","Text":"4 minus x squared dA over R. At this point,"},{"Start":"06:29.090 ","End":"06:35.525","Text":"we have to decide on how to do this double integral over a region."},{"Start":"06:35.525 ","End":"06:38.180","Text":"We want to do it as an iterated integral,"},{"Start":"06:38.180 ","End":"06:40.655","Text":"meaning either dx,dy, or dy,dx."},{"Start":"06:40.655 ","End":"06:44.555","Text":"Which way do we want to slice it, horizontally or vertically?"},{"Start":"06:44.555 ","End":"06:47.945","Text":"From trying both out,"},{"Start":"06:47.945 ","End":"06:52.310","Text":"it turns out that it\u0027s best to slice it horizontally."},{"Start":"06:52.310 ","End":"06:54.320","Text":"Let me back up a moment."},{"Start":"06:54.320 ","End":"06:56.165","Text":"Let me explain what I mean."},{"Start":"06:56.165 ","End":"07:04.500","Text":"In general, we have a double integral over some region of something, dA."},{"Start":"07:04.500 ","End":"07:08.415","Text":"You want to write this in 1 of 2 forms."},{"Start":"07:08.415 ","End":"07:13.880","Text":"We either write it as the integral of x goes from something to something."},{"Start":"07:13.880 ","End":"07:16.205","Text":"Well, we know R, the rectangle in this case,"},{"Start":"07:16.205 ","End":"07:22.600","Text":"we can say x goes from 0-1, and then dx,"},{"Start":"07:22.600 ","End":"07:24.840","Text":"and then for each x,"},{"Start":"07:24.840 ","End":"07:32.189","Text":"y goes from 0-4dy"},{"Start":"07:32.189 ","End":"07:35.430","Text":"of whatever this thing was."},{"Start":"07:35.430 ","End":"07:37.005","Text":"That\u0027s 1 option."},{"Start":"07:37.005 ","End":"07:41.005","Text":"The other option is to do it the other way round."},{"Start":"07:41.005 ","End":"07:43.260","Text":"First, on the outer loop,"},{"Start":"07:43.260 ","End":"07:47.285","Text":"take y from 0-4,"},{"Start":"07:47.285 ","End":"07:51.005","Text":"and for each y we get a horizontal stripe,"},{"Start":"07:51.005 ","End":"08:00.365","Text":"for x going from 0-1 of whatever it is, dx,dy."},{"Start":"08:00.365 ","End":"08:03.485","Text":"I\u0027m just telling you that we\u0027re going to go with this form."},{"Start":"08:03.485 ","End":"08:06.935","Text":"This is the 1 that works better, it\u0027s easier."},{"Start":"08:06.935 ","End":"08:08.660","Text":"You try them both."},{"Start":"08:08.660 ","End":"08:11.030","Text":"You see that 1 of them gets difficult."},{"Start":"08:11.030 ","End":"08:13.925","Text":"I\u0027m just going to continue and say,"},{"Start":"08:13.925 ","End":"08:15.935","Text":"this is my preferred choice."},{"Start":"08:15.935 ","End":"08:20.855","Text":"Here, I\u0027m rewriting this now"},{"Start":"08:20.855 ","End":"08:27.095","Text":"as twice the integral."},{"Start":"08:27.095 ","End":"08:32.280","Text":"Like we said, y goes from 0-4,"},{"Start":"08:32.770 ","End":"08:37.655","Text":"x goes from 0-1."},{"Start":"08:37.655 ","End":"08:46.350","Text":"Now, this is 1 over square root of 4 minus x squared."},{"Start":"08:46.350 ","End":"08:49.395","Text":"We\u0027re going with dx,dy,"},{"Start":"08:49.395 ","End":"08:56.100","Text":"which means that we start out with this, the inner integral."},{"Start":"08:56.100 ","End":"09:02.525","Text":"What I\u0027m going to do is do this highlighted bit as they call it asterisk."},{"Start":"09:02.525 ","End":"09:04.880","Text":"I\u0027ll do this as a side exercise."},{"Start":"09:04.880 ","End":"09:10.924","Text":"The asterisk is the integral from"},{"Start":"09:10.924 ","End":"09:16.969","Text":"0-1 of dx over"},{"Start":"09:16.969 ","End":"09:22.500","Text":"square root of 4 minus x squared."