Double Integrals, Polar Coordinates
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- Double Integrals, Polar Coordinates
- Double Integrals, Polar Coordinates (cont)
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24
- Exercise 25
- Exercise 26
- Exercise 27
- Exercise 28

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[{"Name":"Double Integrals, Polar Coordinates","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Double Integrals, Polar Coordinates","Duration":"11m 51s","ChapterTopicVideoID":8478,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8478.jpeg","UploadDate":"2020-02-26T12:02:33.3330000","DurationForVideoObject":"PT11M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this clip, we\u0027re going to learn about double integrals in"},{"Start":"00:03.360 ","End":"00:06.690","Text":"polar coordinates as opposed to Cartesian coordinates."},{"Start":"00:06.690 ","End":"00:12.450","Text":"I\u0027m going to assume that you\u0027ve already covered the chapter on polar coordinates,"},{"Start":"00:12.450 ","End":"00:16.050","Text":"the basics, and in particular,"},{"Start":"00:16.050 ","End":"00:21.075","Text":"the substitution for x and y in terms of r and Theta,"},{"Start":"00:21.075 ","End":"00:26.955","Text":"that x is r cosine Theta and y is r sine Theta."},{"Start":"00:26.955 ","End":"00:30.870","Text":"I suggest going back to review it if you don\u0027t remember this stuff."},{"Start":"00:30.870 ","End":"00:36.295","Text":"Now, here\u0027s the formula."},{"Start":"00:36.295 ","End":"00:42.925","Text":"If we have an integral of some function of x and y,"},{"Start":"00:42.925 ","End":"00:48.960","Text":"dA, or let\u0027s even say dxdy or dydx."},{"Start":"00:48.960 ","End":"00:52.800","Text":"Any of these forms, dA, dxdy, dydx."},{"Start":"00:52.800 ","End":"00:56.720","Text":"It\u0027s going to be over some region or domain in x and y."},{"Start":"00:56.720 ","End":"00:58.820","Text":"Let me just write x, y in short-term,"},{"Start":"00:58.820 ","End":"01:03.845","Text":"meaning some region described in terms of x and y."},{"Start":"01:03.845 ","End":"01:08.765","Text":"What we\u0027re going to do now is write it as an integral in polar coordinates,"},{"Start":"01:08.765 ","End":"01:10.460","Text":"and it goes as follows."},{"Start":"01:10.460 ","End":"01:15.470","Text":"We have the integral over the same region but"},{"Start":"01:15.470 ","End":"01:21.050","Text":"described in terms of r and Theta of f of,"},{"Start":"01:21.050 ","End":"01:25.175","Text":"now for x, we substitute r cosine Theta."},{"Start":"01:25.175 ","End":"01:30.480","Text":"For y, we substitute r sine Theta,"},{"Start":"01:33.070 ","End":"01:37.575","Text":"at the end, drd Theta."},{"Start":"01:37.575 ","End":"01:40.505","Text":"Then something possibly unexpected,"},{"Start":"01:40.505 ","End":"01:42.770","Text":"an extra r in here."},{"Start":"01:42.770 ","End":"01:44.795","Text":"I\u0027ll return to this in a moment."},{"Start":"01:44.795 ","End":"01:50.075","Text":"I want to say why we would want to use such a conversion."},{"Start":"01:50.075 ","End":"01:55.175","Text":"Typically, it\u0027s when the region has some kind of circular symmetry."},{"Start":"01:55.175 ","End":"02:03.200","Text":"For example, the region might be circular or it might even just be part of a circle,"},{"Start":"02:03.200 ","End":"02:10.625","Text":"let\u0027s say quarter of a circle or it might even be what was called an annulus,"},{"Start":"02:10.625 ","End":"02:16.275","Text":"the bit between 2 circles that might be."},{"Start":"02:16.275 ","End":"02:19.160","Text":"Whenever we have some kind of circular symmetry,"},{"Start":"02:19.160 ","End":"02:23.725","Text":"generally, it\u0027s a good idea to convert to polar."},{"Start":"02:23.725 ","End":"02:33.000","Text":"Now, this matter of this extra r. Let me start by going to a 1-dimensional case,"},{"Start":"02:33.000 ","End":"02:34.565","Text":"the function of 1 variable."},{"Start":"02:34.565 ","End":"02:36.245","Text":"If we had the integral,"},{"Start":"02:36.245 ","End":"02:39.180","Text":"say of f of xdx,"},{"Start":"02:39.640 ","End":"02:43.565","Text":"then you\u0027ve learnt substitution."},{"Start":"02:43.565 ","End":"02:47.185","Text":"Suppose we substitute x equals,"},{"Start":"02:47.185 ","End":"02:49.860","Text":"for example, t squared,"},{"Start":"02:49.860 ","End":"02:52.810","Text":"then what you would do,"},{"Start":"02:52.810 ","End":"02:54.980","Text":"well, you would substitute the limits"},{"Start":"02:54.980 ","End":"03:03.155","Text":"also and you\u0027d get the integral from something else to something else,"},{"Start":"03:03.155 ","End":"03:09.380","Text":"limits on t and that\u0027s the equivalent of converting the region from x,"},{"Start":"03:09.380 ","End":"03:10.685","Text":"y to r Theta."},{"Start":"03:10.685 ","End":"03:15.210","Text":"Then you\u0027d also put t squared instead of x and here,"},{"Start":"03:15.210 ","End":"03:18.060","Text":"you\u0027d have a dt, but that\u0027s not all,"},{"Start":"03:18.060 ","End":"03:19.910","Text":"you\u0027d also have an extra thing,"},{"Start":"03:19.910 ","End":"03:22.590","Text":"the derivative of this 2t."},{"Start":"03:22.800 ","End":"03:31.045","Text":"It\u0027s like the derivative of this with respect to t. This is a 2-dimensional analogy."},{"Start":"03:31.045 ","End":"03:36.610","Text":"In general, when we substitute variables in functions of 2 variables,"},{"Start":"03:36.610 ","End":"03:38.874","Text":"we\u0027ll get something extra."},{"Start":"03:38.874 ","End":"03:43.105","Text":"I\u0027ll just give you the name of this extra now in general,"},{"Start":"03:43.105 ","End":"03:46.265","Text":"it\u0027s called the Jacobian."},{"Start":"03:46.265 ","End":"03:48.550","Text":"That will be in the next chapter,"},{"Start":"03:48.550 ","End":"03:50.725","Text":"but you will have heard the name."},{"Start":"03:50.725 ","End":"03:55.914","Text":"Anyway, just bear in mind that you substitute x, you substitute y,"},{"Start":"03:55.914 ","End":"04:00.150","Text":"you put the rd Theta instead of dA or dxdy,"},{"Start":"04:00.150 ","End":"04:03.520","Text":"and you have this extra r. You also have"},{"Start":"04:03.520 ","End":"04:06.890","Text":"to describe the region that was described in terms of x,"},{"Start":"04:06.890 ","End":"04:09.370","Text":"y now in terms of r and Theta."},{"Start":"04:09.370 ","End":"04:13.460","Text":"It\u0027s typically used when the region is circular in nature,"},{"Start":"04:13.460 ","End":"04:16.980","Text":"or part of a circle or something like a circle."},{"Start":"04:18.070 ","End":"04:23.190","Text":"It\u0027s time for an example where we\u0027ll make things the clearest."},{"Start":"04:23.420 ","End":"04:31.740","Text":"For our example, let\u0027s take the double integral over R,"},{"Start":"04:31.740 ","End":"04:34.080","Text":"and I\u0027ll say what R is in a moment,"},{"Start":"04:34.080 ","End":"04:42.930","Text":"of x squared plus y squared plus 1 dA and R,"},{"Start":"04:42.930 ","End":"04:44.805","Text":"I\u0027ll sketch it for you."},{"Start":"04:44.805 ","End":"04:51.850","Text":"Here is the region R. It\u0027s the area between these 2 circles."},{"Start":"04:51.850 ","End":"04:53.510","Text":"This shape by the way,"},{"Start":"04:53.510 ","End":"04:56.735","Text":"is sometimes called an annulus."},{"Start":"04:56.735 ","End":"05:02.455","Text":"That\u0027s the Latin for a ring-shaped, donut-shaped."},{"Start":"05:02.455 ","End":"05:04.950","Text":"This is a circle of radius 1,"},{"Start":"05:04.950 ","End":"05:06.500","Text":"this is a circle of radius 2."},{"Start":"05:06.500 ","End":"05:08.420","Text":"Let\u0027s give the equations."},{"Start":"05:08.420 ","End":"05:13.745","Text":"The inner 1 would be x squared plus y squared equals 1,"},{"Start":"05:13.745 ","End":"05:19.190","Text":"and the outer 1 x squared plus y squared equals 2 squared or 4."},{"Start":"05:19.190 ","End":"05:23.030","Text":"This would be quite a difficult integral to do if we were going to do it in"},{"Start":"05:23.030 ","End":"05:29.685","Text":"Cartesian terms because where do x and y go from and to?"},{"Start":"05:29.685 ","End":"05:31.410","Text":"We\u0027d have to break it up into regions,"},{"Start":"05:31.410 ","End":"05:34.100","Text":"maybe we\u0027d break it up into 1 piece here,"},{"Start":"05:34.100 ","End":"05:37.145","Text":"then another piece above and below"},{"Start":"05:37.145 ","End":"05:38.990","Text":"these 4 pieces and then"},{"Start":"05:38.990 ","End":"05:42.830","Text":"the upper and lower limits would have square roots and all kinds of things."},{"Start":"05:42.830 ","End":"05:45.860","Text":"It would be quite complicated but in polar,"},{"Start":"05:45.860 ","End":"05:48.460","Text":"it turns out that it\u0027s quite easy."},{"Start":"05:48.460 ","End":"05:53.085","Text":"There\u0027s another formula from polar coordinates."},{"Start":"05:53.085 ","End":"05:55.220","Text":"I actually don\u0027t need the ones I erased,"},{"Start":"05:55.220 ","End":"05:56.390","Text":"x equals r cosine Theta,"},{"Start":"05:56.390 ","End":"05:57.920","Text":"y equals r sine Theta."},{"Start":"05:57.920 ","End":"06:00.350","Text":"There was a third formula that accompanies them,"},{"Start":"06:00.350 ","End":"06:05.145","Text":"and that is that x squared plus y squared equals r squared."},{"Start":"06:05.145 ","End":"06:07.520","Text":"This is going to be the most useful 1 to"},{"Start":"06:07.520 ","End":"06:10.640","Text":"us because we already have x squared plus y squared but, of course,"},{"Start":"06:10.640 ","End":"06:14.820","Text":"you could use the other formulas and then together with the trigonometric identity,"},{"Start":"06:14.820 ","End":"06:16.760","Text":"where cosine squared plus sine squared is 1,"},{"Start":"06:16.760 ","End":"06:20.100","Text":"we wouldn\u0027t have to use this, but it\u0027s easier."},{"Start":"06:21.590 ","End":"06:25.999","Text":"The double integral, now let\u0027s leave the region for the moment."},{"Start":"06:25.999 ","End":"06:27.740","Text":"I\u0027ll fill that in."},{"Start":"06:27.740 ","End":"06:29.895","Text":"That\u0027s constraint on the function."},{"Start":"06:29.895 ","End":"06:31.460","Text":"The function, as we said,"},{"Start":"06:31.460 ","End":"06:34.175","Text":"x squared plus y squared is r squared."},{"Start":"06:34.175 ","End":"06:36.755","Text":"I still have the plus 1,"},{"Start":"06:36.755 ","End":"06:42.060","Text":"and dA is rdrd Theta."},{"Start":"06:45.530 ","End":"06:49.650","Text":"Now, what about the region?"},{"Start":"06:49.650 ","End":"06:51.270","Text":"In case of polar,"},{"Start":"06:51.270 ","End":"06:56.570","Text":"d Theta is the outer integral and dr is the inner integral."},{"Start":"06:56.570 ","End":"07:02.160","Text":"We see what Theta goes from and to."},{"Start":"07:02.160 ","End":"07:09.110","Text":"In this case, it\u0027s clear that Theta is a whole circle from 0-360 degrees,"},{"Start":"07:09.110 ","End":"07:10.684","Text":"but we don\u0027t use degrees."},{"Start":"07:10.684 ","End":"07:14.610","Text":"So it\u0027s from 0-2Pi."},{"Start":"07:14.610 ","End":"07:19.510","Text":"So from 0-2Pi, that\u0027s with the d Theta."},{"Start":"07:21.170 ","End":"07:24.675","Text":"Let\u0027s take a typical ray,"},{"Start":"07:24.675 ","End":"07:30.280","Text":"and let\u0027s say this angle is Theta, this angle here."},{"Start":"07:31.760 ","End":"07:38.420","Text":"It\u0027s like an arrow and pierces it goes in here and out here."},{"Start":"07:38.420 ","End":"07:40.865","Text":"These are the limits for r,"},{"Start":"07:40.865 ","End":"07:43.140","Text":"r goes from, well,"},{"Start":"07:43.140 ","End":"07:48.675","Text":"it\u0027s a constant 1, it\u0027s always from 1-2 wherever you do this."},{"Start":"07:48.675 ","End":"07:52.875","Text":"Here a 1 and here a 2."},{"Start":"07:52.875 ","End":"07:57.385","Text":"This is already the conversion to polar,"},{"Start":"07:57.385 ","End":"08:02.675","Text":"and now we just have to evaluate it but the main theory has been applied here,"},{"Start":"08:02.675 ","End":"08:04.675","Text":"rest of it is just technical."},{"Start":"08:04.675 ","End":"08:06.350","Text":"We do this pretty easily."},{"Start":"08:06.350 ","End":"08:07.640","Text":"Let\u0027s expand."},{"Start":"08:07.640 ","End":"08:09.964","Text":"We\u0027ve got the double integral."},{"Start":"08:09.964 ","End":"08:16.740","Text":"We\u0027ll have r cubed plus r. This is still drd Theta,"},{"Start":"08:16.740 ","End":"08:20.010","Text":"1-2 for r, 0-2Pi,"},{"Start":"08:20.010 ","End":"08:23.760","Text":"for Theta, and we\u0027ll start with the inner integral."},{"Start":"08:23.760 ","End":"08:30.719","Text":"I\u0027ll just highlight it just to emphasize this is what we\u0027re doing first."},{"Start":"08:31.460 ","End":"08:34.110","Text":"This is a polynomial."},{"Start":"08:34.110 ","End":"08:44.669","Text":"So this integral is r^4 over 4 plus r squared over 2 taken from 1-2,"},{"Start":"08:44.669 ","End":"08:50.805","Text":"and yeah, we still have the outer 2Pi d Theta."},{"Start":"08:50.805 ","End":"08:55.170","Text":"If we plug in r equals 2 here,"},{"Start":"08:55.170 ","End":"08:56.700","Text":"what do we get?"},{"Start":"08:56.700 ","End":"09:01.830","Text":"2^4 is 16/4 is 4."},{"Start":"09:01.830 ","End":"09:04.910","Text":"2 squared is 4/4 is 2."},{"Start":"09:04.910 ","End":"09:06.335","Text":"That\u0027s the plus part."},{"Start":"09:06.335 ","End":"09:08.630","Text":"Now, the minus part, plug-in 1,"},{"Start":"09:08.630 ","End":"09:15.865","Text":"it becomes 1/4 plus 1/2."},{"Start":"09:15.865 ","End":"09:23.105","Text":"This is d Theta integral from 0-2Pi."},{"Start":"09:23.105 ","End":"09:27.244","Text":"Next, just evaluate this, just numbers."},{"Start":"09:27.244 ","End":"09:31.835","Text":"What do we have here? 6 minus 3/4."},{"Start":"09:31.835 ","End":"09:34.670","Text":"So I\u0027ll write it as a mixed number first,"},{"Start":"09:34.670 ","End":"09:41.555","Text":"it\u0027s 5 and 1/4 The integral from 0-2Pi d Theta,"},{"Start":"09:41.555 ","End":"09:44.000","Text":"and I think we\u0027ll leave it at that."},{"Start":"09:44.000 ","End":"09:45.740","Text":"That was that example."},{"Start":"09:45.740 ","End":"09:47.845","Text":"Now, let\u0027s do another example."},{"Start":"09:47.845 ","End":"09:49.730","Text":"Then the second example,"},{"Start":"09:49.730 ","End":"09:55.910","Text":"we\u0027re going to keep this function but change the region r. Here it is."},{"Start":"09:55.910 ","End":"09:58.985","Text":"This time, we want a quarter of a circle."},{"Start":"09:58.985 ","End":"10:03.555","Text":"Let\u0027s see how this changes the limits here,"},{"Start":"10:03.555 ","End":"10:06.780","Text":"the ranges for r and Theta."},{"Start":"10:06.780 ","End":"10:14.150","Text":"As I said, we start with Theta as the outer loop and Theta goes from here to here,"},{"Start":"10:14.150 ","End":"10:17.660","Text":"which is from 0-90 degrees,"},{"Start":"10:17.660 ","End":"10:21.080","Text":"but in radians it\u0027s 0-Pi over 2."},{"Start":"10:21.080 ","End":"10:29.110","Text":"This goes, and this goes, and Pi/2, Pi/2."},{"Start":"10:29.110 ","End":"10:30.720","Text":"Now, what about r?"},{"Start":"10:30.720 ","End":"10:34.065","Text":"Well, let\u0027s take a typical Theta."},{"Start":"10:34.065 ","End":"10:35.980","Text":"Say this is Theta,"},{"Start":"10:35.980 ","End":"10:39.440","Text":"just this path, I\u0027ll make it a bit thicker,"},{"Start":"10:39.440 ","End":"10:45.695","Text":"and r goes from here to here and whatever Theta is,"},{"Start":"10:45.695 ","End":"10:48.680","Text":"we see that r goes from 0-2."},{"Start":"10:49.790 ","End":"10:56.490","Text":"Just have to erase this 1 and replace it with 0."},{"Start":"10:56.490 ","End":"10:58.710","Text":"Let\u0027s continue."},{"Start":"10:58.710 ","End":"11:04.500","Text":"Actually, I could have kept the integral. Never mind."},{"Start":"11:04.500 ","End":"11:09.690","Text":"Write it again, r^4 over 4 plus r squared over 2."},{"Start":"11:09.690 ","End":"11:14.220","Text":"This time from 0-2."},{"Start":"11:14.220 ","End":"11:17.649","Text":"I\u0027m just doing this first."},{"Start":"11:19.550 ","End":"11:22.830","Text":"This, when r is 0, gives 0."},{"Start":"11:22.830 ","End":"11:24.785","Text":"We just have to plug in the 2."},{"Start":"11:24.785 ","End":"11:26.600","Text":"So what we get, well,"},{"Start":"11:26.600 ","End":"11:28.279","Text":"it\u0027s the same as what we got before."},{"Start":"11:28.279 ","End":"11:31.700","Text":"We got 16/4 is 4 and here,"},{"Start":"11:31.700 ","End":"11:34.250","Text":"we got 2 squared over 2 is 2,"},{"Start":"11:34.250 ","End":"11:35.975","Text":"and this was 6."},{"Start":"11:35.975 ","End":"11:39.280","Text":"Now, I can go back here."},{"Start":"11:39.280 ","End":"11:43.475","Text":"We\u0027ve got the integral from 0-Pi over 2."},{"Start":"11:43.475 ","End":"11:46.055","Text":"This bit is this bit here."},{"Start":"11:46.055 ","End":"11:49.085","Text":"So we have 6d Theta,"},{"Start":"11:49.085 ","End":"11:52.290","Text":"and I think we\u0027ll leave it at that."}],"ID":8694},{"Watched":false,"Name":"Double Integrals, Polar Coordinates (cont)","Duration":"17m 19s","ChapterTopicVideoID":8477,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8477.jpeg","UploadDate":"2020-02-26T12:02:51.7600000","DurationForVideoObject":"PT17M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"Now let\u0027s make things just a little bit more complicated."},{"Start":"00:04.890 ","End":"00:09.180","Text":"Up till now, we were given the region R in terms of"},{"Start":"00:09.180 ","End":"00:13.620","Text":"a diagram like in this previous example,"},{"Start":"00:13.620 ","End":"00:19.095","Text":"but they were all like that but we didn\u0027t get x and y in terms of limits,"},{"Start":"00:19.095 ","End":"00:20.385","Text":"we just got a picture."},{"Start":"00:20.385 ","End":"00:24.165","Text":"But in practice, this is not the problem you\u0027re given."},{"Start":"00:24.165 ","End":"00:26.085","Text":"You\u0027re actually given well,"},{"Start":"00:26.085 ","End":"00:28.095","Text":"you know what, I\u0027ll just show you."},{"Start":"00:28.095 ","End":"00:32.745","Text":"In this example, I\u0027m not going to give you a picture of a region."},{"Start":"00:32.745 ","End":"00:39.570","Text":"Instead, I\u0027m going to give you the limits of integration from 0 to 2 dx."},{"Start":"00:39.570 ","End":"00:42.405","Text":"Well, yeah, I need to replace the dA,"},{"Start":"00:42.405 ","End":"00:45.270","Text":"and in this case it\u0027s going to be dy dx."},{"Start":"00:45.270 ","End":"00:47.185","Text":"x goes from 0 to 2."},{"Start":"00:47.185 ","End":"00:52.400","Text":"It\u0027s also okay to write x equals 0 to 2 and y"},{"Start":"00:52.400 ","End":"00:57.875","Text":"goes from 0 to square root of 4 minus x squared."},{"Start":"00:57.875 ","End":"01:03.290","Text":"This goes with the dy and then the outer loop is dx."},{"Start":"01:03.290 ","End":"01:05.750","Text":"We still need a picture,"},{"Start":"01:05.750 ","End":"01:09.230","Text":"a sketch, but we have to do it ourselves."},{"Start":"01:09.230 ","End":"01:11.540","Text":"We take a look at this and see."},{"Start":"01:11.540 ","End":"01:18.090","Text":"The outer loop is on x from 0 to 2."},{"Start":"01:18.090 ","End":"01:21.585","Text":"I mark 0 and 2."},{"Start":"01:21.585 ","End":"01:25.655","Text":"For each x, I have to draw 2 functions,"},{"Start":"01:25.655 ","End":"01:30.200","Text":"y equals 0 and y equals square root of 4 minus x squared."},{"Start":"01:30.200 ","End":"01:35.285","Text":"Now y equals 0, that\u0027s just the x axis."},{"Start":"01:35.285 ","End":"01:36.470","Text":"But what about this,"},{"Start":"01:36.470 ","End":"01:39.245","Text":"y equals square root of 4 minus x squared?"},{"Start":"01:39.245 ","End":"01:42.770","Text":"I recognize it as an upper semicircle,"},{"Start":"01:42.770 ","End":"01:46.590","Text":"but if you\u0027re not sure, we can do a side calculation."},{"Start":"01:46.610 ","End":"01:52.400","Text":"y equals square root of 4 minus x squared."},{"Start":"01:52.400 ","End":"01:56.675","Text":"Square both sides, y squared is 4 minus x squared."},{"Start":"01:56.675 ","End":"02:04.110","Text":"x squared plus y squared equals 4 and 4 is 2 squared of course."},{"Start":"02:04.110 ","End":"02:08.525","Text":"This looks like a circle of radius 2,"},{"Start":"02:08.525 ","End":"02:11.600","Text":"but because we have the positive square root,"},{"Start":"02:11.600 ","End":"02:15.780","Text":"then it\u0027s only the upper half of the circle."},{"Start":"02:15.780 ","End":"02:24.870","Text":"But we don\u0027t want the whole semicircle because x only goes from 0 to 2."},{"Start":"02:25.300 ","End":"02:27.920","Text":"I should have shaded this."},{"Start":"02:27.920 ","End":"02:36.060","Text":"Yes, y goes from 0 to square root of 4 minus x squared is y equals 0."},{"Start":"02:36.060 ","End":"02:40.265","Text":"Here\u0027s y equals square root of 4 minus x squared."},{"Start":"02:40.265 ","End":"02:42.350","Text":"I want to shade it,"},{"Start":"02:42.350 ","End":"02:51.010","Text":"and that\u0027s label it R. This is actually the example we had just a few moments ago."},{"Start":"02:51.010 ","End":"02:54.265","Text":"The quarter circle, I\u0027ll just remind you what we do now is"},{"Start":"02:54.265 ","End":"02:59.125","Text":"rewrite this instead of x and y in terms of polar coordinates."},{"Start":"02:59.125 ","End":"03:04.175","Text":"We see that Theta goes from 0 to Pi over 2,"},{"Start":"03:04.175 ","End":"03:06.780","Text":"0 to 90 degrees."},{"Start":"03:06.780 ","End":"03:13.800","Text":"R goes in each case from 0 to 2."},{"Start":"03:13.800 ","End":"03:20.530","Text":"This becomes the integral from 0 to Pi over 2 d Theta,"},{"Start":"03:20.530 ","End":"03:23.200","Text":"like I said, Theta is always the outer loop."},{"Start":"03:23.200 ","End":"03:29.865","Text":"The inner loop is from 0 to 2."},{"Start":"03:29.865 ","End":"03:31.370","Text":"We could write in there,"},{"Start":"03:31.370 ","End":"03:33.305","Text":"sometimes do this just for emphasis,"},{"Start":"03:33.305 ","End":"03:36.920","Text":"r goes from 0 to 2 dr."},{"Start":"03:36.920 ","End":"03:38.765","Text":"The same thing as we had before."},{"Start":"03:38.765 ","End":"03:44.250","Text":"X squared plus y squared is r squared plus 1."},{"Start":"03:44.250 ","End":"03:46.190","Text":"Then do not forget,"},{"Start":"03:46.190 ","End":"03:48.760","Text":"there is this r here,"},{"Start":"03:48.760 ","End":"03:51.270","Text":"dy dx is da,"},{"Start":"03:51.270 ","End":"03:53.180","Text":"which is r dr d Theta."},{"Start":"03:53.180 ","End":"03:57.140","Text":"This thing always occurs as a unit r dr d Theta."},{"Start":"03:57.140 ","End":"04:04.235","Text":"The most important and most difficult part of these exercises is to convert"},{"Start":"04:04.235 ","End":"04:11.510","Text":"the region or domain from x and y terms to r and Theta terms."},{"Start":"04:11.510 ","End":"04:15.275","Text":"The function doesn\u0027t really play an important part."},{"Start":"04:15.275 ","End":"04:17.240","Text":"I mean, if you have to take care of that,"},{"Start":"04:17.240 ","End":"04:19.630","Text":"but that\u0027s not that important."},{"Start":"04:19.630 ","End":"04:23.540","Text":"I\u0027m going to take another example by varying this and I\u0027ll keep our same function,"},{"Start":"04:23.540 ","End":"04:25.970","Text":"x squared plus y squared plus 1."},{"Start":"04:25.970 ","End":"04:29.525","Text":"My next example is going to be a modification of this."},{"Start":"04:29.525 ","End":"04:39.330","Text":"What would happen if I change this 0 to minus 2 is minus 2."},{"Start":"04:39.330 ","End":"04:44.585","Text":"Clearly, this time we want the full upper semicircle."},{"Start":"04:44.585 ","End":"04:47.630","Text":"How would this change?"},{"Start":"04:47.630 ","End":"04:50.900","Text":"Well, the changes just in Thetas that are going from"},{"Start":"04:50.900 ","End":"04:56.370","Text":"0 to 90 degrees up to a 180 degrees or Pi."},{"Start":"04:56.420 ","End":"05:01.035","Text":"Just erase that and that we are."},{"Start":"05:01.035 ","End":"05:04.460","Text":"This is how a change in this can produce a change in this."},{"Start":"05:04.460 ","End":"05:06.350","Text":"Let\u0027s take another example."},{"Start":"05:06.350 ","End":"05:11.480","Text":"What would happen if I now change this to 0?"},{"Start":"05:11.480 ","End":"05:12.950","Text":"How would that change it?"},{"Start":"05:12.950 ","End":"05:16.025","Text":"Well, we\u0027re going from minus 2 to 0."},{"Start":"05:16.025 ","End":"05:20.715","Text":"It\u0027s just this bit here,"},{"Start":"05:20.715 ","End":"05:23.640","Text":"just this left part of it."},{"Start":"05:23.640 ","End":"05:26.835","Text":"You just move the labels over."},{"Start":"05:26.835 ","End":"05:28.560","Text":"How would this change?"},{"Start":"05:28.560 ","End":"05:36.390","Text":"Well, this time, Theta goes from Pi over 2 to Theta equals Pi."},{"Start":"05:36.390 ","End":"05:42.720","Text":"We just have to change the 0 to Pi over 2 here."},{"Start":"05:42.720 ","End":"05:45.375","Text":"That\u0027s another example."},{"Start":"05:45.375 ","End":"05:48.840","Text":"Another example of variation,"},{"Start":"05:48.840 ","End":"05:58.160","Text":"what happens if we change this 0 to minus the square root of 4 minus x squared,"},{"Start":"05:58.160 ","End":"06:00.815","Text":"just the minus of this?"},{"Start":"06:00.815 ","End":"06:04.520","Text":"Note that this is also on the same circle because it doesn\u0027t"},{"Start":"06:04.520 ","End":"06:08.645","Text":"matter if I have here plus or minus."},{"Start":"06:08.645 ","End":"06:11.525","Text":"When I square it, it comes out the same."},{"Start":"06:11.525 ","End":"06:17.565","Text":"What we get is just the reflection of this like so."},{"Start":"06:17.565 ","End":"06:24.680","Text":"This lower semicircle is y equals minus the square root of 4 minus x squared,"},{"Start":"06:24.680 ","End":"06:27.460","Text":"just like it\u0027s written here."},{"Start":"06:27.460 ","End":"06:30.875","Text":"How does this change this?"},{"Start":"06:30.875 ","End":"06:33.950","Text":"Well, we start from Theta equals Pi over 2,"},{"Start":"06:33.950 ","End":"06:35.855","Text":"but this is now wrong."},{"Start":"06:35.855 ","End":"06:38.540","Text":"We need to go all the way up to here."},{"Start":"06:38.540 ","End":"06:43.880","Text":"This is 270 degrees or 3 Pi over 2."},{"Start":"06:43.880 ","End":"06:50.525","Text":"We need to change this to 3 Pi over 2."},{"Start":"06:50.525 ","End":"06:54.140","Text":"Just to clarify, I wanted to go back to this say,"},{"Start":"06:54.140 ","End":"06:56.465","Text":"y is this equal to this."},{"Start":"06:56.465 ","End":"07:00.750","Text":"If you take any particular x from minus 2 to 0,"},{"Start":"07:00.750 ","End":"07:03.755","Text":"say this 1 here, a typical x."},{"Start":"07:03.755 ","End":"07:08.915","Text":"What we do is we take the vertical slice or vertical arrow,"},{"Start":"07:08.915 ","End":"07:14.250","Text":"and we see that it enters here and leaves the region here."},{"Start":"07:14.250 ","End":"07:17.255","Text":"That\u0027s why these are the limits."},{"Start":"07:17.255 ","End":"07:21.470","Text":"This is the minus square root and this is the plus square root."},{"Start":"07:21.470 ","End":"07:23.960","Text":"I just wanted to clarify that."},{"Start":"07:23.960 ","End":"07:25.835","Text":"In case it wasn\u0027t clear."},{"Start":"07:25.835 ","End":"07:27.740","Text":"In the next variation,"},{"Start":"07:27.740 ","End":"07:30.560","Text":"Let\u0027s change this upper limit to 0."},{"Start":"07:30.560 ","End":"07:32.630","Text":"Let\u0027s see what happens now."},{"Start":"07:32.630 ","End":"07:37.370","Text":"We need to replace the upper quarter of a circle with"},{"Start":"07:37.370 ","End":"07:43.170","Text":"the y equals 0. y equals 0 is just the x axis."},{"Start":"07:43.170 ","End":"07:50.440","Text":"This time we\u0027re going from minus the square root of this to 0,"},{"Start":"07:50.440 ","End":"07:53.980","Text":"the quarter circle that\u0027s in the lower left."},{"Start":"07:53.980 ","End":"07:56.740","Text":"How do we modify this?"},{"Start":"07:56.740 ","End":"07:59.455","Text":"Just the starting point."},{"Start":"07:59.455 ","End":"08:02.075","Text":"Instead of starting here."},{"Start":"08:02.075 ","End":"08:04.980","Text":"Theta, we\u0027ll start here,"},{"Start":"08:04.980 ","End":"08:07.260","Text":"which is Theta equals Pi,"},{"Start":"08:07.260 ","End":"08:10.695","Text":"and we go from Pi to 3Pi over 2."},{"Start":"08:10.695 ","End":"08:16.680","Text":"I just have to modify this lower limit to a Pi."},{"Start":"08:16.680 ","End":"08:21.210","Text":"That\u0027s another example, and yet another example."},{"Start":"08:21.210 ","End":"08:25.545","Text":"How about I change this from minus 2, 0,"},{"Start":"08:25.545 ","End":"08:28.545","Text":"I change it to 0, 2,"},{"Start":"08:28.545 ","End":"08:32.355","Text":"from 0 to 2 for x."},{"Start":"08:32.355 ","End":"08:35.535","Text":"Instead of here to here, it\u0027s here to here."},{"Start":"08:35.535 ","End":"08:40.575","Text":"Now, this lower limit is still on the lowest semicircle,"},{"Start":"08:40.575 ","End":"08:43.530","Text":"but it\u0027s just a different part of it."},{"Start":"08:43.530 ","End":"08:47.850","Text":"It\u0027s going to be the missing part here."},{"Start":"08:47.850 ","End":"08:51.690","Text":"This is what I\u0027m referring to or just move the labels over."},{"Start":"08:51.690 ","End":"08:54.645","Text":"You can see x goes from 0 to 2,"},{"Start":"08:54.645 ","End":"08:57.090","Text":"and for each particular x,"},{"Start":"08:57.090 ","End":"09:01.090","Text":"the y goes from"},{"Start":"09:01.850 ","End":"09:08.940","Text":"minus the square root of this to 0."},{"Start":"09:08.940 ","End":"09:12.970","Text":"This is the lower right quarter circle,"},{"Start":"09:12.970 ","End":"09:16.415","Text":"and how this affects this,"},{"Start":"09:16.415 ","End":"09:18.650","Text":"just the Theta is going to change."},{"Start":"09:18.650 ","End":"09:23.765","Text":"This is Theta equals 3Pi over 2,"},{"Start":"09:23.765 ","End":"09:26.185","Text":"and we go all the way."},{"Start":"09:26.185 ","End":"09:32.040","Text":"Now we wouldn\u0027t say Theta equals 0 because we want to increase,"},{"Start":"09:32.040 ","End":"09:34.185","Text":"but 0 is the same as 2Pi,"},{"Start":"09:34.185 ","End":"09:38.920","Text":"so I could call this line Theta equals 2Pi."},{"Start":"09:39.440 ","End":"09:42.885","Text":"I change these limits."},{"Start":"09:42.885 ","End":"09:46.515","Text":"Lower limit, 3Pi over 2,"},{"Start":"09:46.515 ","End":"09:48.840","Text":"the upper limit, 2Pi."},{"Start":"09:48.840 ","End":"09:51.420","Text":"Although there was another way you could have done it,"},{"Start":"09:51.420 ","End":"09:57.030","Text":"you could have said that this is minus Pi over 2 and this is 0. Don\u0027t leave it like this."},{"Start":"09:57.030 ","End":"09:58.740","Text":"But I was just saying as an alternative,"},{"Start":"09:58.740 ","End":"10:02.565","Text":"you could take it from minus Pi over 2 to 0."},{"Start":"10:02.565 ","End":"10:05.325","Text":"Let\u0027s really vary the problem."},{"Start":"10:05.325 ","End":"10:09.270","Text":"Up till now, all our problems have been dy,"},{"Start":"10:09.270 ","End":"10:13.410","Text":"dx, I\u0027m going to try something with a dx, dy."},{"Start":"10:13.410 ","End":"10:16.215","Text":"Note that pretty much all the work,"},{"Start":"10:16.215 ","End":"10:21.260","Text":"at least that we\u0027ve been doing is in converting the limits of"},{"Start":"10:21.260 ","End":"10:26.270","Text":"integration the region from Cartesian to polar."},{"Start":"10:26.270 ","End":"10:29.790","Text":"The actual function didn\u0027t really matter."},{"Start":"10:29.790 ","End":"10:32.775","Text":"It\u0027s important to me actually to start evaluating it,"},{"Start":"10:32.775 ","End":"10:35.205","Text":"but not in the setup phase."},{"Start":"10:35.205 ","End":"10:39.220","Text":"Why don\u0027t I just make it more general again?"},{"Start":"10:40.160 ","End":"10:46.380","Text":"We\u0027ll use the particular form of f or just leave it as f of x and y."},{"Start":"10:46.380 ","End":"10:49.875","Text":"Here, I need a bit more space,"},{"Start":"10:49.875 ","End":"10:51.630","Text":"I need what\u0027s written here,"},{"Start":"10:51.630 ","End":"11:01.485","Text":"f of r cosine Theta, r sine Theta."},{"Start":"11:01.485 ","End":"11:06.280","Text":"As I said, we\u0027ll now do some examples with dx, dy."},{"Start":"11:08.660 ","End":"11:11.895","Text":"This stuff will change also."},{"Start":"11:11.895 ","End":"11:16.560","Text":"The inner loop will be the x goes from something to something,"},{"Start":"11:16.560 ","End":"11:21.429","Text":"and the outer loop will be the loop for y goes from something to something."},{"Start":"11:21.980 ","End":"11:24.900","Text":"These are no longer valid."},{"Start":"11:24.900 ","End":"11:26.340","Text":"But in the polar part,"},{"Start":"11:26.340 ","End":"11:30.030","Text":"it will always be Theta on the outside and r on the inside."},{"Start":"11:30.030 ","End":"11:37.155","Text":"This time I\u0027m going to let the outer loop y go from 0 to 2."},{"Start":"11:37.155 ","End":"11:39.075","Text":"For the inner loop,"},{"Start":"11:39.075 ","End":"11:44.295","Text":"x will go from 0 to"},{"Start":"11:44.295 ","End":"11:50.310","Text":"square root of 4 minus y squared."},{"Start":"11:50.310 ","End":"11:57.135","Text":"How to loop y and loop x?"},{"Start":"11:57.135 ","End":"12:01.980","Text":"This means that we\u0027re going to take horizontal slices this time."},{"Start":"12:01.980 ","End":"12:04.290","Text":"Let\u0027s see if we can sketch this."},{"Start":"12:04.290 ","End":"12:06.330","Text":"Y goes from 0 to 2,"},{"Start":"12:06.330 ","End":"12:09.210","Text":"here is 0, here is 2."},{"Start":"12:09.210 ","End":"12:12.180","Text":"X goes from 0,"},{"Start":"12:12.180 ","End":"12:14.520","Text":"x equals 0 is the y-axis."},{"Start":"12:14.520 ","End":"12:17.354","Text":"Write it over here, x equals 0."},{"Start":"12:17.354 ","End":"12:19.665","Text":"What is this function,"},{"Start":"12:19.665 ","End":"12:22.245","Text":"square root of 4 minus y squared?"},{"Start":"12:22.245 ","End":"12:25.310","Text":"I claim it\u0027s the right semicircle,"},{"Start":"12:25.310 ","End":"12:27.260","Text":"the same radius 2, because look,"},{"Start":"12:27.260 ","End":"12:30.935","Text":"if I just reverse the roles of x and y here,"},{"Start":"12:30.935 ","End":"12:35.815","Text":"we\u0027re on the same radius to circle part of it."},{"Start":"12:35.815 ","End":"12:38.955","Text":"This is now our region r,"},{"Start":"12:38.955 ","End":"12:41.400","Text":"just to show you,"},{"Start":"12:41.400 ","End":"12:45.660","Text":"see that y goes from 0 to 2."},{"Start":"12:45.660 ","End":"12:50.670","Text":"For each y in this range from 0-2,"},{"Start":"12:50.670 ","End":"13:01.180","Text":"we go from 0 up to the semicircle here."},{"Start":"13:01.180 ","End":"13:03.060","Text":"It\u0027s a plus here."},{"Start":"13:03.060 ","End":"13:04.260","Text":"Well, you don\u0027t see the plus,"},{"Start":"13:04.260 ","End":"13:08.670","Text":"which means it\u0027s the right-hand side of the y-axis."},{"Start":"13:08.670 ","End":"13:11.340","Text":"This really is the r region,"},{"Start":"13:11.340 ","End":"13:14.910","Text":"and now we just have to describe it in polar."},{"Start":"13:14.910 ","End":"13:17.010","Text":"Well, we\u0027ve seen this before."},{"Start":"13:17.010 ","End":"13:22.185","Text":"Theta goes from 0 to Pi over 2,"},{"Start":"13:22.185 ","End":"13:25.170","Text":"and r goes from 0 to 2,"},{"Start":"13:25.170 ","End":"13:30.540","Text":"so 0 to 2 for r,"},{"Start":"13:30.540 ","End":"13:34.950","Text":"Theta 0 to Pi over 2."},{"Start":"13:34.950 ","End":"13:38.475","Text":"Now we\u0027ve converted this to polar."},{"Start":"13:38.475 ","End":"13:44.530","Text":"Let\u0027s see what happens to the region if I change this 0 to a minus 2,"},{"Start":"13:45.710 ","End":"13:49.080","Text":"and I\u0027ll put a minus 2 here."},{"Start":"13:49.080 ","End":"13:55.530","Text":"I think it\u0027s pretty clear that this is still the same right semicircle."},{"Start":"13:55.530 ","End":"13:58.470","Text":"It\u0027s just that instead of taking it only from 2 to 0,"},{"Start":"13:58.470 ","End":"14:01.330","Text":"we continue up to minus 2."},{"Start":"14:01.580 ","End":"14:04.380","Text":"This is our region,"},{"Start":"14:04.380 ","End":"14:08.370","Text":"the right semicircle, semi disk."},{"Start":"14:08.370 ","End":"14:14.850","Text":"The question is now how to change the region for Theta and r?"},{"Start":"14:14.850 ","End":"14:17.340","Text":"Well, r still goes from 0 to 2,"},{"Start":"14:17.340 ","End":"14:20.430","Text":"but Theta is now this."},{"Start":"14:20.430 ","End":"14:23.085","Text":"There\u0027s more than 1 way to do this."},{"Start":"14:23.085 ","End":"14:27.465","Text":"If you want to stick to Theta always between 0 and 2Pi,"},{"Start":"14:27.465 ","End":"14:30.330","Text":"then you would have to break it up into 2 bits,"},{"Start":"14:30.330 ","End":"14:32.460","Text":"from 0 to Pi over 2,"},{"Start":"14:32.460 ","End":"14:40.740","Text":"and then from 3Pi over 2 to 2Pi 0 when 2Pi is the same."},{"Start":"14:40.740 ","End":"14:43.605","Text":"1 other way to do it would be to start here,"},{"Start":"14:43.605 ","End":"14:45.390","Text":"at 3Pi over 2."},{"Start":"14:45.390 ","End":"14:48.165","Text":"But then since we\u0027re going in the positive direction,"},{"Start":"14:48.165 ","End":"14:55.860","Text":"this would now be 5Pi over 2, more than 2pi."},{"Start":"14:55.860 ","End":"15:01.800","Text":"The way I would recommend would be to use"},{"Start":"15:01.800 ","End":"15:09.075","Text":"negative values so that this would still be 0 and this would be minus Pi over 2,"},{"Start":"15:09.075 ","End":"15:11.520","Text":"and this would be Pi over 2."},{"Start":"15:11.520 ","End":"15:16.020","Text":"But like I said, you could have taken 3Pi over 2 here,"},{"Start":"15:16.020 ","End":"15:21.150","Text":"and 5Pi over 2 here."},{"Start":"15:21.150 ","End":"15:24.870","Text":"I just don\u0027t like the breaking it up into 2 bits."},{"Start":"15:24.870 ","End":"15:29.955","Text":"Then like I said, I\u0027ll go with minus Pi over 2 to Pi over 2 and get rid of those."},{"Start":"15:29.955 ","End":"15:39.480","Text":"Here, I just have to replace 0 with minus Pi over 2."},{"Start":"15:39.480 ","End":"15:46.170","Text":"Now another variation, suppose I change this 2 to a 0,"},{"Start":"15:46.170 ","End":"15:48.465","Text":"I think you can see what\u0027s going to happen."},{"Start":"15:48.465 ","End":"15:56.865","Text":"We\u0027re taking all these slices like so from here to here."},{"Start":"15:56.865 ","End":"15:58.935","Text":"I forgot to label this one,"},{"Start":"15:58.935 ","End":"16:03.420","Text":"because the y-axis is x equals 0 but this semicircle on"},{"Start":"16:03.420 ","End":"16:08.795","Text":"the right is x equals the square root of what says here,"},{"Start":"16:08.795 ","End":"16:12.004","Text":"4 minus y squared."},{"Start":"16:12.004 ","End":"16:18.410","Text":"If we now have restricted from minus 2 to 0,"},{"Start":"16:18.410 ","End":"16:24.945","Text":"then we\u0027re just staying with this lower right quarter circle."},{"Start":"16:24.945 ","End":"16:28.140","Text":"This is my region."},{"Start":"16:28.140 ","End":"16:30.360","Text":"I have to modify Theta."},{"Start":"16:30.360 ","End":"16:32.070","Text":"R still goes from 0 to 2,"},{"Start":"16:32.070 ","End":"16:35.860","Text":"but Theta no longer goes here."},{"Start":"16:37.070 ","End":"16:42.130","Text":"If I keep the minus Pi over 2, then it goes up to 0."},{"Start":"16:42.390 ","End":"16:44.980","Text":"But of course there are other variations."},{"Start":"16:44.980 ","End":"16:53.030","Text":"I could\u0027ve taken it from 3Pi over 2 to 2Pi."},{"Start":"16:53.160 ","End":"16:56.410","Text":"But I\u0027ll stick with this and this,"},{"Start":"16:56.410 ","End":"17:02.320","Text":"which means that I just have to change this Pi over 2 to a 0."},{"Start":"17:02.320 ","End":"17:04.975","Text":"I think that\u0027s enough of that,"},{"Start":"17:04.975 ","End":"17:06.385","Text":"and I\u0027ll end here."},{"Start":"17:06.385 ","End":"17:09.490","Text":"But I just want to remind you that there are solved exercises."},{"Start":"17:09.490 ","End":"17:14.065","Text":"In fact, there\u0027s a lot of them of this kind following the tutorial,"},{"Start":"17:14.065 ","End":"17:19.220","Text":"and that\u0027s when you\u0027ll really learn this stuff. That\u0027s it."