},{"Start":"09:22.930 ","End":"09:28.550","Text":"I\u0027m going to use the Table of Integrals to look this 1 up."},{"Start":"09:28.550 ","End":"09:31.640","Text":"I\u0027m talking about indefinite integrals for the moment."},{"Start":"09:31.640 ","End":"09:35.045","Text":"I found 1 that\u0027s very close to this."},{"Start":"09:35.045 ","End":"09:45.110","Text":"That says that the integral of dx over the square root of a squared minus x"},{"Start":"09:45.110 ","End":"09:53.005","Text":"squared is equal to arc sine of x over"},{"Start":"09:53.005 ","End":"09:57.370","Text":"a plus a constant because we don\u0027t need the constant because we\u0027re"},{"Start":"09:57.370 ","End":"10:01.990","Text":"going to do a definite integral and this looks very much like this."},{"Start":"10:01.990 ","End":"10:07.195","Text":"If I take a equals 2 because then a squared is 4,"},{"Start":"10:07.195 ","End":"10:12.235","Text":"what I get is from the formula,"},{"Start":"10:12.235 ","End":"10:18.774","Text":"the arc sine of x over 2."},{"Start":"10:18.774 ","End":"10:20.860","Text":"I don\u0027t need the plus c,"},{"Start":"10:20.860 ","End":"10:25.775","Text":"but I do need to evaluate it between 0 and 1,"},{"Start":"10:25.775 ","End":"10:27.690","Text":"which means I plug in 1,"},{"Start":"10:27.690 ","End":"10:30.000","Text":"I plug in 0, and subtract."},{"Start":"10:30.000 ","End":"10:37.340","Text":"What I get is arc sine of 1"},{"Start":"10:37.340 ","End":"10:45.890","Text":"over 2 minus arc sine of 0 over 2, which is 0."},{"Start":"10:45.890 ","End":"10:51.780","Text":"Now, the angle whose sine is 0.5 is,"},{"Start":"10:51.780 ","End":"10:53.090","Text":"and I know it in degrees,"},{"Start":"10:53.090 ","End":"10:55.820","Text":"it\u0027s 30 degrees, but we want it in radians."},{"Start":"10:55.820 ","End":"10:58.714","Text":"30 degrees is Pi over 6."},{"Start":"10:58.714 ","End":"11:03.740","Text":"An arc sine of 0 is just 0 minus 0,"},{"Start":"11:03.740 ","End":"11:06.090","Text":"which is Pi over 6."},{"Start":"11:06.090 ","End":"11:09.755","Text":"Now I\u0027m plugging this asterisk back here."},{"Start":"11:09.755 ","End":"11:14.540","Text":"I get that this is equal to twice the integral from"},{"Start":"11:14.540 ","End":"11:18.930","Text":"0-4 of all this is"},{"Start":"11:18.930 ","End":"11:25.570","Text":"just Pi over 6, and dy."},{"Start":"11:28.790 ","End":"11:31.710","Text":"This is a constant."},{"Start":"11:31.710 ","End":"11:36.095","Text":"In fact, let it I even pull it out front."},{"Start":"11:36.095 ","End":"11:38.615","Text":"To save a line, I\u0027ll just write it here."},{"Start":"11:38.615 ","End":"11:42.525","Text":"Twice Pi over 6 is pi over 3,"},{"Start":"11:42.525 ","End":"11:45.060","Text":"and I\u0027ve got the integral from 0-4."},{"Start":"11:45.060 ","End":"11:48.795","Text":"I\u0027ve just dy or 1dy."},{"Start":"11:48.795 ","End":"11:51.600","Text":"This is Pi over 3."},{"Start":"11:51.600 ","End":"11:55.935","Text":"Now, the integral of 1 is just y,"},{"Start":"11:55.935 ","End":"12:00.660","Text":"and I take this between 0 and 4."},{"Start":"12:00.660 ","End":"12:04.425","Text":"So 4 minus 0 is 4."},{"Start":"12:04.425 ","End":"12:09.015","Text":"I\u0027ve got Pi over 3 times 4."},{"Start":"12:09.015 ","End":"12:15.405","Text":"The final answer is just 4Pi over 3,"},{"Start":"12:15.405 ","End":"12:20.560","Text":"and I\u0027ll highlight that and we\u0027re done."}],"ID":8692}],"Thumbnail":null,"ID":4967}]

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1.1

1