}],"ID":8693},{"Watched":false,"Name":"Exercise 1","Duration":"4m 17s","ChapterTopicVideoID":8490,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8490.jpeg","UploadDate":"2017-02-13T08:40:35.3030000","DurationForVideoObject":"PT4M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this exercise, we have to compute the following double integral,"},{"Start":"00:03.960 ","End":"00:06.764","Text":"where D is as in the picture,"},{"Start":"00:06.764 ","End":"00:12.120","Text":"maybe I\u0027ll label it D. This is in the chapter on polar coordinates,"},{"Start":"00:12.120 ","End":"00:15.105","Text":"which should be a hint that we want to do a polar conversion."},{"Start":"00:15.105 ","End":"00:18.015","Text":"We\u0027re going to use the standard conversion formulas."},{"Start":"00:18.015 ","End":"00:21.944","Text":"Instead of x, we put r cosine Theta,"},{"Start":"00:21.944 ","End":"00:26.580","Text":"instead of y we put r sine Theta,"},{"Start":"00:26.580 ","End":"00:28.590","Text":"and instead of dA,"},{"Start":"00:28.590 ","End":"00:34.530","Text":"we put r dr d Theta."},{"Start":"00:34.530 ","End":"00:41.060","Text":"These are the standard formulas and there\u0027s also a fourth 1 which is usually useful,"},{"Start":"00:41.060 ","End":"00:47.430","Text":"is that x squared plus y squared equals r squared."},{"Start":"00:47.500 ","End":"00:58.260","Text":"The next thing to do is to describe the region or domain D in terms of r and Theta."},{"Start":"00:59.180 ","End":"01:04.440","Text":"It\u0027s clear here that we have a full circle,"},{"Start":"01:04.440 ","End":"01:06.860","Text":"so we could start from here where Theta equals"},{"Start":"01:06.860 ","End":"01:10.430","Text":"0 and work our way around counterclockwise,"},{"Start":"01:10.430 ","End":"01:15.230","Text":"which is the positive mathematical direction and start from Theta equals"},{"Start":"01:15.230 ","End":"01:21.255","Text":"0 and end at Theta equals 360 degrees only that\u0027s 2Pi."},{"Start":"01:21.255 ","End":"01:24.950","Text":"For any particular Theta, doesn\u0027t matter where,"},{"Start":"01:24.950 ","End":"01:32.040","Text":"we\u0027re always going to get the same limits on r from 0-4,"},{"Start":"01:32.120 ","End":"01:36.180","Text":"given that the radius here is 4."},{"Start":"01:36.180 ","End":"01:39.310","Text":"We go from 0-4."},{"Start":"01:40.880 ","End":"01:43.175","Text":"After doing all this,"},{"Start":"01:43.175 ","End":"01:45.725","Text":"we get the following conversion,"},{"Start":"01:45.725 ","End":"01:50.345","Text":"we get the double integral and instead of D,"},{"Start":"01:50.345 ","End":"01:53.149","Text":"we have it described in polar terms,"},{"Start":"01:53.149 ","End":"01:57.420","Text":"where r goes from 0-4,"},{"Start":"01:57.500 ","End":"02:02.100","Text":"Theta goes from 0-2Pi,"},{"Start":"02:02.100 ","End":"02:04.069","Text":"that\u0027s describe the region."},{"Start":"02:04.069 ","End":"02:09.095","Text":"Now we have the square root and then we\u0027re going to use this useful ones of"},{"Start":"02:09.095 ","End":"02:18.480","Text":"r-squared and dA is r dr d Theta."},{"Start":"02:20.420 ","End":"02:26.800","Text":"Now the square root of r squared is r and r times r is r squared,"},{"Start":"02:26.800 ","End":"02:35.445","Text":"so I get the double integral of r squared dr d Theta,"},{"Start":"02:35.445 ","End":"02:40.010","Text":"write the limits 0-4 and 0-2Pi."},{"Start":"02:40.010 ","End":"02:46.310","Text":"As usual, we do the inner integral first, this 1 here."},{"Start":"02:46.310 ","End":"02:48.225","Text":"I\u0027ll do this 1,"},{"Start":"02:48.225 ","End":"02:49.890","Text":"the side, I\u0027ll call it asterisk."},{"Start":"02:49.890 ","End":"02:51.330","Text":"I\u0027ll do this over here,"},{"Start":"02:51.330 ","End":"02:54.295","Text":"not to mess with the main development."},{"Start":"02:54.295 ","End":"02:57.870","Text":"This would be, let\u0027s see,"},{"Start":"02:57.870 ","End":"03:03.300","Text":"integral of r squared dr is just 1/3 r cubed and"},{"Start":"03:03.300 ","End":"03:09.450","Text":"we have to evaluate this between 0 and 4."},{"Start":"03:09.450 ","End":"03:11.744","Text":"When we plug in 4,"},{"Start":"03:11.744 ","End":"03:18.690","Text":"we get 4 cubed is 64/3, when we plug in 0,"},{"Start":"03:18.690 ","End":"03:23.595","Text":"we just get 0 and so putting this back here,"},{"Start":"03:23.595 ","End":"03:29.280","Text":"we get the integral from 0-2Pi of"},{"Start":"03:29.280 ","End":"03:35.040","Text":"64 over 3 d Theta."},{"Start":"03:35.040 ","End":"03:44.900","Text":"Of course, I can take the 64/3 outside the integral and get the"},{"Start":"03:44.900 ","End":"03:48.755","Text":"integral of 1d Theta is just"},{"Start":"03:48.755 ","End":"03:55.605","Text":"Theta evaluated between 0 and 2Pi."},{"Start":"03:55.605 ","End":"03:59.310","Text":"This is just 2Pi minus 0, which is 2Pi,"},{"Start":"03:59.310 ","End":"04:04.960","Text":"so I get 64/3 times 2Pi."},{"Start":"04:05.600 ","End":"04:12.850","Text":"I can write this as 128Pi over 3,"},{"Start":"04:12.850 ","End":"04:17.580","Text":"and I\u0027ll highlight it and that is our answer and we\u0027re done."}],"ID":8714},{"Watched":false,"Name":"Exercise 2","Duration":"3m 14s","ChapterTopicVideoID":8491,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8491.jpeg","UploadDate":"2017-02-13T08:40:57.2870000","DurationForVideoObject":"PT3M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.560","Text":"In this exercise, we have to compute the following double integral."},{"Start":"00:05.720 ","End":"00:09.450","Text":"Where D is the domain described here."},{"Start":"00:09.450 ","End":"00:12.540","Text":"It\u0027s a quarter circle of radius 4."},{"Start":"00:12.540 ","End":"00:15.960","Text":"We\u0027re going to do it using polar coordinates."},{"Start":"00:15.960 ","End":"00:18.480","Text":"The standard formulas are,"},{"Start":"00:18.480 ","End":"00:21.780","Text":"I just copy-pasted them from the previous exercise."},{"Start":"00:21.780 ","End":"00:23.100","Text":"These 3 are mandatory,"},{"Start":"00:23.100 ","End":"00:26.739","Text":"this is useful for the equation."},{"Start":"00:26.750 ","End":"00:33.970","Text":"What we have to do is to define D in terms of r and Theta."},{"Start":"00:34.100 ","End":"00:42.570","Text":"This is the domain D. Let\u0027s look at Theta first quarter circle."},{"Start":"00:42.570 ","End":"00:45.885","Text":"Why don\u0027t I just take Theta this way."},{"Start":"00:45.885 ","End":"00:49.445","Text":"Here Theta is 0."},{"Start":"00:49.445 ","End":"00:52.340","Text":"The positive direction in mathematics is counterclockwise."},{"Start":"00:52.340 ","End":"00:53.810","Text":"Here Theta\u0027s 90 degrees,"},{"Start":"00:53.810 ","End":"00:55.585","Text":"but we\u0027re working in radians."},{"Start":"00:55.585 ","End":"00:58.940","Text":"Here we have Theta is Pi over 2,"},{"Start":"00:58.940 ","End":"01:00.575","Text":"that gives us Theta."},{"Start":"01:00.575 ","End":"01:06.230","Text":"For any particular Theta r always goes from 0 to 4."},{"Start":"01:06.230 ","End":"01:08.810","Text":"The constant doesn\u0027t depend on Theta."},{"Start":"01:08.810 ","End":"01:11.440","Text":"Now we can do the conversion."},{"Start":"01:11.440 ","End":"01:17.090","Text":"Describing D as an iterated integral in dr_d Theta,"},{"Start":"01:17.090 ","End":"01:25.240","Text":"we have that r goes from 0 to 4."},{"Start":"01:25.240 ","End":"01:29.739","Text":"Theta goes from 0 to 90 degrees,"},{"Start":"01:29.739 ","End":"01:32.075","Text":"which is Pi over 2."},{"Start":"01:32.075 ","End":"01:36.375","Text":"Square root of x squared plus y squared here\u0027s this useful 1."},{"Start":"01:36.375 ","End":"01:41.625","Text":"It\u0027s the square root of R squared and"},{"Start":"01:41.625 ","End":"01:48.130","Text":"dA is r dr_d Theta."},{"Start":"01:48.130 ","End":"01:50.135","Text":"Let\u0027s just rewrite this."},{"Start":"01:50.135 ","End":"01:51.530","Text":"Square root of r squared is r,"},{"Start":"01:51.530 ","End":"01:53.495","Text":"r with r is r squared."},{"Start":"01:53.495 ","End":"01:57.965","Text":"We get the integral from 0 to Pi over 2,"},{"Start":"01:57.965 ","End":"02:08.050","Text":"integral from 0-4 of r squared dr_d Theta."},{"Start":"02:08.050 ","End":"02:12.355","Text":"We evaluate the inner integral first."},{"Start":"02:12.355 ","End":"02:15.850","Text":"I mean this 1."},{"Start":"02:16.370 ","End":"02:19.039","Text":"This inner bit we\u0027ve done previously,"},{"Start":"02:19.039 ","End":"02:20.930","Text":"but even if not, we can do it in our heads."},{"Start":"02:20.930 ","End":"02:23.495","Text":"Look, this is r cubed over 3."},{"Start":"02:23.495 ","End":"02:25.819","Text":"When we put in 4 and 0,"},{"Start":"02:25.819 ","End":"02:28.835","Text":"we get 64 over 3 minus 0."},{"Start":"02:28.835 ","End":"02:33.030","Text":"This bit is just 64 over 3."},{"Start":"02:33.030 ","End":"02:38.825","Text":"We have the integral from 0 to Pi over 2,"},{"Start":"02:38.825 ","End":"02:42.290","Text":"64 over 3d Theta."},{"Start":"02:42.290 ","End":"02:45.860","Text":"The 64 over 3 comes out in front."},{"Start":"02:45.860 ","End":"02:52.255","Text":"The integral from 0 to Pi over 2 of 1d Theta is just Pi over 2 minus 0,"},{"Start":"02:52.255 ","End":"02:55.810","Text":"which is Pi over 2."},{"Start":"02:58.400 ","End":"03:00.675","Text":"The final answer would be,"},{"Start":"03:00.675 ","End":"03:04.455","Text":"2 goes into 64, 32 times."},{"Start":"03:04.455 ","End":"03:10.265","Text":"We get 32 Pi over 3."},{"Start":"03:10.265 ","End":"03:14.100","Text":"I\u0027ll highlight it and we are done."}],"ID":8715},{"Watched":false,"Name":"Exercise 3","Duration":"2m 27s","ChapterTopicVideoID":8492,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8492.jpeg","UploadDate":"2017-02-13T08:41:14.1570000","DurationForVideoObject":"PT2M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"Here we have to compute this double integral where d is as in the sketch,"},{"Start":"00:04.860 ","End":"00:09.570","Text":"it\u0027s the semicircle, I\u0027ll just label it D,"},{"Start":"00:09.570 ","End":"00:12.445","Text":"and we\u0027re going to do it in polar coordinates."},{"Start":"00:12.445 ","End":"00:15.900","Text":"I copy pasted all the formulas we need from"},{"Start":"00:15.900 ","End":"00:20.400","Text":"the previous exercise and we just have to decide what is D,"},{"Start":"00:20.400 ","End":"00:23.310","Text":"how to describe it in terms of r and Theta."},{"Start":"00:23.310 ","End":"00:28.620","Text":"Well, Theta, we could take from here to here."},{"Start":"00:28.620 ","End":"00:34.410","Text":"Here Theta is equal to 0 and here Theta is 180 degrees,"},{"Start":"00:34.410 ","End":"00:35.625","Text":"in other words, Pi,"},{"Start":"00:35.625 ","End":"00:37.335","Text":"because we work in radians."},{"Start":"00:37.335 ","End":"00:40.854","Text":"As for r, for any given Theta,"},{"Start":"00:40.854 ","End":"00:48.245","Text":"r goes from the origin where r is 0 up to 4 because we\u0027re told that the radius is 4."},{"Start":"00:48.245 ","End":"00:54.460","Text":"Now we can write this as the double integral,"},{"Start":"00:54.460 ","End":"00:58.380","Text":"and I\u0027ll write the limits"},{"Start":"00:58.380 ","End":"01:06.330","Text":"that r goes from 0 to 4."},{"Start":"01:06.330 ","End":"01:10.420","Text":"Theta goes from 0 to Pi,"},{"Start":"01:10.420 ","End":"01:13.850","Text":"the square root of x squared plus y squared,"},{"Start":"01:13.850 ","End":"01:15.650","Text":"which is r squared,"},{"Start":"01:15.650 ","End":"01:22.095","Text":"and then dA is rdr, d Theta."},{"Start":"01:22.095 ","End":"01:28.070","Text":"In other words, if I just write the square root of r-squared is r combined with the dr,"},{"Start":"01:28.070 ","End":"01:30.280","Text":"so we\u0027ve got r squared here."},{"Start":"01:30.280 ","End":"01:34.995","Text":"We\u0027ve got r squared dr, d Theta,"},{"Start":"01:34.995 ","End":"01:38.580","Text":"and the integral from 0 to 4,"},{"Start":"01:38.580 ","End":"01:43.045","Text":"that\u0027s the r and from zero to Pi, that\u0027s the Theta."},{"Start":"01:43.045 ","End":"01:49.685","Text":"As usual, we do the inside 1 first, that\u0027s this 1."},{"Start":"01:49.685 ","End":"01:51.560","Text":"We\u0027ve already done this,"},{"Start":"01:51.560 ","End":"01:52.925","Text":"but I\u0027ll remind you,"},{"Start":"01:52.925 ","End":"01:55.520","Text":"the integral is r cubed over 3."},{"Start":"01:55.520 ","End":"01:59.660","Text":"When I plug in 4, I get 64 over 3 minus 0,"},{"Start":"01:59.660 ","End":"02:08.925","Text":"so this bit is 64 over 3 d Theta integral from 0 to Pi."},{"Start":"02:08.925 ","End":"02:12.560","Text":"The 64 over 3 is a constant,"},{"Start":"02:12.560 ","End":"02:14.540","Text":"it comes out front."},{"Start":"02:14.540 ","End":"02:19.390","Text":"The integral from zero to Pi of d Theta is just Pi minus 0,"},{"Start":"02:19.390 ","End":"02:23.780","Text":"which is Pi and I could leave the answer like this."},{"Start":"02:23.780 ","End":"02:28.590","Text":"But I\u0027d just like to highlight before declaring that we are done."}],"ID":8716},{"Watched":false,"Name":"Exercise 4","Duration":"3m 29s","ChapterTopicVideoID":8493,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8493.jpeg","UploadDate":"2017-02-13T08:41:38.6170000","DurationForVideoObject":"PT3M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we have to compute the following double integral."},{"Start":"00:03.450 ","End":"00:05.385","Text":"D is described in the sketch."},{"Start":"00:05.385 ","End":"00:10.110","Text":"This is D. This shape actually has a name in mathematics."},{"Start":"00:10.110 ","End":"00:17.320","Text":"It\u0027s called an annulus ring shaped between 2 concentric circles."},{"Start":"00:17.330 ","End":"00:20.880","Text":"We\u0027re going to use polar coordinates."},{"Start":"00:20.880 ","End":"00:24.975","Text":"I copied the formulas we need from the previous exercise."},{"Start":"00:24.975 ","End":"00:30.045","Text":"We have to find how to describe D in terms of r and Theta."},{"Start":"00:30.045 ","End":"00:32.985","Text":"Theta makes a full circle."},{"Start":"00:32.985 ","End":"00:39.015","Text":"Let me start at Theta equals 0 and go all the way around to here, counterclockwise."},{"Start":"00:39.015 ","End":"00:41.850","Text":"Here Theta equals 0 at the start."},{"Start":"00:41.850 ","End":"00:46.220","Text":"At the end, Theta equals 2 Pi, that\u0027s 360 degrees."},{"Start":"00:46.220 ","End":"00:52.040","Text":"As for r in any given radius,"},{"Start":"00:52.040 ","End":"00:56.510","Text":"we go from here to here."},{"Start":"00:56.510 ","End":"00:58.130","Text":"It doesn\u0027t depend on Theta."},{"Start":"00:58.130 ","End":"01:00.425","Text":"It\u0027s always from 1-4."},{"Start":"01:00.425 ","End":"01:05.970","Text":"This is 1 and this is 4 as given in the picture."},{"Start":"01:05.970 ","End":"01:12.180","Text":"We can now write this as the double integral,"},{"Start":"01:12.180 ","End":"01:18.030","Text":"Theta goes from 0-2 Pi."},{"Start":"01:18.030 ","End":"01:21.135","Text":"Actually, I\u0027ll do the Theta on the outside."},{"Start":"01:21.135 ","End":"01:25.310","Text":"Usually I preferred it that way to start off with the r,"},{"Start":"01:25.310 ","End":"01:28.800","Text":"which is from 1-4."},{"Start":"01:28.880 ","End":"01:34.460","Text":"Then this, which is the square root of x squared plus y squared from here"},{"Start":"01:34.460 ","End":"01:41.890","Text":"is r squared and dA is rdr d Theta."},{"Start":"01:41.890 ","End":"01:44.960","Text":"In principle, we could use r d Theta dr,"},{"Start":"01:44.960 ","End":"01:49.735","Text":"but it usually seems to work best with dr d Theta."},{"Start":"01:49.735 ","End":"01:55.950","Text":"We get the integral from 0-2 Pi as Theta, integral from 1-4."},{"Start":"01:55.950 ","End":"02:03.725","Text":"That\u0027s for r. Square root of r squared times r is r times r is r squared dr d Theta."},{"Start":"02:03.725 ","End":"02:06.655","Text":"We do the inner integral first."},{"Start":"02:06.655 ","End":"02:08.650","Text":"I\u0027d like to do this 1 at the slide."},{"Start":"02:08.650 ","End":"02:13.620","Text":"I\u0027ll call it asterisk."},{"Start":"02:13.620 ","End":"02:19.535","Text":"r squared gives me 1/3 r cubed."},{"Start":"02:19.535 ","End":"02:23.300","Text":"It takes this between 1 and 4."},{"Start":"02:26.570 ","End":"02:32.915","Text":"If I plug-in 4, I get 64 over 3,"},{"Start":"02:32.915 ","End":"02:36.010","Text":"I plug in 1, I get 1 over 3."},{"Start":"02:36.010 ","End":"02:41.730","Text":"I subtract, what I get is 63 over 3,"},{"Start":"02:41.730 ","End":"02:44.565","Text":"which is a whole number, 21."},{"Start":"02:44.565 ","End":"02:46.715","Text":"Now going back here,"},{"Start":"02:46.715 ","End":"02:49.060","Text":"this bit which is 21,"},{"Start":"02:49.060 ","End":"02:50.725","Text":"I can actually put in front,"},{"Start":"02:50.725 ","End":"02:58.960","Text":"so what I get is 21 times the integral from 0-2 Pi of d Theta."},{"Start":"02:58.960 ","End":"03:01.225","Text":"Let\u0027s write it as 1 d Theta."},{"Start":"03:01.225 ","End":"03:03.220","Text":"The integral of 1 is just Theta."},{"Start":"03:03.220 ","End":"03:08.850","Text":"I get Pi minus 0 is Pi. That\u0027s 2 Pi."},{"Start":"03:08.850 ","End":"03:11.250","Text":"Sorry, forgive me."},{"Start":"03:11.250 ","End":"03:13.830","Text":"2 Pi minus 0 is 2 Pi."},{"Start":"03:13.830 ","End":"03:17.175","Text":"I get 21 times 2 Pi,"},{"Start":"03:17.175 ","End":"03:24.190","Text":"and that is equal to 42 Pi."},{"Start":"03:24.740 ","End":"03:29.800","Text":"This is our answer. We\u0027re done."}],"ID":8717},{"Watched":false,"Name":"Exercise 5","Duration":"10m 5s","ChapterTopicVideoID":8494,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8494.jpeg","UploadDate":"2017-02-13T08:42:53.9630000","DurationForVideoObject":"PT10M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"Here we have to compute this double integral."},{"Start":"00:03.060 ","End":"00:07.875","Text":"D is as in the picture, it\u0027s this semicircle."},{"Start":"00:07.875 ","End":"00:10.530","Text":"There\u0027s more than 1 way to do this."},{"Start":"00:10.530 ","End":"00:14.610","Text":"Let me start out with a naive way which is not necessarily the best,"},{"Start":"00:14.610 ","End":"00:16.785","Text":"but it\u0027s 1 way."},{"Start":"00:16.785 ","End":"00:23.235","Text":"For reference I\u0027ve copy pasted the standard formulas that we\u0027re always using."},{"Start":"00:23.235 ","End":"00:28.725","Text":"Our first goal is to describe D in polar coordinates and r and Theta."},{"Start":"00:28.725 ","End":"00:35.000","Text":"The most simple-minded is to always take Theta as going from 0 to 2Pi,"},{"Start":"00:35.000 ","End":"00:36.800","Text":"which starts at the x-axis,"},{"Start":"00:36.800 ","End":"00:38.960","Text":"goes counterclockwise and 2Pi."},{"Start":"00:38.960 ","End":"00:40.550","Text":"If we did that,"},{"Start":"00:40.550 ","End":"00:43.325","Text":"then this would be Theta equals 0."},{"Start":"00:43.325 ","End":"00:51.560","Text":"This would be Theta equals Pi over 2 but this would be Theta equals 270 degrees,"},{"Start":"00:51.560 ","End":"00:54.345","Text":"which is 3Pi over 2."},{"Start":"00:54.345 ","End":"00:58.055","Text":"What we would have to do would be taken in 2 bits."},{"Start":"00:58.055 ","End":"01:01.640","Text":"We\u0027d first of all take it from here to here,"},{"Start":"01:01.640 ","End":"01:05.670","Text":"from 0 to Pi over 2 and then complete it here blank,"},{"Start":"01:05.670 ","End":"01:09.545","Text":"blank and then complete it with a bit from here to here."},{"Start":"01:09.545 ","End":"01:11.330","Text":"To take it in 2 pieces."},{"Start":"01:11.330 ","End":"01:12.890","Text":"As I said at the end,"},{"Start":"01:12.890 ","End":"01:15.690","Text":"I\u0027ll show you a shorter way of doing this."},{"Start":"01:15.830 ","End":"01:19.800","Text":"The limits for r, of course are always from"},{"Start":"01:19.800 ","End":"01:25.910","Text":"0-4 and that holds true whether we\u0027re in this part or in this part,"},{"Start":"01:25.910 ","End":"01:29.010","Text":"it\u0027s still from 0-4."},{"Start":"01:29.140 ","End":"01:37.090","Text":"We get the double integral for this bit,"},{"Start":"01:38.930 ","End":"01:41.639","Text":"I\u0027d like to take r inside,"},{"Start":"01:41.639 ","End":"01:46.335","Text":"r goes from 0-4,"},{"Start":"01:46.335 ","End":"01:53.610","Text":"Theta goes from 0 to Pi over 2 and"},{"Start":"01:53.610 ","End":"02:01.040","Text":"then the square root x-squared plus y-squared from here is r squared,"},{"Start":"02:01.040 ","End":"02:06.515","Text":"dA is rd rd Theta."},{"Start":"02:06.515 ","End":"02:13.425","Text":"We need rdrd Theta plus,"},{"Start":"02:13.425 ","End":"02:19.730","Text":"and then the other integral is going to be from 3Pi over 2."},{"Start":"02:19.730 ","End":"02:28.035","Text":"Here 0 is the same as 2Pi that means equivalent to."},{"Start":"02:28.035 ","End":"02:35.835","Text":"We take the integral from 3Pi over 2 to 2Pi,"},{"Start":"02:35.835 ","End":"02:37.440","Text":"everything else is the same,"},{"Start":"02:37.440 ","End":"02:39.660","Text":"I\u0027ll just write same."},{"Start":"02:39.660 ","End":"02:42.980","Text":"No need to do things twice."},{"Start":"02:42.980 ","End":"02:46.020","Text":"Let\u0027s just work on this 1."},{"Start":"02:46.390 ","End":"02:50.280","Text":"We start with the inner integral."},{"Start":"02:51.800 ","End":"02:55.545","Text":"Let me do this 1 at the side."},{"Start":"02:55.545 ","End":"02:58.980","Text":"Sorry, not quite yet, we\u0027re going to simplify first."},{"Start":"02:58.980 ","End":"03:03.105","Text":"Going to write this as 0 to Pi over 2."},{"Start":"03:03.105 ","End":"03:05.355","Text":"From 0-4."},{"Start":"03:05.355 ","End":"03:06.795","Text":"The square root of r-squared is r,"},{"Start":"03:06.795 ","End":"03:13.035","Text":"r times r is r squared so we have r squared dr d Theta"},{"Start":"03:13.035 ","End":"03:21.425","Text":"plus integral from 3Pi over 2 to 2Pi of the same thing."},{"Start":"03:21.425 ","End":"03:24.770","Text":"You should probably on an exam not do it like me,"},{"Start":"03:24.770 ","End":"03:25.955","Text":"write the whole thing out,"},{"Start":"03:25.955 ","End":"03:28.910","Text":"but I just want to save time."},{"Start":"03:28.910 ","End":"03:33.860","Text":"Now this is the bit I want to integrate,"},{"Start":"03:33.860 ","End":"03:35.330","Text":"I was a bit premature there."},{"Start":"03:35.330 ","End":"03:37.120","Text":"I\u0027ll do that at the side."},{"Start":"03:37.120 ","End":"03:42.750","Text":"We have the integral from 0-4 of r squared dr is"},{"Start":"03:42.750 ","End":"03:50.265","Text":"equal to r cubed over 3, from 0-4."},{"Start":"03:50.265 ","End":"03:53.160","Text":"We\u0027ve seen this before, 4 cubed is 64."},{"Start":"03:53.160 ","End":"03:57.465","Text":"This is 64 over 3."},{"Start":"03:57.465 ","End":"04:02.869","Text":"Now back here, this part I highlighted as a constant,"},{"Start":"04:02.869 ","End":"04:06.570","Text":"64 over 3 so I can pull it out front."},{"Start":"04:06.710 ","End":"04:11.655","Text":"I get all this is equal to, equal to,"},{"Start":"04:11.655 ","End":"04:17.330","Text":"64 over 3 times the integral from"},{"Start":"04:17.330 ","End":"04:24.600","Text":"0 to Pi over 2 of 1d Theta."},{"Start":"04:26.120 ","End":"04:28.995","Text":"I might as well write this part out,"},{"Start":"04:28.995 ","End":"04:35.180","Text":"this is same thing almost except that instead of 0 to pi over 2,"},{"Start":"04:35.180 ","End":"04:43.735","Text":"I get from 3Pi over 2 to 2Pi also 1d Theta."},{"Start":"04:43.735 ","End":"04:46.500","Text":"The integral of 1 between 2 limits,"},{"Start":"04:46.500 ","End":"04:51.090","Text":"it\u0027s just the upper minus the lower because this is just Theta Pi over 2 minus 0."},{"Start":"04:51.090 ","End":"04:56.505","Text":"I get 64 over 3Pi over 2 minus 0."},{"Start":"04:56.505 ","End":"04:59.039","Text":"Here 64 over 3,"},{"Start":"04:59.039 ","End":"05:04.530","Text":"2Pi minus 3Pi over 2."},{"Start":"05:04.530 ","End":"05:12.240","Text":"Now this bit is just Pi over 2 and this bit is also Pi over 2."},{"Start":"05:12.240 ","End":"05:15.585","Text":"Pi over 2 plus Pi over 2 is just Pi."},{"Start":"05:15.585 ","End":"05:17.820","Text":"I have 64 over 3 twice."},{"Start":"05:17.820 ","End":"05:20.240","Text":"The final answer, write it over here is"},{"Start":"05:20.240 ","End":"05:25.735","Text":"64 over 3 times Pi"},{"Start":"05:25.735 ","End":"05:30.290","Text":"and that is the answer but we\u0027re not quite done yet."},{"Start":"05:30.290 ","End":"05:34.790","Text":"I said I\u0027d show you an alternative way or maybe even more than 1."},{"Start":"05:34.790 ","End":"05:39.655","Text":"The idea is not to have to break this up into 2 bits."},{"Start":"05:39.655 ","End":"05:42.410","Text":"1 of the standard things that we can do with"},{"Start":"05:42.410 ","End":"05:46.805","Text":"Theta is that multiples of 2Pi don\u0027t make a difference."},{"Start":"05:46.805 ","End":"05:50.845","Text":"An angle is not uniquely specified just like 0 is 2Pi."},{"Start":"05:50.845 ","End":"05:54.365","Text":"3Pi over 2 could also be looked at"},{"Start":"05:54.365 ","End":"05:58.820","Text":"as minus Pi over 2 because it\u0027s like going Pi over 2 the other way."},{"Start":"05:58.820 ","End":"06:01.760","Text":"I could write this is minus Pi over 2."},{"Start":"06:01.760 ","End":"06:04.430","Text":"This again is Pi over 2."},{"Start":"06:04.430 ","End":"06:09.690","Text":"But now I can get a single integral from minus Pi over 2 to Pi over 2."},{"Start":"06:11.600 ","End":"06:15.810","Text":"It\u0027s going to go off screen but I\u0027ll write it."},{"Start":"06:15.810 ","End":"06:17.870","Text":"The alternative is as I said,"},{"Start":"06:17.870 ","End":"06:24.495","Text":"not to break it up into 2 bits but to take all in 1,"},{"Start":"06:24.495 ","End":"06:27.480","Text":"r goes from 0-4 as usual,"},{"Start":"06:27.480 ","End":"06:31.609","Text":"the only difference is in the Theta is that now we\u0027re going to get Theta"},{"Start":"06:31.609 ","End":"06:37.185","Text":"from minus Pi over 2 to Pi over 2."},{"Start":"06:37.185 ","End":"06:40.390","Text":"Everything else would be the same."},{"Start":"06:40.790 ","End":"06:43.440","Text":"You would end up with this."},{"Start":"06:43.440 ","End":"06:51.630","Text":"We get 2r squared drd Theta and then we would continue and"},{"Start":"06:51.630 ","End":"07:01.130","Text":"say this inner integral here is exactly the same as what we had before."},{"Start":"07:01.130 ","End":"07:03.680","Text":"We even still have it up here."},{"Start":"07:03.680 ","End":"07:09.150","Text":"This part is 64 over 3 which comes out front,"},{"Start":"07:09.150 ","End":"07:14.150","Text":"64 over 3 the integral from minus Pi over 2 to"},{"Start":"07:14.150 ","End":"07:19.500","Text":"Pi over 2 times 1d Theta."},{"Start":"07:19.500 ","End":"07:21.680","Text":"Like we said, when we have an integral of 1,"},{"Start":"07:21.680 ","End":"07:27.800","Text":"it\u0027s just the upper limit minus the lower limit so it comes out to be 64 over 3,"},{"Start":"07:27.800 ","End":"07:30.905","Text":"Pi over 2 minus,"},{"Start":"07:30.905 ","End":"07:33.885","Text":"minus Pi over 2,"},{"Start":"07:33.885 ","End":"07:35.985","Text":"but I put an extra brackets here."},{"Start":"07:35.985 ","End":"07:42.050","Text":"This minus this is Pi so we get exactly the same answer as before,"},{"Start":"07:42.050 ","End":"07:44.815","Text":"64 Pi over 3."},{"Start":"07:44.815 ","End":"07:48.490","Text":"There is even a 1/3 way of doing this,"},{"Start":"07:48.490 ","End":"07:49.870","Text":"or even any number of ways,"},{"Start":"07:49.870 ","End":"07:53.980","Text":"because like I said the angles are not uniquely determined,"},{"Start":"07:53.980 ","End":"07:55.540","Text":"only up to multiples of 2Pi."},{"Start":"07:55.540 ","End":"07:59.470","Text":"So someone could have said maybe I don\u0027t like"},{"Start":"07:59.470 ","End":"08:04.595","Text":"minus angles and would say here we have 3Pi over 2."},{"Start":"08:04.595 ","End":"08:07.980","Text":"But I want to do it in 1 bit and continue."},{"Start":"08:07.980 ","End":"08:11.305","Text":"If I went around another 1/2 circle,"},{"Start":"08:11.305 ","End":"08:19.500","Text":"I\u0027ll end up with 5Pi over 2 here which is exactly Pi over 2 plus 2Pi."},{"Start":"08:19.500 ","End":"08:22.750","Text":"Then we would get the same thing as before,"},{"Start":"08:22.750 ","End":"08:24.700","Text":"but instead of from here to here,"},{"Start":"08:24.700 ","End":"08:27.350","Text":"let me go down."},{"Start":"08:29.470 ","End":"08:33.410","Text":"What I\u0027m doing is basically copying this,"},{"Start":"08:33.410 ","End":"08:38.944","Text":"except for the Theta still go from r equals 0-4 that never changes,"},{"Start":"08:38.944 ","End":"08:41.630","Text":"r squared drd Theta."},{"Start":"08:41.630 ","End":"08:51.150","Text":"This time Theta we said was from 3Pi over 2 to 5Pi over 2."},{"Start":"08:51.790 ","End":"08:54.790","Text":"Let me start on the D here."},{"Start":"08:54.790 ","End":"08:59.820","Text":"This is equal to 64 over 3"},{"Start":"08:59.820 ","End":"09:06.165","Text":"integral from 3Pi over 2 to 5Pi over 2,"},{"Start":"09:06.165 ","End":"09:10.755","Text":"1d Theta 64 over 3."},{"Start":"09:10.755 ","End":"09:14.715","Text":"Like I said we subtract the top minus the bottom so this time"},{"Start":"09:14.715 ","End":"09:22.845","Text":"5Pi over 2 minus 3Pi over 2."},{"Start":"09:22.845 ","End":"09:26.415","Text":"Just like here, this minus this was Pi,"},{"Start":"09:26.415 ","End":"09:31.590","Text":"this minus this is also Pi it\u0027s 2Pi over 2."},{"Start":"09:31.590 ","End":"09:35.510","Text":"Just like before this and this altogether gave us Pi."},{"Start":"09:35.510 ","End":"09:38.400","Text":"We still end up with 64 over 3."},{"Start":"09:38.400 ","End":"09:43.110","Text":"Obviously doing things different method should give the same answer."},{"Start":"09:44.370 ","End":"09:50.315","Text":"I would prefer, if I had to choose from all the different methods."},{"Start":"09:50.315 ","End":"09:54.860","Text":"I think the most natural to go from minus Pi over 2 to Pi over"},{"Start":"09:54.860 ","End":"09:59.900","Text":"2 so this one would be my choice."},{"Start":"09:59.900 ","End":"10:02.590","Text":"But all of them are good."},{"Start":"10:02.590 ","End":"10:06.160","Text":"We\u0027re done. We still have this."}],"ID":8718},{"Watched":false,"Name":"Exercise 6","Duration":"5m 10s","ChapterTopicVideoID":8495,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8495.jpeg","UploadDate":"2017-02-13T08:43:33.7770000","DurationForVideoObject":"PT5M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.655","Text":"In this exercise, we have to compute this double integral where D is as in the sketch,"},{"Start":"00:05.655 ","End":"00:13.139","Text":"let me label it D. We\u0027re going to do it with polar coordinates."},{"Start":"00:13.139 ","End":"00:15.430","Text":"Let me bring in the formulas."},{"Start":"00:15.430 ","End":"00:20.310","Text":"Just copy, paste them from previous exercises."},{"Start":"00:20.360 ","End":"00:26.900","Text":"The first thing we have to do is to describe D in polar terms in Theta and"},{"Start":"00:26.900 ","End":"00:30.020","Text":"r. We\u0027ve already discussed this in"},{"Start":"00:30.020 ","End":"00:33.665","Text":"a previous exercise that there\u0027s more than 1 way to describe an angle."},{"Start":"00:33.665 ","End":"00:35.660","Text":"For example, this angle."},{"Start":"00:35.660 ","End":"00:37.955","Text":"If you\u0027re okay with negative angles,"},{"Start":"00:37.955 ","End":"00:40.760","Text":"and we can go 90 degrees clockwise,"},{"Start":"00:40.760 ","End":"00:42.485","Text":"which is the negative direction."},{"Start":"00:42.485 ","End":"00:44.240","Text":"90 degrees is Pi over 2."},{"Start":"00:44.240 ","End":"00:48.380","Text":"This could be minus Pi over 2."},{"Start":"00:48.380 ","End":"00:51.305","Text":"But if you don\u0027t like negative angles,"},{"Start":"00:51.305 ","End":"00:58.010","Text":"then you would call it 3Pi over 2 at 270 degrees."},{"Start":"00:58.010 ","End":"01:03.030","Text":"The thing is if you do it as 3Pi over 2,"},{"Start":"01:03.260 ","End":"01:07.780","Text":"you might have to do it as 2 separate bits, we could say,"},{"Start":"01:07.780 ","End":"01:11.465","Text":"we\u0027re going from a token degrees for the moment, we can say,"},{"Start":"01:11.465 ","End":"01:18.910","Text":"we\u0027re going to go from 0-180 degrees and then separately from 270-360."},{"Start":"01:18.910 ","End":"01:21.590","Text":"But that makes us do it in 2 pieces."},{"Start":"01:21.590 ","End":"01:24.079","Text":"You actually have a couple of other choices."},{"Start":"01:24.079 ","End":"01:27.635","Text":"We could start at 3Pi over 2 and then"},{"Start":"01:27.635 ","End":"01:33.500","Text":"continue and this angle would be instead of 180 degrees,"},{"Start":"01:33.500 ","End":"01:35.930","Text":"you could call it 540 degrees,"},{"Start":"01:35.930 ","End":"01:38.735","Text":"which would be actually 3Pi."},{"Start":"01:38.735 ","End":"01:45.110","Text":"But usually what we do is when we have to go backwards,"},{"Start":"01:45.110 ","End":"01:47.660","Text":"we use a negative that\u0027s minus Pi over 2,"},{"Start":"01:47.660 ","End":"01:54.545","Text":"and this one will just be Pi and I\u0027ll just not bother with this."},{"Start":"01:54.545 ","End":"01:56.680","Text":"Although as I said, there are ways of doing it,"},{"Start":"01:56.680 ","End":"02:00.155","Text":"either you break it up into 2 pieces or you use different angles."},{"Start":"02:00.155 ","End":"02:03.730","Text":"The most natural is to go from minus Pi over 2 to Pi."},{"Start":"02:03.730 ","End":"02:05.555","Text":"This is how I\u0027m going to do it,"},{"Start":"02:05.555 ","End":"02:08.750","Text":"just to tell you that there are options."},{"Start":"02:08.750 ","End":"02:12.305","Text":"Now that I\u0027ve chosen where my Theta goes to,"},{"Start":"02:12.305 ","End":"02:15.510","Text":"of course the r, wherever Theta is,"},{"Start":"02:15.510 ","End":"02:18.630","Text":"r still goes from 0 at the center,"},{"Start":"02:18.630 ","End":"02:22.160","Text":"to 4 the edge and it\u0027s the same for all Theta."},{"Start":"02:22.160 ","End":"02:26.965","Text":"What we get is the double integral."},{"Start":"02:26.965 ","End":"02:34.250","Text":"Now D in polar terms is where r goes from 0-4"},{"Start":"02:34.250 ","End":"02:42.270","Text":"and Theta we chose to take it from minus Pi over 2 to Pi."},{"Start":"02:42.270 ","End":"02:49.145","Text":"Then the square root of x squared plus y squared from this formula is r squared."},{"Start":"02:49.145 ","End":"02:55.585","Text":"DA from here is r dr d Theta."},{"Start":"02:55.585 ","End":"02:59.165","Text":"As you\u0027ve seen, we\u0027ve seen this before."},{"Start":"02:59.165 ","End":"03:05.060","Text":"We go from minus Pi over 2 to Pi."},{"Start":"03:05.060 ","End":"03:08.935","Text":"This I\u0027m just copying 0-4."},{"Start":"03:08.935 ","End":"03:10.980","Text":"Square root of r squared is r,"},{"Start":"03:10.980 ","End":"03:16.585","Text":"r times r is r squared dr d Theta."},{"Start":"03:16.585 ","End":"03:20.165","Text":"We do the middle bit first."},{"Start":"03:20.165 ","End":"03:29.555","Text":"That\u0027s the integral dr. we\u0027ve done many times before and the answer came out 64/3."},{"Start":"03:29.555 ","End":"03:33.125","Text":"But if you haven\u0027t seen it before, just think about it."},{"Start":"03:33.125 ","End":"03:37.025","Text":"Integral of r squared is r cubed over 3."},{"Start":"03:37.025 ","End":"03:41.085","Text":"If I plug in 4, 2r cubed over 3,"},{"Start":"03:41.085 ","End":"03:43.800","Text":"4 cubed over 3 is 64/3."},{"Start":"03:43.800 ","End":"03:46.095","Text":"Plugging in the 0 gives of course nothing,"},{"Start":"03:46.095 ","End":"03:51.705","Text":"so 64/3 minus nothing is just 64/3."},{"Start":"03:51.705 ","End":"03:53.030","Text":"I\u0027m not going to do it at the side."},{"Start":"03:53.030 ","End":"03:56.360","Text":"We\u0027ve done it before and you could do it yourself. Let\u0027s continue."},{"Start":"03:56.360 ","End":"03:58.900","Text":"The 64/3 is a constant,"},{"Start":"03:58.900 ","End":"04:04.490","Text":"so I can pull it out in front of the integral sign and we have the integral from"},{"Start":"04:04.490 ","End":"04:10.050","Text":"minus Pi over 2 to Pi of just 1,"},{"Start":"04:10.050 ","End":"04:13.255","Text":"just d Theta or 1 d Theta."},{"Start":"04:13.255 ","End":"04:15.020","Text":"Whenever we have the integral of 1,"},{"Start":"04:15.020 ","End":"04:17.675","Text":"it\u0027s just the upper limit minus the lower limit."},{"Start":"04:17.675 ","End":"04:23.490","Text":"We have 64/3 Pi minus,"},{"Start":"04:23.490 ","End":"04:26.680","Text":"minus Pi over 2."},{"Start":"04:26.760 ","End":"04:33.550","Text":"Pi minus minus Pi over 2 is just 3Pi over 2."},{"Start":"04:33.550 ","End":"04:39.735","Text":"This bit is 3Pi over 2."},{"Start":"04:39.735 ","End":"04:43.095","Text":"What I end up with is, let\u0027s see,"},{"Start":"04:43.095 ","End":"04:48.960","Text":"64/3 times 3Pi over 2."},{"Start":"04:48.960 ","End":"04:50.130","Text":"Do some canceling."},{"Start":"04:50.130 ","End":"05:03.080","Text":"Two into 64 goes 32 and 3 cancels with 3."},{"Start":"05:03.080 ","End":"05:06.255","Text":"The final answer is 32 Pi."},{"Start":"05:06.255 ","End":"05:10.010","Text":"I\u0027ll highlight it and we are done."}],"ID":8719},{"Watched":false,"Name":"Exercise 7","Duration":"4m 7s","ChapterTopicVideoID":8496,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8496.jpeg","UploadDate":"2017-02-13T08:44:05.9800000","DurationForVideoObject":"PT4M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we need to compute the following integral."},{"Start":"00:04.530 ","End":"00:07.065","Text":"D is the domain in the sketch,"},{"Start":"00:07.065 ","End":"00:10.470","Text":"this is D. The question is,"},{"Start":"00:10.470 ","End":"00:13.830","Text":"how do we do this in polar coordinates?"},{"Start":"00:13.830 ","End":"00:16.890","Text":"First of all, let me give the formulas."},{"Start":"00:16.890 ","End":"00:24.690","Text":"The thing that we have to do is to convert or describe the domain D in polar terms."},{"Start":"00:24.690 ","End":"00:30.375","Text":"Now, we know that this line is where Theta equals 0."},{"Start":"00:30.375 ","End":"00:34.230","Text":"This line here, Theta equals,"},{"Start":"00:34.230 ","End":"00:38.220","Text":"well, I\u0027ll leave a question mark there for the moment and we\u0027ll return to that."},{"Start":"00:38.220 ","End":"00:40.470","Text":"Anyway, whatever Theta is,"},{"Start":"00:40.470 ","End":"00:45.295","Text":"the r goes from 0 to 4,"},{"Start":"00:45.295 ","End":"00:48.420","Text":"and this is assumed to be the arc of a circle."},{"Start":"00:49.070 ","End":"00:52.385","Text":"The question is, what is this angle?"},{"Start":"00:52.385 ","End":"00:57.295","Text":"We know the equation of the line in Cartesian coordinates y equals x."},{"Start":"00:57.295 ","End":"01:03.245","Text":"In general, when we have an equation of the form something x plus something,"},{"Start":"01:03.245 ","End":"01:06.315","Text":"then this is the slope."},{"Start":"01:06.315 ","End":"01:11.100","Text":"In our case, we have m equals 1."},{"Start":"01:11.100 ","End":"01:15.050","Text":"But the slope is also the tangent of the angle that\u0027s known,"},{"Start":"01:15.050 ","End":"01:20.790","Text":"so we have that tangent of this Theta is equal to 1."},{"Start":"01:20.790 ","End":"01:26.660","Text":"You either know that Theta is 45 degrees because you remember"},{"Start":"01:26.660 ","End":"01:33.530","Text":"the important angles or you do it on the calculator inverse tangent or shift tangent."},{"Start":"01:33.530 ","End":"01:36.790","Text":"If you do it on the calculator,"},{"Start":"01:36.790 ","End":"01:38.040","Text":"we get that Theta."},{"Start":"01:38.040 ","End":"01:43.550","Text":"I\u0027ll just call it arc tangent of 1 and this is equal."},{"Start":"01:43.550 ","End":"01:45.019","Text":"If you set it for degrees,"},{"Start":"01:45.019 ","End":"01:46.370","Text":"you get 45 degrees."},{"Start":"01:46.370 ","End":"01:50.615","Text":"But we\u0027re working in radians and you should get Pi over 4."},{"Start":"01:50.615 ","End":"01:56.330","Text":"I\u0027ll just go back here and replace the question mark by Pi over 4."},{"Start":"01:56.330 ","End":"01:58.585","Text":"Okay, now we\u0027re all set."},{"Start":"01:58.585 ","End":"02:06.245","Text":"We get the double integral and we know it\u0027s going to be dA,"},{"Start":"02:06.245 ","End":"02:14.625","Text":"I\u0027ll start there, rdrd Theta and the region D in terms of polar."},{"Start":"02:14.625 ","End":"02:16.124","Text":"We need the r first,"},{"Start":"02:16.124 ","End":"02:18.885","Text":"it goes from 0 to 4."},{"Start":"02:18.885 ","End":"02:21.060","Text":"Theta, we just said,"},{"Start":"02:21.060 ","End":"02:24.460","Text":"goes from 0 to 45 degrees,"},{"Start":"02:24.460 ","End":"02:28.465","Text":"but we\u0027re working in radians so it\u0027s Pi over 4."},{"Start":"02:28.465 ","End":"02:33.360","Text":"What we have here is the square root of x squared plus y squared,"},{"Start":"02:33.360 ","End":"02:34.680","Text":"but from this formula,"},{"Start":"02:34.680 ","End":"02:37.065","Text":"this is r squared."},{"Start":"02:37.065 ","End":"02:42.665","Text":"We get the integral from 0 to Pi over 4, that\u0027s for Theta,"},{"Start":"02:42.665 ","End":"02:50.545","Text":"the integral from 0 to 4 for r. This expression here gives us r squared drd Theta."},{"Start":"02:50.545 ","End":"02:57.830","Text":"The inner integrals, what we do first and we\u0027ve done it several times before,"},{"Start":"02:57.830 ","End":"02:59.930","Text":"so I\u0027ll just quote the answer."},{"Start":"02:59.930 ","End":"03:06.790","Text":"It came out to be 64 Pi over 3."},{"Start":"03:07.700 ","End":"03:11.310","Text":"Just 64 over 3, I don\u0027t know why I put the Pi in."},{"Start":"03:11.310 ","End":"03:15.930","Text":"Forgive me. If you want me to remind you how we did this,"},{"Start":"03:15.930 ","End":"03:17.090","Text":"we did it at the side."},{"Start":"03:17.090 ","End":"03:18.995","Text":"We got r cubed over 3."},{"Start":"03:18.995 ","End":"03:22.525","Text":"We plugged in 4 and we got 4 cubed over 3,"},{"Start":"03:22.525 ","End":"03:24.615","Text":"4 cubed is 64 over 3."},{"Start":"03:24.615 ","End":"03:26.540","Text":"When you plug in 0, we don\u0027t get anything,"},{"Start":"03:26.540 ","End":"03:27.875","Text":"so this was that."},{"Start":"03:27.875 ","End":"03:32.195","Text":"Now we can continue and 64 over 3 is a constant,"},{"Start":"03:32.195 ","End":"03:41.580","Text":"so it comes out in front of the integral from 0 to Pi over 4 of just d Theta, 1 d Theta."},{"Start":"03:41.580 ","End":"03:43.140","Text":"When you have the integral of 1,"},{"Start":"03:43.140 ","End":"03:45.695","Text":"it\u0027s just the upper limit minus the lower limit."},{"Start":"03:45.695 ","End":"03:50.870","Text":"It\u0027s 64 over 3 times Pi over 4."},{"Start":"03:50.870 ","End":"03:53.540","Text":"Pi over 4 minus 0, of course, is Pi over 4."},{"Start":"03:53.540 ","End":"03:57.875","Text":"If we simplify it, 4 into 64 goes 16 times,"},{"Start":"03:57.875 ","End":"04:04.225","Text":"so we\u0027ve got 16 Pi over 3 or 16 over 3 Pi."},{"Start":"04:04.225 ","End":"04:07.590","Text":"I\u0027ll just highlight it and we\u0027re done."}],"ID":8720},{"Watched":false,"Name":"Exercise 8","Duration":"5m 45s","ChapterTopicVideoID":8497,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8497.jpeg","UploadDate":"2017-02-13T08:44:48.7530000","DurationForVideoObject":"PT5M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this exercise, we have to compute this double"},{"Start":"00:02.850 ","End":"00:07.500","Text":"integral over D. D is the domain in the sketch."},{"Start":"00:07.500 ","End":"00:09.300","Text":"This is it here, D,"},{"Start":"00:09.300 ","End":"00:13.200","Text":"and we\u0027re going to use polar coordinates."},{"Start":"00:13.200 ","End":"00:19.410","Text":"Here, I copy pasted the formulas for polar conversion."},{"Start":"00:19.410 ","End":"00:25.150","Text":"What we still need to do is describe D in polar terms."},{"Start":"00:25.820 ","End":"00:30.300","Text":"Now, we know this is the arc of a circle with radius 4,"},{"Start":"00:30.300 ","End":"00:32.310","Text":"at least that\u0027s what we\u0027re presuming."},{"Start":"00:32.310 ","End":"00:34.590","Text":"So that for any given Theta,"},{"Start":"00:34.590 ","End":"00:38.085","Text":"we know that r goes from 0 to 4,"},{"Start":"00:38.085 ","End":"00:40.515","Text":"but what about Theta?"},{"Start":"00:40.515 ","End":"00:48.195","Text":"I know that this first line is given by Theta equals something."},{"Start":"00:48.195 ","End":"00:50.330","Text":"We are got to figure this out still,"},{"Start":"00:50.330 ","End":"00:51.590","Text":"because it goes through the origin,"},{"Start":"00:51.590 ","End":"00:54.250","Text":"it\u0027s a fixed angle and so is this one,"},{"Start":"00:54.250 ","End":"00:56.240","Text":"Theta equals something else."},{"Start":"00:56.240 ","End":"00:58.790","Text":"But how do I find these 2 angles?"},{"Start":"00:58.790 ","End":"01:04.115","Text":"Well, we have the equations of the lines in Cartesian coordinates,"},{"Start":"01:04.115 ","End":"01:07.785","Text":"and we know that the slope of y equals mx,"},{"Start":"01:07.785 ","End":"01:14.570","Text":"or mx plus something is just the m. I know that this one, for the first line,"},{"Start":"01:14.570 ","End":"01:19.385","Text":"let\u0027s call the slope m1,"},{"Start":"01:19.385 ","End":"01:24.800","Text":"and this is equal to 1 over the square root of 3."},{"Start":"01:24.800 ","End":"01:26.270","Text":"For the other line,"},{"Start":"01:26.270 ","End":"01:30.125","Text":"I have an m2, which is square root of 3."},{"Start":"01:30.125 ","End":"01:37.420","Text":"Let\u0027s, for the moment, call these angles Theta 1 and Theta 2."},{"Start":"01:37.420 ","End":"01:40.205","Text":"We know that the slope is the tangent of the angle."},{"Start":"01:40.205 ","End":"01:45.740","Text":"I have the tangent of Theta 1 is 1 over root 3,"},{"Start":"01:45.740 ","End":"01:50.100","Text":"and the tangent of Theta 2 is root 3."},{"Start":"01:50.100 ","End":"01:54.215","Text":"Now, some of you might remember that the tangents are some special angles."},{"Start":"01:54.215 ","End":"01:59.360","Text":"For example, I recognize that this is the tangent of 30 degrees and this is 60 degrees."},{"Start":"01:59.360 ","End":"02:02.580","Text":"But if you didn\u0027t, we could, first of all, write,"},{"Start":"02:02.580 ","End":"02:08.150","Text":"Theta 1 is the arc tangent of 1 over root 3,"},{"Start":"02:08.150 ","End":"02:15.960","Text":"the inverse tangent, and that Theta 2 is the arc tangent of square root of 3."},{"Start":"02:15.960 ","End":"02:17.330","Text":"Then if you didn\u0027t remember,"},{"Start":"02:17.330 ","End":"02:19.955","Text":"you could do it on the calculator."},{"Start":"02:19.955 ","End":"02:24.850","Text":"If you set your calculator to radians,"},{"Start":"02:24.850 ","End":"02:27.990","Text":"you\u0027ll get Theta 1 equals,"},{"Start":"02:27.990 ","End":"02:29.070","Text":"depending on the calculator,"},{"Start":"02:29.070 ","End":"02:32.145","Text":"do inverse tangent, or shift tangent."},{"Start":"02:32.145 ","End":"02:36.244","Text":"If you did it in degrees, you\u0027d get 30 degrees."},{"Start":"02:36.244 ","End":"02:41.105","Text":"I could even say for the moment that it\u0027s 30 degrees,"},{"Start":"02:41.105 ","End":"02:42.889","Text":"if you didn\u0027t set it to radians,"},{"Start":"02:42.889 ","End":"02:44.270","Text":"but if you set to radians,"},{"Start":"02:44.270 ","End":"02:47.135","Text":"the answer would be Pi over 6."},{"Start":"02:47.135 ","End":"02:50.270","Text":"I wanted to just emphasize that I\u0027m not going to use degrees,"},{"Start":"02:50.270 ","End":"02:55.325","Text":"just in case you did, you\u0027d multiply this by Pi over 180, anyway."},{"Start":"02:55.325 ","End":"02:57.380","Text":"As for Theta 2,"},{"Start":"02:57.380 ","End":"02:59.860","Text":"we did it on the calculator."},{"Start":"02:59.860 ","End":"03:04.490","Text":"We should get the answer of Pi over 3,"},{"Start":"03:04.490 ","End":"03:05.975","Text":"which is 60 degrees."},{"Start":"03:05.975 ","End":"03:09.530","Text":"If you got 60 degrees, because you set it to degrees,"},{"Start":"03:09.530 ","End":"03:12.350","Text":"you would multiply by Pi over a 180,"},{"Start":"03:12.350 ","End":"03:15.065","Text":"and this is Pi over 3."},{"Start":"03:15.065 ","End":"03:18.010","Text":"But I like to sometimes just have insight, that I know it\u0027s"},{"Start":"03:18.010 ","End":"03:21.415","Text":"30 degrees, 60 degrees, but we should use radians."},{"Start":"03:21.415 ","End":"03:25.360","Text":"Now, let me just replace Theta 1 and Theta 2."},{"Start":"03:25.360 ","End":"03:27.800","Text":"I don\u0027t even have to do that,"},{"Start":"03:28.010 ","End":"03:31.335","Text":"I\u0027ll just emphasize them."},{"Start":"03:31.335 ","End":"03:33.555","Text":"Now I\u0027ve got the double integral,"},{"Start":"03:33.555 ","End":"03:36.990","Text":"and I know that Theta goes from,"},{"Start":"03:36.990 ","End":"03:39.300","Text":"I won\u0027t say 30-60 degrees,"},{"Start":"03:39.300 ","End":"03:43.635","Text":"I\u0027ll say Pi over 6 to Pi over 3."},{"Start":"03:43.635 ","End":"03:47.830","Text":"R goes from 0-4,"},{"Start":"03:49.550 ","End":"03:53.695","Text":"and the square root,"},{"Start":"03:53.695 ","End":"03:58.355","Text":"x squared plus y squared, from this formula, is r squared."},{"Start":"03:58.355 ","End":"04:05.220","Text":"The last thing is dA, which is r dr d Theta."},{"Start":"04:05.220 ","End":"04:09.740","Text":"So we get, square root of r squared is rr times r is r squared."},{"Start":"04:09.740 ","End":"04:15.755","Text":"We have the integral from Pi over 6 to Pi over 3."},{"Start":"04:15.755 ","End":"04:24.620","Text":"The integral from 0-4 of r squared drd Theta."},{"Start":"04:24.620 ","End":"04:27.410","Text":"We always do the integrals from inside out,"},{"Start":"04:27.410 ","End":"04:31.490","Text":"so we have this one, and we\u0027ve seen this many times before."},{"Start":"04:31.490 ","End":"04:32.810","Text":"I\u0027ll just give you the answer,"},{"Start":"04:32.810 ","End":"04:35.210","Text":"it\u0027s 64 over 3."},{"Start":"04:35.210 ","End":"04:37.405","Text":"This is a constant,"},{"Start":"04:37.405 ","End":"04:44.660","Text":"so I can pull it out in front of the integral and get 64 over 3 times the integral from"},{"Start":"04:44.660 ","End":"04:53.730","Text":"Pi over 6 to Pi over 3 of just d Theta or 1 times d Theta."},{"Start":"04:53.730 ","End":"04:55.920","Text":"Now that we have the integral of 1,"},{"Start":"04:55.920 ","End":"04:59.975","Text":"we just have to subtract the upper limit minus the lower limit."},{"Start":"04:59.975 ","End":"05:03.385","Text":"We get 64 over 3,"},{"Start":"05:03.385 ","End":"05:08.595","Text":"Pi over 3 minus Pi over 6."},{"Start":"05:08.595 ","End":"05:11.535","Text":"But Pi over 3 minus Pi over 6,"},{"Start":"05:11.535 ","End":"05:13.700","Text":"like 1/3 minus 6 is 6."},{"Start":"05:13.700 ","End":"05:16.050","Text":"This is just Pi over 6."},{"Start":"05:16.050 ","End":"05:28.000","Text":"What we get is, 64 over 3 times Pi over 6."},{"Start":"05:28.160 ","End":"05:32.015","Text":"Only thing I can cancel is by 2,"},{"Start":"05:32.015 ","End":"05:37.280","Text":"32 here 3, and so I get, let\u0027s see,"},{"Start":"05:37.280 ","End":"05:42.220","Text":"32 over 3 times 3 is 9 Pi."},{"Start":"05:42.220 ","End":"05:44.144","Text":"This is our answer,"},{"Start":"05:44.144 ","End":"05:46.480","Text":"and we are done.4"}],"ID":8721},{"Watched":false,"Name":"Exercise 9","Duration":"7m 33s","ChapterTopicVideoID":8498,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8498.jpeg","UploadDate":"2017-02-13T08:48:34.9470000","DurationForVideoObject":"PT7M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this exercise, we\u0027re given the following double integral to compute."},{"Start":"00:04.230 ","End":"00:07.245","Text":"D is the domain described in the sketch."},{"Start":"00:07.245 ","End":"00:08.940","Text":"It\u0027s a bit small,"},{"Start":"00:08.940 ","End":"00:12.990","Text":"I think I\u0027d like to enlarge it. That\u0027s a bit better."},{"Start":"00:12.990 ","End":"00:19.330","Text":"This is going to be my region D. Same here."},{"Start":"00:19.370 ","End":"00:21.925","Text":"I have to say this,"},{"Start":"00:21.925 ","End":"00:25.430","Text":"this is given in the section on polar coordinates and the question"},{"Start":"00:25.430 ","End":"00:28.700","Text":"stated that we should do it with polar coordinates."},{"Start":"00:28.700 ","End":"00:30.185","Text":"Normally, I wouldn\u0027t."},{"Start":"00:30.185 ","End":"00:33.680","Text":"There is no circular symmetry to it."},{"Start":"00:33.680 ","End":"00:36.860","Text":"There\u0027s no arcs of circles or anything."},{"Start":"00:36.860 ","End":"00:40.145","Text":"Normally, I wouldn\u0027t, and we\u0027re going to do it in polar"},{"Start":"00:40.145 ","End":"00:43.760","Text":"just as an exercise and because we were told to."},{"Start":"00:43.760 ","End":"00:50.900","Text":"The difficult part of this exercise is describing domain D in terms"},{"Start":"00:50.900 ","End":"00:58.025","Text":"of polar r and Theta because it doesn\u0027t have that circular symmetry."},{"Start":"00:58.025 ","End":"01:00.275","Text":"Anyway, let\u0027s make a beginning."},{"Start":"01:00.275 ","End":"01:07.080","Text":"We say we have the double integral and we\u0027ll have Theta going."},{"Start":"01:07.080 ","End":"01:09.000","Text":"That\u0027ll be the easier part."},{"Start":"01:09.000 ","End":"01:13.365","Text":"Theta will go from 0 to something,"},{"Start":"01:13.365 ","End":"01:15.285","Text":"we\u0027ll compute that in a moment,"},{"Start":"01:15.285 ","End":"01:18.480","Text":"and r will also go,"},{"Start":"01:18.480 ","End":"01:21.335","Text":"it will also be from 0 to something."},{"Start":"01:21.335 ","End":"01:23.510","Text":"Let me just show you what I mean."},{"Start":"01:23.510 ","End":"01:28.415","Text":"So far, we\u0027ve identified this line as Theta equals 0,"},{"Start":"01:28.415 ","End":"01:35.900","Text":"and here this line will be Theta equals something and we\u0027ll figure that out in a moment."},{"Start":"01:35.900 ","End":"01:40.190","Text":"Now, for each Theta between 0 and whatever this is,"},{"Start":"01:40.190 ","End":"01:43.550","Text":"it will come out to be 60 degrees Pi over 3,"},{"Start":"01:43.550 ","End":"01:45.155","Text":"but I\u0027m jumping the gun."},{"Start":"01:45.155 ","End":"01:48.075","Text":"But for each Theta,"},{"Start":"01:48.075 ","End":"01:50.849","Text":"r will go from 0,"},{"Start":"01:50.849 ","End":"01:52.280","Text":"that\u0027s not a problem."},{"Start":"01:52.280 ","End":"01:56.510","Text":"But this also a question mark isn\u0027t just a question mark,"},{"Start":"01:56.510 ","End":"02:00.160","Text":"it depends on Theta because as Theta changes,"},{"Start":"02:00.160 ","End":"02:03.980","Text":"the upper limit of r is also going to change and this is"},{"Start":"02:03.980 ","End":"02:08.965","Text":"different than the previous exercise where it was always some constant like 4."},{"Start":"02:08.965 ","End":"02:14.585","Text":"Anyway, back here I know that r goes from 0 to something and we\u0027ll fill this in also."},{"Start":"02:14.585 ","End":"02:16.430","Text":"Still, this part we can do,"},{"Start":"02:16.430 ","End":"02:22.815","Text":"the square root of x squared plus y squared is the square root of r squared,"},{"Start":"02:22.815 ","End":"02:29.640","Text":"and dA is r dr d Theta."},{"Start":"02:29.640 ","End":"02:32.015","Text":"We have to still fill in here and here."},{"Start":"02:32.015 ","End":"02:33.590","Text":"Let\u0027s do the easy part first."},{"Start":"02:33.590 ","End":"02:35.630","Text":"What is this Theta?"},{"Start":"02:35.630 ","End":"02:39.979","Text":"Going to use some basic trigonometry to find these unknowns."},{"Start":"02:39.979 ","End":"02:42.675","Text":"For example, what is this Theta?"},{"Start":"02:42.675 ","End":"02:51.595","Text":"Well, I know that the tangent of it is root 3 over 1."},{"Start":"02:51.595 ","End":"02:54.875","Text":"Why don\u0027t I just give them labels?"},{"Start":"02:54.875 ","End":"02:59.720","Text":"Let\u0027s call this angle Theta_1 and this,"},{"Start":"02:59.720 ","End":"03:01.910","Text":"I don\u0027t want to reuse the letter r,"},{"Start":"03:01.910 ","End":"03:03.590","Text":"so I\u0027ll call it r_1,"},{"Start":"03:03.590 ","End":"03:08.430","Text":"but actually, this will be a function of Theta."},{"Start":"03:09.470 ","End":"03:12.140","Text":"Back to Theta_1, like I was saying,"},{"Start":"03:12.140 ","End":"03:15.890","Text":"the tangent of this angle Theta_1,"},{"Start":"03:15.890 ","End":"03:17.360","Text":"we\u0027re given the x and the y,"},{"Start":"03:17.360 ","End":"03:18.935","Text":"it\u0027s the y over the x,"},{"Start":"03:18.935 ","End":"03:20.660","Text":"is root 3 over 1,"},{"Start":"03:20.660 ","End":"03:22.835","Text":"which is just root 3."},{"Start":"03:22.835 ","End":"03:26.120","Text":"If we do the arctangent of root 3,"},{"Start":"03:26.120 ","End":"03:28.325","Text":"or if you remember the special angles,"},{"Start":"03:28.325 ","End":"03:32.300","Text":"then Theta_1 is 60 degrees."},{"Start":"03:32.300 ","End":"03:34.160","Text":"That\u0027s the angle whose tangent is root 3."},{"Start":"03:34.160 ","End":"03:37.219","Text":"We don\u0027t want to put it in degrees. I\u0027m not going to write 60."},{"Start":"03:37.219 ","End":"03:39.875","Text":"I\u0027m going to write Pi over 3,"},{"Start":"03:39.875 ","End":"03:45.740","Text":"so that I can already put in here, Pi over 3."},{"Start":"03:45.740 ","End":"03:54.360","Text":"Now the thing with the r. This is my general varying Theta."},{"Start":"03:54.360 ","End":"04:01.065","Text":"This one will be Theta as it goes from 0-Pi over 3."},{"Start":"04:01.065 ","End":"04:04.810","Text":"What I will have is using trigonometry,"},{"Start":"04:04.810 ","End":"04:13.100","Text":"I want to use the cosine because this r_1 is the hypotenuse."},{"Start":"04:13.100 ","End":"04:17.720","Text":"I know that cosine of"},{"Start":"04:17.720 ","End":"04:27.395","Text":"this angle Theta is equal to the adjacent over the hypotenuse."},{"Start":"04:27.395 ","End":"04:29.555","Text":"This adjacent, this is 1,"},{"Start":"04:29.555 ","End":"04:33.360","Text":"is 1, the hypotenuse is r_1."},{"Start":"04:33.360 ","End":"04:36.405","Text":"It\u0027s a function of Theta, but I don\u0027t have to write that."},{"Start":"04:36.405 ","End":"04:39.030","Text":"If I extract r_1,"},{"Start":"04:39.030 ","End":"04:45.675","Text":"I\u0027ve got that r_1 is 1 over cosine Theta."},{"Start":"04:45.675 ","End":"04:51.070","Text":"This is the r_1 that I want to put as the upper limit that depends on Theta."},{"Start":"04:51.070 ","End":"04:59.505","Text":"I can now replace it by 1 over cosine Theta."},{"Start":"04:59.505 ","End":"05:01.650","Text":"As I said, it depends on Theta,"},{"Start":"05:01.650 ","End":"05:03.790","Text":"it\u0027s a function of Theta."},{"Start":"05:03.860 ","End":"05:06.190","Text":"Continuing."},{"Start":"05:06.190 ","End":"05:10.160","Text":"Now we just have technical a double integral."},{"Start":"05:10.160 ","End":"05:14.860","Text":"This is equal to the integral from 0-Pi over 3"},{"Start":"05:14.860 ","End":"05:23.630","Text":"of the integral from 0-1 over cosine Theta."},{"Start":"05:23.630 ","End":"05:25.760","Text":"You could call the secant of Theta,"},{"Start":"05:25.760 ","End":"05:28.385","Text":"but I\u0027ll leave it as 1 over cosine Theta."},{"Start":"05:28.385 ","End":"05:31.220","Text":"This we\u0027ve had before, and it\u0027s,"},{"Start":"05:31.220 ","End":"05:38.260","Text":"r times r is r squared dr d Theta."},{"Start":"05:38.260 ","End":"05:43.180","Text":"As usual, we do our integrals from the inside out."},{"Start":"05:43.180 ","End":"05:45.865","Text":"This is the one we want to do first."},{"Start":"05:45.865 ","End":"05:48.880","Text":"Now, in the phrasing of the question in the exercise book,"},{"Start":"05:48.880 ","End":"05:52.500","Text":"it actually said do not evaluate the integral obtained."},{"Start":"05:52.500 ","End":"05:56.260","Text":"We could stop here and say, \"Okay, we\u0027re done.\""},{"Start":"05:56.260 ","End":"05:59.239","Text":"But I\u0027d just like to continue a little bit more"},{"Start":"05:59.239 ","End":"06:03.550","Text":"and show you why you weren\u0027t required to complete it,"},{"Start":"06:03.550 ","End":"06:06.265","Text":"because it becomes a bit of a difficult integral."},{"Start":"06:06.265 ","End":"06:09.890","Text":"But let\u0027s continue a little bit more anyway for those who want."},{"Start":"06:09.890 ","End":"06:14.030","Text":"I\u0027m going to compute this bit I shaded, call it asterisk."},{"Start":"06:14.030 ","End":"06:16.070","Text":"I\u0027ll do it at the side maybe here."},{"Start":"06:16.070 ","End":"06:22.850","Text":"The asterisk. What we have is the integral of r squared is"},{"Start":"06:22.850 ","End":"06:27.710","Text":"1/3 r cubed evaluated"},{"Start":"06:27.710 ","End":"06:33.680","Text":"between 0 and 1 over cosine Theta,"},{"Start":"06:33.680 ","End":"06:35.810","Text":"and this is equal to,"},{"Start":"06:35.810 ","End":"06:37.805","Text":"if I plug in the upper limit,"},{"Start":"06:37.805 ","End":"06:41.610","Text":"I get 1 over,"},{"Start":"06:41.610 ","End":"06:43.715","Text":"I\u0027ll put the 3 on the bottom,"},{"Start":"06:43.715 ","End":"06:48.435","Text":"and I get cosine cubed Theta,"},{"Start":"06:48.435 ","End":"06:52.965","Text":"and I\u0027ll put in the 0, it\u0027s minus 0."},{"Start":"06:52.965 ","End":"06:57.409","Text":"But returning here, if we continued,"},{"Start":"06:57.409 ","End":"07:00.725","Text":"what we would get would be this."},{"Start":"07:00.725 ","End":"07:06.245","Text":"This part stays the same, 0-Pi/3."},{"Start":"07:06.245 ","End":"07:09.470","Text":"The 1/3, I could pull in front,"},{"Start":"07:09.470 ","End":"07:17.105","Text":"but I have the integral of 1 over cosine cubed Theta d Theta."},{"Start":"07:17.105 ","End":"07:19.730","Text":"This is a bit of a difficult integral to compute"},{"Start":"07:19.730 ","End":"07:22.810","Text":"unless you have integral tables and so on,"},{"Start":"07:22.810 ","End":"07:26.210","Text":"and so that\u0027s why we even stopped before,"},{"Start":"07:26.210 ","End":"07:31.325","Text":"but I just continued it a bit further for those who were wondering."},{"Start":"07:31.325 ","End":"07:33.900","Text":"Now I\u0027m going to stop."}],"ID":8722},{"Watched":false,"Name":"Exercise 10","Duration":"5m 54s","ChapterTopicVideoID":8499,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8499.jpeg","UploadDate":"2017-02-13T08:50:19.2570000","DurationForVideoObject":"PT5M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.425","Text":"In this exercise, we want to compute this double integral,"},{"Start":"00:04.425 ","End":"00:08.970","Text":"but we\u0027re told to do it by converting to polar coordinates."},{"Start":"00:08.970 ","End":"00:12.495","Text":"What I want to do is first of all,"},{"Start":"00:12.495 ","End":"00:15.810","Text":"to write it as the double integral over"},{"Start":"00:15.810 ","End":"00:26.230","Text":"some region of dydx or even throw in 1 to show us the function dydx."},{"Start":"00:26.330 ","End":"00:30.380","Text":"I\u0027ll convert this region to polar coordinates."},{"Start":"00:30.380 ","End":"00:33.110","Text":"Let\u0027s see what this region might be."},{"Start":"00:33.110 ","End":"00:38.180","Text":"We can describe the region D by saying that x goes from minus 1 to"},{"Start":"00:38.180 ","End":"00:43.700","Text":"1 and y goes from 0 to this expression."},{"Start":"00:43.700 ","End":"00:49.430","Text":"Let me write this out as minus 1 less than or equal to x,"},{"Start":"00:49.430 ","End":"00:59.970","Text":"less than or equal to 1 and y between 0 and square root of 1 minus x squared."},{"Start":"00:59.970 ","End":"01:01.920","Text":"Now, some of this is clear."},{"Start":"01:01.920 ","End":"01:06.495","Text":"The x goes from 1 to minus 1."},{"Start":"01:06.495 ","End":"01:12.455","Text":"The y goes from 0, which is the x axis to some positive or non-negative function."},{"Start":"01:12.455 ","End":"01:14.030","Text":"But what is this function?"},{"Start":"01:14.030 ","End":"01:15.620","Text":"Square root of 1 minus x squared."},{"Start":"01:15.620 ","End":"01:17.585","Text":"Let\u0027s do at the side."},{"Start":"01:17.585 ","End":"01:20.600","Text":"If y equals the square root of"},{"Start":"01:20.600 ","End":"01:24.500","Text":"1 minus x squared and I want to see what it looks like, let\u0027s see."},{"Start":"01:24.500 ","End":"01:25.835","Text":"Square both sides."},{"Start":"01:25.835 ","End":"01:32.270","Text":"We\u0027ve got y squared equals 1 minus x squared and then bring the x squared over."},{"Start":"01:32.270 ","End":"01:38.150","Text":"I have x squared plus y squared equals 1 and I can write 1 as 1 squared."},{"Start":"01:38.150 ","End":"01:41.170","Text":"This is a circle of radius 1,"},{"Start":"01:41.170 ","End":"01:47.680","Text":"but here the square root is only the positive or non-negative,"},{"Start":"01:47.680 ","End":"01:53.145","Text":"so I need only an upper semicircle and here we are."},{"Start":"01:53.145 ","End":"01:57.460","Text":"Since y goes between 0 and the semicircle,"},{"Start":"01:57.460 ","End":"02:03.775","Text":"it\u0027s shaded region. Shade it a bit."},{"Start":"02:03.775 ","End":"02:10.960","Text":"There we go, and this will be our region D. Now,"},{"Start":"02:10.960 ","End":"02:14.345","Text":"I want to write this in polar form."},{"Start":"02:14.345 ","End":"02:17.005","Text":"Polar form, this is pretty simple."},{"Start":"02:17.005 ","End":"02:21.580","Text":"The semicircle will take the angle Theta from here to here."},{"Start":"02:21.580 ","End":"02:25.310","Text":"In other words, we start from Theta equals 0 and"},{"Start":"02:25.310 ","End":"02:30.420","Text":"go counterclockwise up to Theta equals a 180 degrees."},{"Start":"02:30.420 ","End":"02:33.485","Text":"Only, we work with radians Theta equals Pi."},{"Start":"02:33.485 ","End":"02:36.215","Text":"For each given Theta,"},{"Start":"02:36.215 ","End":"02:41.405","Text":"r always goes from 0 to 1."},{"Start":"02:41.405 ","End":"02:44.210","Text":"We can say that"},{"Start":"02:44.210 ","End":"02:54.420","Text":"Theta goes from 0"},{"Start":"02:54.420 ","End":"03:00.240","Text":"to Pi and that r goes from 0 to 1."},{"Start":"03:00.240 ","End":"03:06.780","Text":"I can rewrite this now as double integral,"},{"Start":"03:06.780 ","End":"03:09.195","Text":"but this time in polar."},{"Start":"03:09.195 ","End":"03:13.155","Text":"Theta goes from 0 to Pi,"},{"Start":"03:13.155 ","End":"03:17.520","Text":"r goes from 0 to 1."},{"Start":"03:17.520 ","End":"03:20.190","Text":"You have to remember that dxdy,"},{"Start":"03:20.190 ","End":"03:29.670","Text":"which is dA in polar convert to r dr, d Theta."},{"Start":"03:29.670 ","End":"03:31.200","Text":"I\u0027m assuming you know all the formulas."},{"Start":"03:31.200 ","End":"03:36.315","Text":"I didn\u0027t bother copying them all, but that\u0027s dA."},{"Start":"03:36.315 ","End":"03:39.435","Text":"Now, we just have to integrate this."},{"Start":"03:39.435 ","End":"03:46.495","Text":"As usual, we do the innermost first. That\u0027s this."},{"Start":"03:46.495 ","End":"03:51.680","Text":"In fact, it\u0027s simple enough to do in our heads because the integral of r is a 1/2 r"},{"Start":"03:51.680 ","End":"03:58.340","Text":"squared and if I substitute a 1/2 r-squared once r equals 1 and once r equals 0,"},{"Start":"03:58.340 ","End":"04:00.845","Text":"I just end up with 1/2."},{"Start":"04:00.845 ","End":"04:04.200","Text":"This bit is 1/2."},{"Start":"04:07.000 ","End":"04:12.400","Text":"I can take the 1/2 out front and I\u0027ve got the integral from Theta"},{"Start":"04:12.400 ","End":"04:21.345","Text":"equals 0 to Pi of just 1 d Theta."},{"Start":"04:21.345 ","End":"04:24.335","Text":"Just d Theta, but like to write the 1 in."},{"Start":"04:24.335 ","End":"04:26.420","Text":"Whenever we have the integral of 1,"},{"Start":"04:26.420 ","End":"04:28.530","Text":"it\u0027s just the upper minus the lower."},{"Start":"04:28.530 ","End":"04:31.600","Text":"It\u0027s Pi minus 0, it\u0027s Pi."},{"Start":"04:31.600 ","End":"04:37.110","Text":"This is equal to 1/2 Pi,"},{"Start":"04:37.110 ","End":"04:38.495","Text":"and that is the answer."},{"Start":"04:38.495 ","End":"04:40.310","Text":"But don\u0027t go yet."},{"Start":"04:40.310 ","End":"04:46.310","Text":"I just want to show you another way we could have got to this answer if it hadn\u0027t said,"},{"Start":"04:46.310 ","End":"04:48.394","Text":"we have to convert to polar."},{"Start":"04:48.394 ","End":"04:53.485","Text":"The double integral of 1 dA"},{"Start":"04:53.485 ","End":"04:58.865","Text":"or dydx is just the area of the region D. On the other hand,"},{"Start":"04:58.865 ","End":"05:05.030","Text":"this is equal to the area of D. Now,"},{"Start":"05:05.030 ","End":"05:06.530","Text":"what is the area of D?"},{"Start":"05:06.530 ","End":"05:08.255","Text":"It\u0027s half a circle."},{"Start":"05:08.255 ","End":"05:14.325","Text":"In general, a circle is Pi r squared."},{"Start":"05:14.325 ","End":"05:17.070","Text":"That\u0027s for a circle."},{"Start":"05:17.070 ","End":"05:21.100","Text":"But we have a 1/2 circle of radius 1."},{"Start":"05:21.100 ","End":"05:26.940","Text":"Our area is going to be 1/2 for the half circle,"},{"Start":"05:26.940 ","End":"05:29.890","Text":"Pi and r is 1,"},{"Start":"05:29.890 ","End":"05:32.355","Text":"in our case squared."},{"Start":"05:32.355 ","End":"05:38.930","Text":"This is also equal to Pi over 2 or a 1/2 Pi."},{"Start":"05:38.930 ","End":"05:41.240","Text":"We got the same answer."},{"Start":"05:41.240 ","End":"05:47.090","Text":"Let me just rewrite it. That\u0027s 1/2 Pi because I wanted to really look like this."},{"Start":"05:47.090 ","End":"05:48.950","Text":"That was an alternative way,"},{"Start":"05:48.950 ","End":"05:51.665","Text":"but we did practice polar coordinates."},{"Start":"05:51.665 ","End":"05:53.910","Text":"Anyway, I\u0027m done."}],"ID":8695},{"Watched":false,"Name":"Exercise 11","Duration":"6m 38s","ChapterTopicVideoID":8500,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8500.jpeg","UploadDate":"2017-02-13T08:52:05.5300000","DurationForVideoObject":"PT6M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.190","Text":"In this exercise, we want to compute the integral by converting to polar coordinates."},{"Start":"00:08.190 ","End":"00:13.140","Text":"The strategy is, we\u0027ll be to write it as the double integral"},{"Start":"00:13.140 ","End":"00:19.300","Text":"over a region D of the same thing, dydx."},{"Start":"00:21.650 ","End":"00:25.140","Text":"Once we found what the region D looks like,"},{"Start":"00:25.140 ","End":"00:28.500","Text":"then we\u0027ll express D in polar terms."},{"Start":"00:28.500 ","End":"00:30.735","Text":"Let\u0027s see what D might look like."},{"Start":"00:30.735 ","End":"00:34.650","Text":"The outer limits from minus 1 to 1,"},{"Start":"00:34.650 ","End":"00:39.920","Text":"that\u0027s x. I know that x goes from minus 1-1,"},{"Start":"00:39.920 ","End":"00:42.125","Text":"and for each sets x,"},{"Start":"00:42.125 ","End":"00:50.020","Text":"y goes between minus the square root of 1 minus x squared,"},{"Start":"00:50.020 ","End":"00:55.675","Text":"and plus the square root of 1 minus x squared."},{"Start":"00:55.675 ","End":"00:59.750","Text":"You should have seen this enough times before to recognize that this is"},{"Start":"00:59.750 ","End":"01:03.290","Text":"actually an upper semicircle and the lowest semicircle,"},{"Start":"01:03.290 ","End":"01:04.535","Text":"in other words the circle."},{"Start":"01:04.535 ","End":"01:06.670","Text":"Let me bring a sketch in."},{"Start":"01:06.670 ","End":"01:09.680","Text":"If you\u0027re not sure why, I\u0027ll justify this in a moment,"},{"Start":"01:09.680 ","End":"01:11.270","Text":"let me just continue."},{"Start":"01:11.270 ","End":"01:16.720","Text":"We have that x goes between 1 and minus 1,"},{"Start":"01:16.720 ","End":"01:19.610","Text":"and for each x in this,"},{"Start":"01:19.610 ","End":"01:21.995","Text":"if we take a vertical slice,"},{"Start":"01:21.995 ","End":"01:29.490","Text":"like so, then y goes from below to this point and above to this point,"},{"Start":"01:29.490 ","End":"01:32.990","Text":"and I\u0027m claiming that this lower semicircle is"},{"Start":"01:32.990 ","End":"01:38.385","Text":"exactly y equals minus root of 1 minus x squared,"},{"Start":"01:38.385 ","End":"01:42.435","Text":"and the upper 1 is y equals root 1 minus x squared."},{"Start":"01:42.435 ","End":"01:44.400","Text":"I\u0027ll tell you, you should have seen this before,"},{"Start":"01:44.400 ","End":"01:46.040","Text":"let me just show you again why."},{"Start":"01:46.040 ","End":"01:47.840","Text":"Actually, let me do it backwards."},{"Start":"01:47.840 ","End":"01:50.330","Text":"Let\u0027s start with the circle of radius 1,"},{"Start":"01:50.330 ","End":"01:57.060","Text":"which we know is x squared plus y squared equals radius squared, 1 squared."},{"Start":"01:57.060 ","End":"02:00.240","Text":"Then I\u0027ll bring the x squared over to the other side,"},{"Start":"02:00.240 ","End":"02:03.540","Text":"so I get y squared equals 1 minus x squared,"},{"Start":"02:03.540 ","End":"02:05.565","Text":"and then I take the square root."},{"Start":"02:05.565 ","End":"02:11.810","Text":"The y could be either plus or minus the square root of 1 minus x squared."},{"Start":"02:11.810 ","End":"02:14.075","Text":"Obviously, the plus is this 1,"},{"Start":"02:14.075 ","End":"02:18.020","Text":"the minus is this 1, and anything between,"},{"Start":"02:18.020 ","End":"02:24.020","Text":"like this is the between and I could continue shading the whole circle,"},{"Start":"02:24.020 ","End":"02:27.170","Text":"so actually we have a disk on it, precisely a circle."},{"Start":"02:27.170 ","End":"02:30.740","Text":"It\u0027s actually called a disk and we have the inside as well,"},{"Start":"02:30.740 ","End":"02:39.960","Text":"and this is what we call D. Now I want to express this D in polar terms."},{"Start":"02:39.960 ","End":"02:42.830","Text":"I cleaned up a bit and now we want to do it,"},{"Start":"02:42.830 ","End":"02:45.170","Text":"not in Cartesian, but in polar."},{"Start":"02:45.170 ","End":"02:49.160","Text":"A whole circle means that we could take Theta,"},{"Start":"02:49.160 ","End":"02:57.030","Text":"here Theta equals 0 and we can go the whole way around and end up the same place,"},{"Start":"02:57.030 ","End":"03:00.435","Text":"but I could call this Theta also 2Pi,"},{"Start":"03:00.435 ","End":"03:03.135","Text":"and for each particular Theta,"},{"Start":"03:03.135 ","End":"03:05.340","Text":"let\u0027s say this is a typical Theta,"},{"Start":"03:05.340 ","End":"03:11.265","Text":"r always from 0-1 because the radius is 1."},{"Start":"03:11.265 ","End":"03:19.020","Text":"Now I can write the same constraints instead of this way, in polar terms,"},{"Start":"03:19.020 ","End":"03:23.710","Text":"it would go as Theta"},{"Start":"03:23.710 ","End":"03:30.530","Text":"goes between 0 and 360 degrees only we use radians,"},{"Start":"03:30.530 ","End":"03:39.030","Text":"and r doesn\u0027t matter for what Theta doesn\u0027t depend on it as always between 0 and 1."},{"Start":"03:39.030 ","End":"03:47.115","Text":"Now I\u0027m going to rewrite this integral as the integral, let\u0027s see."},{"Start":"03:47.115 ","End":"03:51.285","Text":"Theta goes from 0-2Pi,"},{"Start":"03:51.285 ","End":"03:56.055","Text":"r goes from 0-1,"},{"Start":"03:56.055 ","End":"04:01.260","Text":"and dydx, which is dA in"},{"Start":"04:01.260 ","End":"04:07.155","Text":"polar form, is rdrd Theta."},{"Start":"04:07.155 ","End":"04:11.390","Text":"Now I\u0027ve expressed the same integral in polar coordinates,"},{"Start":"04:11.390 ","End":"04:14.430","Text":"and now let\u0027s actually compute the thing."},{"Start":"04:15.260 ","End":"04:18.705","Text":"As usual, we work from inside out,"},{"Start":"04:18.705 ","End":"04:21.680","Text":"so we\u0027ll do the inner 1 1st."},{"Start":"04:21.680 ","End":"04:25.159","Text":"Let me do this as a side exercise asterisk,"},{"Start":"04:25.159 ","End":"04:26.765","Text":"I\u0027ll do this at the side."},{"Start":"04:26.765 ","End":"04:33.845","Text":"What we have basically is the integral of r is a 1/2r squared."},{"Start":"04:33.845 ","End":"04:38.950","Text":"This will have to evaluate between 0-1,"},{"Start":"04:38.950 ","End":"04:41.475","Text":"so this is equal to, if I plug in 1,"},{"Start":"04:41.475 ","End":"04:47.025","Text":"I get 1/2, 1 squared minus 1/2 times 0 squared,"},{"Start":"04:47.025 ","End":"04:51.805","Text":"and this is equal to just 1/2."},{"Start":"04:51.805 ","End":"04:57.480","Text":"I take this and just make a note to myself that this is 1/2,"},{"Start":"04:57.480 ","End":"04:59.570","Text":"and being a constant,"},{"Start":"04:59.570 ","End":"05:01.790","Text":"I can take this thing in front,"},{"Start":"05:01.790 ","End":"05:03.830","Text":"so I\u0027ve got 1/2,"},{"Start":"05:03.830 ","End":"05:12.780","Text":"the integral from 0-2Pi of just d Theta or 1dTheta."},{"Start":"05:15.230 ","End":"05:18.045","Text":"Whenever we have the integral of 1,"},{"Start":"05:18.045 ","End":"05:20.655","Text":"it\u0027s just the upper minus the lower,"},{"Start":"05:20.655 ","End":"05:26.640","Text":"so it\u0027s 1/2 times 2Pi minus 0 is just 2Pi,"},{"Start":"05:26.640 ","End":"05:30.990","Text":"and a half times 2Pi is Pi,"},{"Start":"05:30.990 ","End":"05:33.890","Text":"and that is our answer."},{"Start":"05:33.890 ","End":"05:38.270","Text":"Although we\u0027re done, I\u0027d like to just show you another way it could have been done."},{"Start":"05:38.270 ","End":"05:41.840","Text":"If the question hadn\u0027t said use polar coordinates,"},{"Start":"05:41.840 ","End":"05:43.175","Text":"we could have done it more easily."},{"Start":"05:43.175 ","End":"05:44.570","Text":"Let me just copy that."},{"Start":"05:44.570 ","End":"05:48.140","Text":"If I have the double integral of,"},{"Start":"05:48.140 ","End":"05:51.340","Text":"we can add a 1 here, of course,"},{"Start":"05:53.020 ","End":"05:58.370","Text":"of 1 dA over a region D,"},{"Start":"05:58.370 ","End":"06:02.300","Text":"the integral of 1 it\u0027s well known, we\u0027ve studied this,"},{"Start":"06:02.300 ","End":"06:08.805","Text":"is just the area of the region D. Now,"},{"Start":"06:08.805 ","End":"06:13.550","Text":"D is a circle and it\u0027s well-known that the area of the circle of"},{"Start":"06:13.550 ","End":"06:18.470","Text":"radius r is Pi r squared, which in our case,"},{"Start":"06:18.470 ","End":"06:23.880","Text":"since the radius is 1, is Pi times 1 squared, which is Pi,"},{"Start":"06:23.880 ","End":"06:26.180","Text":"so we got the same answer in"},{"Start":"06:26.180 ","End":"06:29.060","Text":"a much simpler way by using"},{"Start":"06:29.060 ","End":"06:33.275","Text":"the property that the integral of 1 over a region is its area."},{"Start":"06:33.275 ","End":"06:35.135","Text":"That was just an extra,"},{"Start":"06:35.135 ","End":"06:37.710","Text":"and we are really done."}],"ID":8696},{"Watched":false,"Name":"Exercise 12","Duration":"6m 23s","ChapterTopicVideoID":8501,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8501.jpeg","UploadDate":"2017-02-13T08:53:46.4530000","DurationForVideoObject":"PT6M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"In this exercise, we have to compute the following"},{"Start":"00:02.790 ","End":"00:06.555","Text":"integral by converting to polar coordinates."},{"Start":"00:06.555 ","End":"00:09.430","Text":"Notice that it\u0027s dx, dy,"},{"Start":"00:09.430 ","End":"00:13.700","Text":"which in general is da,"},{"Start":"00:13.700 ","End":"00:17.340","Text":"but it means that it\u0027s horizontal slices."},{"Start":"00:17.340 ","End":"00:18.810","Text":"I\u0027ll explain what I mean."},{"Start":"00:18.810 ","End":"00:20.265","Text":"It\u0027s a type 2 region."},{"Start":"00:20.265 ","End":"00:23.865","Text":"If I write this as an integral over a region,"},{"Start":"00:23.865 ","End":"00:26.310","Text":"over D of the same thing,"},{"Start":"00:26.310 ","End":"00:30.900","Text":"x^ squared plus y^ squared dx, dy,"},{"Start":"00:30.900 ","End":"00:33.990","Text":"well, as I said,"},{"Start":"00:33.990 ","End":"00:36.105","Text":"I should really write this da."},{"Start":"00:36.105 ","End":"00:39.870","Text":"But this indicates that y does"},{"Start":"00:39.870 ","End":"00:44.690","Text":"the outward loop and then x for each y travels from something to something,"},{"Start":"00:44.690 ","End":"00:48.410","Text":"so it\u0027s a type 2 region horizontal slices."},{"Start":"00:48.410 ","End":"00:50.930","Text":"Let\u0027s see what this D would look like."},{"Start":"00:50.930 ","End":"00:56.020","Text":"Now, y, which is the outer loop, goes from 0-1,"},{"Start":"00:56.020 ","End":"00:58.940","Text":"and x on the inner loop goes from"},{"Start":"00:58.940 ","End":"01:03.995","Text":"0 to the square root of 1 minus y squared for that particular y."},{"Start":"01:03.995 ","End":"01:07.295","Text":"Let me write that. Once again,"},{"Start":"01:07.295 ","End":"01:08.585","Text":"on the outer loop,"},{"Start":"01:08.585 ","End":"01:12.195","Text":"y runs from 0-1,"},{"Start":"01:12.195 ","End":"01:14.375","Text":"and for each such y,"},{"Start":"01:14.375 ","End":"01:23.840","Text":"x horizontally travels from 0 to the positive square root of 1 minus y squared."},{"Start":"01:23.840 ","End":"01:27.145","Text":"Now let\u0027s try and sketch this."},{"Start":"01:27.145 ","End":"01:30.095","Text":"Here\u0027s a pair of axis and you know what?"},{"Start":"01:30.095 ","End":"01:33.335","Text":"I\u0027ll sketch it and then explain how I arrived at it."},{"Start":"01:33.335 ","End":"01:38.030","Text":"I claim it\u0027s just exactly this quarter circle and"},{"Start":"01:38.030 ","End":"01:42.200","Text":"the interior between this and the axis. How do I figure this?"},{"Start":"01:42.200 ","End":"01:44.470","Text":"Y goes from 0-1,"},{"Start":"01:44.470 ","End":"01:47.610","Text":"so here\u0027s 0 and here\u0027s 1."},{"Start":"01:47.610 ","End":"01:49.620","Text":"For each particular y,"},{"Start":"01:49.620 ","End":"01:58.300","Text":"x goes from 0 up to this function of y."},{"Start":"01:58.300 ","End":"01:59.890","Text":"It\u0027s a 1/4 of a circle,"},{"Start":"01:59.890 ","End":"02:01.105","Text":"it\u0027s part of a circle."},{"Start":"02:01.105 ","End":"02:07.350","Text":"We\u0027ve seen this before, but I\u0027ll show you that this part is where x is equal to,"},{"Start":"02:07.350 ","End":"02:11.110","Text":"and if x is equal to square root of 1 minus y squared,"},{"Start":"02:11.110 ","End":"02:12.490","Text":"and I square both sides,"},{"Start":"02:12.490 ","End":"02:17.320","Text":"x squared is 1 minus y squared or x squared plus y squared equals 1,"},{"Start":"02:17.320 ","End":"02:19.074","Text":"so it\u0027s part of a circle."},{"Start":"02:19.074 ","End":"02:24.280","Text":"The positive square root means it\u0027s the right side and we\u0027re limiting x from 0 -1,"},{"Start":"02:24.280 ","End":"02:27.020","Text":"so we get 1/4 of a circle."},{"Start":"02:27.140 ","End":"02:32.430","Text":"I\u0027ve shaded it, and let\u0027s label it D. Now,"},{"Start":"02:32.430 ","End":"02:36.630","Text":"I want to describe this D in polar terms."},{"Start":"02:36.740 ","End":"02:40.400","Text":"This was the cartesian description of D,"},{"Start":"02:40.400 ","End":"02:42.470","Text":"now I want to describe it in terms of r and Theta,"},{"Start":"02:42.470 ","End":"02:46.050","Text":"so I\u0027m going to erase what I don\u0027t need, there."},{"Start":"02:46.050 ","End":"02:48.390","Text":"I\u0027ll just write a 1 here."},{"Start":"02:48.390 ","End":"02:52.585","Text":"Now we want to describe it in terms of r and Theta."},{"Start":"02:52.585 ","End":"02:54.800","Text":"This is an arc of a circle."},{"Start":"02:54.800 ","End":"03:00.995","Text":"This line here is where we start and we go counter-clockwise up to here."},{"Start":"03:00.995 ","End":"03:03.815","Text":"Theta is going from here to here."},{"Start":"03:03.815 ","End":"03:07.970","Text":"This line here is where Theta equals 0."},{"Start":"03:07.970 ","End":"03:10.915","Text":"This vertical line was where Theta equals,"},{"Start":"03:10.915 ","End":"03:12.660","Text":"won\u0027t say 90 degrees,"},{"Start":"03:12.660 ","End":"03:15.705","Text":"I\u0027ll say Pi over 2 because we\u0027re in radians."},{"Start":"03:15.705 ","End":"03:21.470","Text":"For each particular Theta in this range from 0 to Pi over 2,"},{"Start":"03:21.470 ","End":"03:27.575","Text":"r goes from 0-1 always, so that\u0027s constant."},{"Start":"03:27.575 ","End":"03:34.070","Text":"I can describe this D in polar terms as"},{"Start":"03:34.070 ","End":"03:43.265","Text":"the outer loop is Theta between 0 and I say 90 degrees and I write Pi over 2,"},{"Start":"03:43.265 ","End":"03:49.695","Text":"then r goes between same always 0 and 1."},{"Start":"03:49.695 ","End":"03:53.655","Text":"Now this is a polar region,"},{"Start":"03:53.655 ","End":"03:57.910","Text":"and now we can write this in polar terms."},{"Start":"03:59.810 ","End":"04:06.060","Text":"We usually take Theta as the outer loop from 0 to Pi over 2,"},{"Start":"04:06.060 ","End":"04:08.204","Text":"that maybe Theta equals,"},{"Start":"04:08.204 ","End":"04:10.680","Text":"and that will be here, D Theta,"},{"Start":"04:10.680 ","End":"04:19.895","Text":"and then r from 0-1 dr. Well,"},{"Start":"04:19.895 ","End":"04:25.765","Text":"it doesn\u0027t quite work that way because da is actually r dr d Theta,"},{"Start":"04:25.765 ","End":"04:30.395","Text":"but at least this tells us the order that Theta\u0027s the outer and r is the inner."},{"Start":"04:30.395 ","End":"04:32.450","Text":"Next, we have to convert this bit,"},{"Start":"04:32.450 ","End":"04:34.130","Text":"x squared plus y squared."},{"Start":"04:34.130 ","End":"04:37.100","Text":"If you remember your formulas and you should,"},{"Start":"04:37.100 ","End":"04:40.440","Text":"then x squared plus y squared is r squared."},{"Start":"04:40.440 ","End":"04:46.070","Text":"If you forgot, refer to your formula sheet for polar conversion."},{"Start":"04:46.110 ","End":"04:55.700","Text":"We have this, I\u0027m going to save a line and instead of r squared times r,"},{"Start":"04:55.700 ","End":"04:59.250","Text":"I\u0027ll write it as r cubed to save a line."},{"Start":"04:59.480 ","End":"05:03.565","Text":"We do the calculations from the inside out."},{"Start":"05:03.565 ","End":"05:05.735","Text":"First of all, we do this."},{"Start":"05:05.735 ","End":"05:08.960","Text":"I like to do the inner 1 as a side exercise or call it"},{"Start":"05:08.960 ","End":"05:13.025","Text":"asterisk and do it at the side and then return here."},{"Start":"05:13.025 ","End":"05:19.565","Text":"What I have is the integral from 0-1 of r cubed dr,"},{"Start":"05:19.565 ","End":"05:24.360","Text":"and that is equal to 1/4 r to the 4,"},{"Start":"05:24.360 ","End":"05:27.285","Text":"taken between 0 and 1."},{"Start":"05:27.285 ","End":"05:32.615","Text":"When I put in 1, I just get 1 to the 4 times a 1/4 is 1/4."},{"Start":"05:32.615 ","End":"05:37.065","Text":"When I put in 0 it\u0027s just 0."},{"Start":"05:37.065 ","End":"05:38.760","Text":"It\u0027s a 1/4 minus 0,"},{"Start":"05:38.760 ","End":"05:40.530","Text":"which is just a 1/4."},{"Start":"05:40.530 ","End":"05:43.659","Text":"When I go back here,"},{"Start":"05:46.970 ","End":"05:50.720","Text":"this thing, which is 1/4 from there,"},{"Start":"05:50.720 ","End":"05:54.320","Text":"I\u0027ll pull outside the integral because it\u0027s a constant,"},{"Start":"05:54.320 ","End":"06:02.010","Text":"so I get 1/4 times the integral from 0-Pi over 2 of just d Theta,"},{"Start":"06:02.010 ","End":"06:04.425","Text":"I\u0027ll write it as 1 d Theta."},{"Start":"06:04.425 ","End":"06:06.950","Text":"As you know, when you have the integral of 1,"},{"Start":"06:06.950 ","End":"06:09.695","Text":"it\u0027s just the upper limit minus the lower limit,"},{"Start":"06:09.695 ","End":"06:15.735","Text":"so what I get is 1/4 Pi over 2 minus 0 is Pi over 2."},{"Start":"06:15.735 ","End":"06:19.125","Text":"I get Pi over 8 or 1/8 Pi,"},{"Start":"06:19.125 ","End":"06:21.225","Text":"whichever way you prefer."},{"Start":"06:21.225 ","End":"06:24.790","Text":"I\u0027ll highlight that and declare that we\u0027re"}],"ID":8697},{"Watched":false,"Name":"Exercise 13","Duration":"5m 51s","ChapterTopicVideoID":8502,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8502.jpeg","UploadDate":"2017-02-13T08:55:15.1200000","DurationForVideoObject":"PT5M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we have to compute the following integral,"},{"Start":"00:03.450 ","End":"00:06.285","Text":"but we have to convert it to polar coordinates."},{"Start":"00:06.285 ","End":"00:09.750","Text":"What we\u0027re going to do, is write this as the double"},{"Start":"00:09.750 ","End":"00:16.620","Text":"integral over a region, or domain D of the same thing,"},{"Start":"00:16.620 ","End":"00:20.980","Text":"x squared plus y squared dx, dy."},{"Start":"00:20.980 ","End":"00:24.365","Text":"We\u0027re going to sketch D, and then we\u0027ll convert it to polar."},{"Start":"00:24.365 ","End":"00:28.340","Text":"Remember that dx, dy is just 1 of the ways of saying da."},{"Start":"00:28.340 ","End":"00:32.970","Text":"It means the outer loop is y and the inner loop is x."},{"Start":"00:33.050 ","End":"00:36.150","Text":"Let\u0027s see if we can sketch this now."},{"Start":"00:36.150 ","End":"00:38.735","Text":"If I write the equations for this,"},{"Start":"00:38.735 ","End":"00:45.980","Text":"this says that outwardly y goes from minus 1 to 1."},{"Start":"00:45.980 ","End":"00:50.580","Text":"For each such, y the inner loop x goes"},{"Start":"00:50.580 ","End":"00:58.355","Text":"from this function of y, square root of 1 minus y squared above and below,"},{"Start":"00:58.355 ","End":"01:02.745","Text":"minus the square root of 1 minus y squared."},{"Start":"01:02.745 ","End":"01:04.970","Text":"This describes the region D,"},{"Start":"01:04.970 ","End":"01:07.595","Text":"at least in Cartesian coordinates."},{"Start":"01:07.595 ","End":"01:13.470","Text":"I want to find a different description of D in polar coordinates,"},{"Start":"01:13.470 ","End":"01:16.235","Text":"so let\u0027s introduce a sketch,"},{"Start":"01:16.235 ","End":"01:18.970","Text":"I\u0027ll just move this out of the way."},{"Start":"01:18.970 ","End":"01:22.175","Text":"I brought in the picture, actually,"},{"Start":"01:22.175 ","End":"01:24.950","Text":"solved already, and I\u0027m going to explain how I got to it."},{"Start":"01:24.950 ","End":"01:29.074","Text":"I\u0027m claiming that this just describes the unit circle."},{"Start":"01:29.074 ","End":"01:32.355","Text":"Well, look, y goes from -1 to 1."},{"Start":"01:32.355 ","End":"01:35.200","Text":"So we start off at minus 1 and end at 1."},{"Start":"01:35.200 ","End":"01:41.640","Text":"Now, for each y, let\u0027s say this is a typical y,"},{"Start":"01:42.760 ","End":"01:45.410","Text":"we take a horizontal slice."},{"Start":"01:45.410 ","End":"01:53.985","Text":"So x is traveling from 1 function to another function from this 1 to this 1."},{"Start":"01:53.985 ","End":"02:00.710","Text":"Now, I\u0027m claiming that these 2 are just simply the left and right halves of a circle."},{"Start":"02:00.710 ","End":"02:10.860","Text":"I say that this is the minus, and this is the plus."},{"Start":"02:11.080 ","End":"02:13.550","Text":"We\u0027ve seen this enough times,"},{"Start":"02:13.550 ","End":"02:15.365","Text":"but I\u0027ll show you again."},{"Start":"02:15.365 ","End":"02:17.300","Text":"It\u0027s easier to show it in reverse."},{"Start":"02:17.300 ","End":"02:22.940","Text":"The circle is x squared plus y squared equals 1."},{"Start":"02:22.940 ","End":"02:28.225","Text":"So y squared is 1 minus x squared."},{"Start":"02:28.225 ","End":"02:33.605","Text":"Y is plus or minus the square root of 1 minus x squared."},{"Start":"02:33.605 ","End":"02:36.050","Text":"The right-hand side would be the plus,"},{"Start":"02:36.050 ","End":"02:39.200","Text":"and the left-hand side would be the minus."},{"Start":"02:39.200 ","End":"02:43.325","Text":"This goes from square root of 1 minus y squared."},{"Start":"02:43.325 ","End":"02:49.100","Text":"This is where x is traveling down to minus the square root of 1 minus y squared."},{"Start":"02:49.100 ","End":"02:55.085","Text":"Now, as I go from minus 1 to 1, and sweeping out the whole circle, so that this, in fact,"},{"Start":"02:55.085 ","End":"03:02.210","Text":"is our domain D. Now, I\u0027m going to clean this up a bit."},{"Start":"03:02.210 ","End":"03:07.610","Text":"I would like to do is describe the same region D in terms of polar."},{"Start":"03:07.610 ","End":"03:09.665","Text":"We\u0027ve done this enough times already."},{"Start":"03:09.665 ","End":"03:12.680","Text":"We take Theta all the way around,"},{"Start":"03:12.680 ","End":"03:16.760","Text":"and I\u0027m not even going to draw it all the way up to back to here."},{"Start":"03:16.760 ","End":"03:23.870","Text":"From Theta, from 0, all the way around to 2 Pi."},{"Start":"03:23.870 ","End":"03:27.770","Text":"For each Theta, for any typical Theta,"},{"Start":"03:27.770 ","End":"03:29.210","Text":"r is going to be the same."},{"Start":"03:29.210 ","End":"03:33.620","Text":"It\u0027s always going to go from 0-1, because this is a circle."},{"Start":"03:33.620 ","End":"03:38.525","Text":"Now, we can describe D in polar terms."},{"Start":"03:38.525 ","End":"03:45.885","Text":"The outer loop, Theta from 0 to 2 Pi,"},{"Start":"03:45.885 ","End":"03:51.765","Text":"and the inner loop are from 0 to 1."},{"Start":"03:51.765 ","End":"03:58.500","Text":"We can now do the conversion and convert this to polar,"},{"Start":"03:58.760 ","End":"04:05.674","Text":"and we get the outer integral is Theta equals 0-2 Pi,"},{"Start":"04:05.674 ","End":"04:11.150","Text":"the inner integral, r equals 0-1,"},{"Start":"04:11.150 ","End":"04:17.750","Text":"x squared plus y squared from the formula sheet is r squared and dx,"},{"Start":"04:17.750 ","End":"04:28.290","Text":"dy or da is just r dr, d Theta."},{"Start":"04:29.480 ","End":"04:35.685","Text":"I\u0027m going to save a line by writing r cubed instead of r squared r,"},{"Start":"04:35.685 ","End":"04:40.025","Text":"as usual, we do the inner integral first."},{"Start":"04:40.025 ","End":"04:44.180","Text":"We\u0027ve done this before and the answer comes out to be a quarter,"},{"Start":"04:44.180 ","End":"04:46.475","Text":"but now, I\u0027ll remind you how we did this."},{"Start":"04:46.475 ","End":"04:49.084","Text":"Let me do this at the side, I\u0027ll put an asterisk,"},{"Start":"04:49.084 ","End":"04:50.585","Text":"do it over here."},{"Start":"04:50.585 ","End":"04:54.949","Text":"The integral from 0-1 of r cubed,"},{"Start":"04:54.949 ","End":"05:00.005","Text":"dr is just r cubed gives us 1 quarter,"},{"Start":"05:00.005 ","End":"05:01.684","Text":"r to the fourth,"},{"Start":"05:01.684 ","End":"05:03.815","Text":"taken between 0 and 1."},{"Start":"05:03.815 ","End":"05:05.630","Text":"When I plug in 1,"},{"Start":"05:05.630 ","End":"05:09.815","Text":"I get 1 quarter times 1 to the fourth is 1 quarter."},{"Start":"05:09.815 ","End":"05:12.035","Text":"When I plug in 0, I just get 0."},{"Start":"05:12.035 ","End":"05:14.790","Text":"That\u0027s why this is a quarter."},{"Start":"05:14.870 ","End":"05:18.290","Text":"Continuing, I get all the quarters."},{"Start":"05:18.290 ","End":"05:20.810","Text":"A constant comes out in front of the integral,"},{"Start":"05:20.810 ","End":"05:25.275","Text":"the integral from 0-2 Pi of just d Theta,"},{"Start":"05:25.275 ","End":"05:27.980","Text":"which I prefer to write as 1d Theta."},{"Start":"05:27.980 ","End":"05:31.280","Text":"As we know, whenever you have the integral of 1,"},{"Start":"05:31.280 ","End":"05:33.875","Text":"it\u0027s just the upper limit minus the lower limit."},{"Start":"05:33.875 ","End":"05:38.590","Text":"I get 1 quarter of 2 Pi minus 0."},{"Start":"05:38.590 ","End":"05:43.565","Text":"The final answer is a quarter times 2 Pi is just a half Pi."},{"Start":"05:43.565 ","End":"05:47.465","Text":"Half Pi or Pi over 2, whatever you prefer."},{"Start":"05:47.465 ","End":"05:49.520","Text":"This is the answer,"},{"Start":"05:49.520 ","End":"05:51.690","Text":"and we are done."}],"ID":8698},{"Watched":false,"Name":"Exercise 14","Duration":"7m 7s","ChapterTopicVideoID":8503,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8503.jpeg","UploadDate":"2017-02-13T08:57:00.1570000","DurationForVideoObject":"PT7M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"In this exercise, we have to compute the following"},{"Start":"00:02.520 ","End":"00:05.925","Text":"integral by converting to polar coordinates."},{"Start":"00:05.925 ","End":"00:08.325","Text":"What I\u0027m going to do is first,"},{"Start":"00:08.325 ","End":"00:12.630","Text":"I\u0027m going to sketch the region D such that this double integral"},{"Start":"00:12.630 ","End":"00:17.860","Text":"is the integral over the region D of the same thing, dydx."},{"Start":"00:18.170 ","End":"00:20.700","Text":"I like to write the 1 in,"},{"Start":"00:20.700 ","End":"00:24.820","Text":"and remembering that this dydx is also dA."},{"Start":"00:24.980 ","End":"00:29.040","Text":"I\u0027m going to sketch the region D and then write it in polar."},{"Start":"00:29.040 ","End":"00:33.570","Text":"Now, we\u0027ve seen very similar to this before many times,"},{"Start":"00:33.570 ","End":"00:38.250","Text":"but with 1 instead of A. I\u0027ll just introduce the sketch."},{"Start":"00:38.250 ","End":"00:42.470","Text":"Here\u0027s a sketch of the region D. It\u0027s just a circle centered at"},{"Start":"00:42.470 ","End":"00:46.565","Text":"the origin with radius a, and I\u0027ll explain."},{"Start":"00:46.565 ","End":"00:50.000","Text":"What we have here is the outer limit,"},{"Start":"00:50.000 ","End":"00:53.965","Text":"which is variable x going from minus a to a."},{"Start":"00:53.965 ","End":"01:02.180","Text":"For each such x, y goes vertically from minus this square root to plus this square root."},{"Start":"01:02.180 ","End":"01:03.980","Text":"If I write this,"},{"Start":"01:03.980 ","End":"01:13.160","Text":"then D can be described as x going from minus a to a,"},{"Start":"01:13.160 ","End":"01:15.559","Text":"and for each such x,"},{"Start":"01:15.559 ","End":"01:18.935","Text":"y goes between square root of"},{"Start":"01:18.935 ","End":"01:24.125","Text":"a squared minus x squared and minus square root of a squared minus x squared."},{"Start":"01:24.125 ","End":"01:26.795","Text":"You should by now recognize this as"},{"Start":"01:26.795 ","End":"01:31.490","Text":"the upper semicircle and this is the lower semicircle."},{"Start":"01:31.490 ","End":"01:34.805","Text":"But I\u0027ll show you again why."},{"Start":"01:34.805 ","End":"01:38.030","Text":"We know that an equation of a circle of radius a at"},{"Start":"01:38.030 ","End":"01:41.810","Text":"the origin is x squared plus y squared equals a squared."},{"Start":"01:41.810 ","End":"01:44.929","Text":"If I just bring the x to the other side,"},{"Start":"01:44.929 ","End":"01:48.635","Text":"I get y squared equals a squared minus x squared,"},{"Start":"01:48.635 ","End":"01:55.670","Text":"and so y is plus or minus the square root of a squared minus x squared."},{"Start":"01:55.670 ","End":"02:01.585","Text":"The plus gives the upper semicircle and the minus gives the lower semicircle."},{"Start":"02:01.585 ","End":"02:05.430","Text":"X travels from minus a to a."},{"Start":"02:05.430 ","End":"02:08.355","Text":"Let\u0027s say this is a typical x,"},{"Start":"02:08.355 ","End":"02:16.399","Text":"then we take a vertical slice through this and y goes from this lower semicircle,"},{"Start":"02:16.399 ","End":"02:22.925","Text":"which is minus the square root to the top which is plus the square root."},{"Start":"02:22.925 ","End":"02:24.100","Text":"I could even write it,"},{"Start":"02:24.100 ","End":"02:29.840","Text":"this is y equals square root of a squared minus x squared,"},{"Start":"02:29.840 ","End":"02:34.920","Text":"and y equals minus square root of a squared minus x squared."},{"Start":"02:35.330 ","End":"02:38.720","Text":"This explains how we\u0027ve converted"},{"Start":"02:38.720 ","End":"02:43.310","Text":"this region D to a type 1 region in Cartesian coordinates."},{"Start":"02:43.310 ","End":"02:48.804","Text":"Now we want to get away from Cartesian and move to polar."},{"Start":"02:48.804 ","End":"02:51.255","Text":"I\u0027ve cleaned up what I don\u0027t need,"},{"Start":"02:51.255 ","End":"02:54.285","Text":"and I have here the same domain D,"},{"Start":"02:54.285 ","End":"02:58.340","Text":"it\u0027s a circle of radius a. I want to express it in polar."},{"Start":"02:58.340 ","End":"03:08.220","Text":"A typical angle Theta would be here."},{"Start":"03:08.240 ","End":"03:12.710","Text":"This Theta starts, we can take it to stop from here and"},{"Start":"03:12.710 ","End":"03:18.330","Text":"go all the way round and end up here."},{"Start":"03:18.330 ","End":"03:26.950","Text":"We start off with Theta equals 0 to all the way around and end up with Theta equals 2Pi."},{"Start":"03:26.950 ","End":"03:29.440","Text":"For each such Theta,"},{"Start":"03:29.440 ","End":"03:32.135","Text":"the radius goes from here to here."},{"Start":"03:32.135 ","End":"03:34.790","Text":"Here we have r equals 0,"},{"Start":"03:34.790 ","End":"03:38.600","Text":"and here we have r equals a."},{"Start":"03:38.600 ","End":"03:41.825","Text":"We can now rewrite this region."},{"Start":"03:41.825 ","End":"03:50.520","Text":"In polar, the outer loop Theta goes from 0 to 2Pi,"},{"Start":"03:50.520 ","End":"03:55.920","Text":"and the inner loop a goes from 0 to a."},{"Start":"03:55.920 ","End":"03:59.945","Text":"Now I can do this conversion to polar"},{"Start":"03:59.945 ","End":"04:09.555","Text":"and let\u0027s write Theta equals 0 to 2Pi,"},{"Start":"04:09.555 ","End":"04:15.840","Text":"r goes from 0-1 and now we need the conversion formulas,"},{"Start":"04:15.840 ","End":"04:23.685","Text":"dydx, which is dA is just rdrd Theta,"},{"Start":"04:23.685 ","End":"04:25.530","Text":"the 1 I don\u0027t need."},{"Start":"04:25.530 ","End":"04:27.510","Text":"Now I have to compute."},{"Start":"04:27.510 ","End":"04:31.225","Text":"Sorry, it\u0027s from 0 to a."},{"Start":"04:31.225 ","End":"04:34.010","Text":"In many previous exercises,"},{"Start":"04:34.010 ","End":"04:36.380","Text":"it was from 0-1, sorry."},{"Start":"04:36.380 ","End":"04:39.499","Text":"We do the inner integral first,"},{"Start":"04:39.499 ","End":"04:42.840","Text":"which is this 1,"},{"Start":"04:43.940 ","End":"04:46.980","Text":"and I like to do it at the side,"},{"Start":"04:46.980 ","End":"04:48.060","Text":"I call it asterisk,"},{"Start":"04:48.060 ","End":"04:50.370","Text":"can do it as a side exercise."},{"Start":"04:50.370 ","End":"04:52.460","Text":"The side exercise, I want the"},{"Start":"04:52.460 ","End":"05:02.315","Text":"integral from 0 to a of rdr."},{"Start":"05:02.315 ","End":"05:09.225","Text":"This is equal to the integral of r is 1/2r squared."},{"Start":"05:09.225 ","End":"05:16.085","Text":"I have 1/2, I can take it outside of r squared from 0 to a."},{"Start":"05:16.085 ","End":"05:18.800","Text":"If I plug in a, I get a squared."},{"Start":"05:18.800 ","End":"05:26.880","Text":"If I plug in 0, I get nothing so I end up with 1/2a squared."},{"Start":"05:26.880 ","End":"05:28.680","Text":"Now I go back here,"},{"Start":"05:28.680 ","End":"05:34.475","Text":"I\u0027ll just write that this whole asterisk is 1/2a squared."},{"Start":"05:34.475 ","End":"05:36.410","Text":"Now, that\u0027s a constant,"},{"Start":"05:36.410 ","End":"05:40.855","Text":"I pull it out in front I get 1/2a"},{"Start":"05:40.855 ","End":"05:47.390","Text":"squared times the integral from 0 to 2Pi of d Theta."},{"Start":"05:47.390 ","End":"05:49.505","Text":"I prefer to write 1 in here."},{"Start":"05:49.505 ","End":"05:51.650","Text":"Whenever we have the integral of 1,"},{"Start":"05:51.650 ","End":"05:55.340","Text":"it\u0027s just the upper limit minus the lower limit."},{"Start":"05:55.340 ","End":"06:05.145","Text":"What I get is 1/2a squared times 2Pi minus 0 is 2Pi."},{"Start":"06:05.145 ","End":"06:07.175","Text":"I multiply them together,"},{"Start":"06:07.175 ","End":"06:12.910","Text":"I get Pi a squared."},{"Start":"06:13.970 ","End":"06:16.160","Text":"This is the final answer,"},{"Start":"06:16.160 ","End":"06:17.660","Text":"but don\u0027t go yet."},{"Start":"06:17.660 ","End":"06:20.030","Text":"I\u0027d like to show you another way we could have got to"},{"Start":"06:20.030 ","End":"06:24.260","Text":"this result if we hadn\u0027t been told to use the polar coordinates."},{"Start":"06:24.260 ","End":"06:30.499","Text":"In Cartesian, whenever we have the integral of 1dA over a region,"},{"Start":"06:30.499 ","End":"06:34.615","Text":"I\u0027ll call this double asterisk and continue down here."},{"Start":"06:34.615 ","End":"06:40.380","Text":"The integral of 1 is always just the area of the region,"},{"Start":"06:40.380 ","End":"06:49.515","Text":"so we have the area of D. Now the area of D is a circle of radius a."},{"Start":"06:49.515 ","End":"06:54.859","Text":"In general, the area of a circle is Pi times radius squared."},{"Start":"06:54.859 ","End":"06:59.060","Text":"But our radius in this case is a so it\u0027s Pi a squared,"},{"Start":"06:59.060 ","End":"07:01.535","Text":"so that\u0027s just another way of looking at it."},{"Start":"07:01.535 ","End":"07:02.870","Text":"Area of a circle,"},{"Start":"07:02.870 ","End":"07:07.560","Text":"of radius a. Now we\u0027re done."}],"ID":8699},{"Watched":false,"Name":"Exercise 15","Duration":"4m 41s","ChapterTopicVideoID":8504,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8504.jpeg","UploadDate":"2017-02-13T08:58:11.0730000","DurationForVideoObject":"PT4M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"This exercise is similar to ones we\u0027ve done before."},{"Start":"00:03.240 ","End":"00:05.220","Text":"I\u0027m going to increase the pace."},{"Start":"00:05.220 ","End":"00:10.985","Text":"What I want to do is to compute this integral by converting to polar coordinates."},{"Start":"00:10.985 ","End":"00:17.270","Text":"What I\u0027m going to do is write it as the double integral over a region d of x"},{"Start":"00:17.270 ","End":"00:25.850","Text":"squared plus y squared dxdy and dxdy is also dA."},{"Start":"00:25.850 ","End":"00:31.620","Text":"I\u0027m going to express d in Cartesian terms."},{"Start":"00:31.620 ","End":"00:37.270","Text":"If we look at it, the outer loop is y going from 0 to 2."},{"Start":"00:37.700 ","End":"00:40.725","Text":"For each such y,"},{"Start":"00:40.725 ","End":"00:50.490","Text":"we have limits on x. X goes from 0 to the square root of 4 minus y squared."},{"Start":"00:50.490 ","End":"00:54.540","Text":"Now, I\u0027m going to introduce the sketch of this,"},{"Start":"00:54.540 ","End":"00:59.190","Text":"and then I\u0027ll explain if necessary. Here\u0027s the sketch."},{"Start":"00:59.190 ","End":"01:05.325","Text":"I say it\u0027s 1/4 circle in the first quadrant and that this is 2, and this is 2,"},{"Start":"01:05.325 ","End":"01:10.920","Text":"and you can see this because for any y between 0 and 2,"},{"Start":"01:10.920 ","End":"01:13.560","Text":"let\u0027s say y is here,"},{"Start":"01:13.560 ","End":"01:18.999","Text":"and we draw a horizontal segment and look at the limits,"},{"Start":"01:19.310 ","End":"01:27.765","Text":"x equals 0 is here and x equals square root of 4 minus y squared is here."},{"Start":"01:27.765 ","End":"01:33.675","Text":"It\u0027s on the circle and I\u0027ll show you again,"},{"Start":"01:33.675 ","End":"01:39.045","Text":"this part here where x equals the square root of 4 minus y squared."},{"Start":"01:39.045 ","End":"01:42.570","Text":"If I square both sides and bring the y squared over,"},{"Start":"01:42.570 ","End":"01:45.825","Text":"I get x squared plus y squared equals 4,"},{"Start":"01:45.825 ","End":"01:50.340","Text":"which is the circle of radius 2 because 4 is 2 squared."},{"Start":"01:50.340 ","End":"01:55.770","Text":"But obviously, y is only going on non-negative and so is x,"},{"Start":"01:55.770 ","End":"01:58.440","Text":"so we only get the 1/4 of the circle."},{"Start":"01:58.440 ","End":"02:01.275","Text":"Now we have d in Cartesian coordinates."},{"Start":"02:01.275 ","End":"02:04.125","Text":"Let\u0027s express it in polar coordinates."},{"Start":"02:04.125 ","End":"02:09.330","Text":"In polar coordinates, if we take the angle Theta,"},{"Start":"02:09.330 ","End":"02:11.625","Text":"then Theta goes from here,"},{"Start":"02:11.625 ","End":"02:15.615","Text":"dt equals zero all the way up to here,"},{"Start":"02:15.615 ","End":"02:18.660","Text":"where Theta equals 90 degrees,"},{"Start":"02:18.660 ","End":"02:21.225","Text":"but that\u0027s Pi over 2 in radians."},{"Start":"02:21.225 ","End":"02:27.320","Text":"As for r, it goes from 0 to 2,"},{"Start":"02:27.320 ","End":"02:32.570","Text":"that\u0027s for r. We can rewrite this region in"},{"Start":"02:32.570 ","End":"02:39.885","Text":"polar terms as Theta going from 0 to 90,"},{"Start":"02:39.885 ","End":"02:41.775","Text":"whichever is just Pi over 2,"},{"Start":"02:41.775 ","End":"02:46.395","Text":"and r going from 0 to 2."},{"Start":"02:46.395 ","End":"02:51.330","Text":"Now I convert the region to polar using the standard conversion formulas."},{"Start":"02:51.330 ","End":"02:52.590","Text":"I\u0027ve got the integral,"},{"Start":"02:52.590 ","End":"02:54.029","Text":"first of all, the region,"},{"Start":"02:54.029 ","End":"02:57.945","Text":"we have Theta from 0 to Pi over 2,"},{"Start":"02:57.945 ","End":"03:01.875","Text":"we have r from 0 to 2,"},{"Start":"03:01.875 ","End":"03:05.070","Text":"x squared plus y squared is r squared,"},{"Start":"03:05.070 ","End":"03:11.430","Text":"dA is rdrd Theta."},{"Start":"03:11.430 ","End":"03:14.340","Text":"Now, we have this integral to compute."},{"Start":"03:14.340 ","End":"03:19.185","Text":"I\u0027m going to save a line by writing this as r cubed."},{"Start":"03:19.185 ","End":"03:22.380","Text":"We start by computing the inner integral,"},{"Start":"03:22.380 ","End":"03:27.090","Text":"the dr integral, which I\u0027d like to do at the side,"},{"Start":"03:27.090 ","End":"03:30.120","Text":"I\u0027ll call it asterisk and I\u0027ll do it over here somewhere."},{"Start":"03:30.120 ","End":"03:37.665","Text":"What we have is the integral from 0 to 2 r cubed dr."},{"Start":"03:37.665 ","End":"03:46.950","Text":"This is equal to 1/4 r to the 4 between 0 and 2."},{"Start":"03:46.950 ","End":"03:51.360","Text":"If we plug in 2, we get 2 to the 4 is 16,"},{"Start":"03:51.360 ","End":"03:54.975","Text":"16 over 4 is 4."},{"Start":"03:54.975 ","End":"03:58.485","Text":"0 just gives us 0,"},{"Start":"03:58.485 ","End":"04:00.750","Text":"so we just get 4."},{"Start":"04:00.750 ","End":"04:03.990","Text":"I\u0027ll remind myself that this whole integral is 4,"},{"Start":"04:03.990 ","End":"04:05.775","Text":"doesn\u0027t even depend on Theta."},{"Start":"04:05.775 ","End":"04:10.080","Text":"Continuing, I pull the constant out in front."},{"Start":"04:10.080 ","End":"04:17.740","Text":"I have 4 times the integral from 0 to Pi over 2 of just d Theta,"},{"Start":"04:17.740 ","End":"04:20.430","Text":"but I like to write it as 1 d Theta."},{"Start":"04:20.430 ","End":"04:22.380","Text":"Now, where we have the integral of 1,"},{"Start":"04:22.380 ","End":"04:25.095","Text":"it\u0027s just the upper limit minus the lower limit."},{"Start":"04:25.095 ","End":"04:30.490","Text":"It\u0027s Pi over 2 minus 0 is just Pi over 2."},{"Start":"04:30.490 ","End":"04:33.575","Text":"Then after canceling, 4 over 2 is 2."},{"Start":"04:33.575 ","End":"04:35.915","Text":"The answer is 2 Pi."},{"Start":"04:35.915 ","End":"04:41.160","Text":"I\u0027ll highlight it and we are done."}],"ID":8700},{"Watched":false,"Name":"Exercise 16","Duration":"12m 58s","ChapterTopicVideoID":8505,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8505.jpeg","UploadDate":"2017-02-13T09:01:28.2270000","DurationForVideoObject":"PT12M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this exercise, we have to compute this integral using"},{"Start":"00:03.960 ","End":"00:08.820","Text":"polar coordinates and you\u0027ll see after I\u0027ve drawn the sketch,"},{"Start":"00:08.820 ","End":"00:11.609","Text":"it is not something I would normally do in polar coordinates,"},{"Start":"00:11.609 ","End":"00:14.490","Text":"but the tester or the examiner has the right"},{"Start":"00:14.490 ","End":"00:17.955","Text":"to ask you to do it in polar and it\u0027s good for practice."},{"Start":"00:17.955 ","End":"00:22.350","Text":"Let\u0027s see, my strategy is to write this as a double"},{"Start":"00:22.350 ","End":"00:27.375","Text":"integral over a region D of the same thing,"},{"Start":"00:27.375 ","End":"00:31.890","Text":"dxdy and remembering that dxdy is"},{"Start":"00:31.890 ","End":"00:38.155","Text":"dA and I\u0027ll sketch the region in Cartesian form and then we\u0027ll switch to polar."},{"Start":"00:38.155 ","End":"00:41.780","Text":"Note that the outer iteration is on y,"},{"Start":"00:41.780 ","End":"00:44.360","Text":"which goes from 0 to 6 and for"},{"Start":"00:44.360 ","End":"00:51.490","Text":"each such y we have an integral dx and x goes from 0 to that particular y"},{"Start":"00:51.490 ","End":"00:56.180","Text":"and so I can describe D using the formulas that"},{"Start":"00:56.180 ","End":"01:05.615","Text":"the outer loop is y goes from 0 to 6 and for each such y,"},{"Start":"01:05.615 ","End":"01:10.930","Text":"x goes from 0 up to this y here."},{"Start":"01:10.930 ","End":"01:14.585","Text":"Now I\u0027ll sketch it and then I\u0027ll explain the sketch."},{"Start":"01:14.585 ","End":"01:17.300","Text":"I start with a pair of axis and I notice I only need"},{"Start":"01:17.300 ","End":"01:20.814","Text":"the first quadrant because x and y are bigger or equal to 0."},{"Start":"01:20.814 ","End":"01:31.770","Text":"I need y going from 0 to 6 and I know x goes from 0 to y so I need the line x equals y."},{"Start":"01:31.770 ","End":"01:35.130","Text":"Part of the line, this is x equals while y."},{"Start":"01:35.130 ","End":"01:38.029","Text":"Also note that the y-axis is x equals 0."},{"Start":"01:38.029 ","End":"01:45.235","Text":"I\u0027m going to complete the triangle like so and I\u0027m going to shade this triangle."},{"Start":"01:45.235 ","End":"01:50.015","Text":"Now I claim that this is the region D that we\u0027re talking about."},{"Start":"01:50.015 ","End":"01:52.250","Text":"Just a quick explanation."},{"Start":"01:52.250 ","End":"01:54.200","Text":"This is a Type 2 region."},{"Start":"01:54.200 ","End":"01:59.390","Text":"It\u0027s dxdy which means that for a given y, let\u0027s say this is y."},{"Start":"01:59.390 ","End":"02:04.980","Text":"Then I take horizontal slices like so and I look"},{"Start":"02:04.980 ","End":"02:11.115","Text":"at where it intersects and this intersection point is 0,"},{"Start":"02:11.115 ","End":"02:14.145","Text":"x is 0 and here x is y."},{"Start":"02:14.145 ","End":"02:18.900","Text":"Clearly as y sweeps from 0 to 6 and x sweeps from 0 to y,"},{"Start":"02:18.900 ","End":"02:23.160","Text":"then this is the region D. I cleared stuff I"},{"Start":"02:23.160 ","End":"02:28.085","Text":"don\u0027t need and now I\u0027m going to describe D in polar terms."},{"Start":"02:28.085 ","End":"02:31.940","Text":"Now, the angle Theta goes from here to here."},{"Start":"02:31.940 ","End":"02:34.550","Text":"This is Theta equals,"},{"Start":"02:34.550 ","End":"02:39.785","Text":"will see in a moment and this is Theta equals Pi over 2."},{"Start":"02:39.785 ","End":"02:41.540","Text":"Perhaps I erased too much."},{"Start":"02:41.540 ","End":"02:43.640","Text":"I should have kept this equation,"},{"Start":"02:43.640 ","End":"02:48.770","Text":"x equals y, which I can write as y equals x so the slope is 1."},{"Start":"02:48.770 ","End":"02:56.065","Text":"Clearly this is 45 degrees but we\u0027re working in radians so it\u0027s Pi over 4."},{"Start":"02:56.065 ","End":"02:58.850","Text":"It\u0027s the arctangent of 1 because y equals x,"},{"Start":"02:58.850 ","End":"03:01.805","Text":"that\u0027s y equals 1x, just a slope of 1."},{"Start":"03:01.805 ","End":"03:05.720","Text":"We\u0027ll have Theta and I\u0027ll write it here."},{"Start":"03:05.720 ","End":"03:09.590","Text":"This is not the description of d in polar that Theta goes"},{"Start":"03:09.590 ","End":"03:21.720","Text":"from Pi over 4 to Pi over 2."},{"Start":"03:21.720 ","End":"03:24.045","Text":"Now what about r?"},{"Start":"03:24.045 ","End":"03:31.760","Text":"Certainly, if I take any particular Theta in this range,"},{"Start":"03:31.760 ","End":"03:37.670","Text":"that r goes from 0 to,"},{"Start":"03:37.670 ","End":"03:40.380","Text":"here\u0027s the tricky bit."},{"Start":"03:40.540 ","End":"03:47.855","Text":"It goes from 0 but up to and we want to find out what this is."},{"Start":"03:47.855 ","End":"03:50.545","Text":"Why don\u0027t I give it a letter?"},{"Start":"03:50.545 ","End":"03:53.490","Text":"Let me call it r1."},{"Start":"03:53.490 ","End":"03:55.770","Text":"R1 will depend on Theta."},{"Start":"03:55.770 ","End":"04:00.300","Text":"It\u0027s a function of Theta as Theta goes from Pi over 4 to Pi over 2."},{"Start":"04:00.300 ","End":"04:04.890","Text":"I get different lengths and finally I get length 6."},{"Start":"04:04.890 ","End":"04:12.275","Text":"Let\u0027s see, here I wrote the equations for the conversion from Cartesian to polar."},{"Start":"04:12.275 ","End":"04:15.625","Text":"I want to use this equation now."},{"Start":"04:15.625 ","End":"04:18.095","Text":"Now at this point here,"},{"Start":"04:18.095 ","End":"04:25.010","Text":"I have that y is equal to 6 but y is r sine Theta or in our case,"},{"Start":"04:25.010 ","End":"04:31.910","Text":"r1 sine Theta for this particular Theta is equal to"},{"Start":"04:31.910 ","End":"04:38.940","Text":"6 and so r1 is 6 over sine Theta."},{"Start":"04:38.940 ","End":"04:40.250","Text":"Like I said, it\u0027s a function of Theta,"},{"Start":"04:40.250 ","End":"04:41.539","Text":"it depends on Theta,"},{"Start":"04:41.539 ","End":"04:48.365","Text":"but I can write this here now as 6 over sine Theta."},{"Start":"04:48.365 ","End":"04:51.845","Text":"Now this is the same region described in polar."},{"Start":"04:51.845 ","End":"04:57.275","Text":"Now I can write a polar equivalent of this integral."},{"Start":"04:57.275 ","End":"05:02.765","Text":"I\u0027ve got the double integral and now"},{"Start":"05:02.765 ","End":"05:11.600","Text":"the outer loop is Theta from Pi over 4 to Pi over 2."},{"Start":"05:11.600 ","End":"05:20.690","Text":"R goes from 0 to 6 over sine Theta."},{"Start":"05:20.690 ","End":"05:25.955","Text":"X will be, I\u0027ll use this equation this time."},{"Start":"05:25.955 ","End":"05:29.040","Text":"R cosine Theta."},{"Start":"05:31.850 ","End":"05:36.145","Text":"That was this equation and I\u0027m going to use this equation for dA,"},{"Start":"05:36.145 ","End":"05:43.070","Text":"which gives me rdrd Theta."},{"Start":"05:43.070 ","End":"05:48.115","Text":"This is now the integral in polar form."},{"Start":"05:48.115 ","End":"05:50.710","Text":"I\u0027d like to just simplify it a bit."},{"Start":"05:50.710 ","End":"05:57.460","Text":"I\u0027ve got the integral from Theta equals Pi over 4 to Pi over 2."},{"Start":"05:57.460 ","End":"05:59.650","Text":"Now, the inner integral is dr,"},{"Start":"05:59.650 ","End":"06:03.820","Text":"that\u0027s the one we\u0027re doing first and cosine Theta is a constant as far as r"},{"Start":"06:03.820 ","End":"06:08.440","Text":"goes so I can take cosine Theta outside this integral,"},{"Start":"06:08.440 ","End":"06:15.380","Text":"which is from 0 to 6 over sine Theta and what am I left with?"},{"Start":"06:15.380 ","End":"06:22.435","Text":"I\u0027m left with r squared dr d Theta."},{"Start":"06:22.435 ","End":"06:26.955","Text":"As usual, we do the inner integral first,"},{"Start":"06:26.955 ","End":"06:31.845","Text":"that\u0027s the dr and I like to do this one at the side."},{"Start":"06:31.845 ","End":"06:34.295","Text":"Say where do I have room here?"},{"Start":"06:34.295 ","End":"06:42.064","Text":"I have the integral from 0 to 6 over"},{"Start":"06:42.064 ","End":"06:46.130","Text":"sine Theta of r squared"},{"Start":"06:46.130 ","End":"06:53.405","Text":"dr and this is equal to the integral of r squared is 1 third r cubed,"},{"Start":"06:53.405 ","End":"06:58.820","Text":"it\u0027s 1 third and I can just take the third outside and the r cubed I need to"},{"Start":"06:58.820 ","End":"07:05.710","Text":"evaluate between 0 and 6 over sine Theta."},{"Start":"07:05.710 ","End":"07:07.800","Text":"Now, if I plug in r equals 0,"},{"Start":"07:07.800 ","End":"07:16.990","Text":"I get 0 so I only have to plug in the top limit and what we get is"},{"Start":"07:21.280 ","End":"07:28.455","Text":"we have 1/3 and then 6 over"},{"Start":"07:28.455 ","End":"07:34.244","Text":"sine Theta cubed and"},{"Start":"07:34.244 ","End":"07:39.410","Text":"if I just slightly rewrite it,"},{"Start":"07:39.410 ","End":"07:42.940","Text":"I will get that sine 6 cubed."},{"Start":"07:42.940 ","End":"07:46.680","Text":"I could just cancel 1 of the 6s by 3 instead 6 times 6 times 6,"},{"Start":"07:46.680 ","End":"07:48.300","Text":"I have 2 times 6 times 6,"},{"Start":"07:48.300 ","End":"07:56.325","Text":"I make it 72 over sine cubed Theta."},{"Start":"07:56.325 ","End":"07:59.150","Text":"I\u0027m just going to rewrite it so it\u0027ll be close at hand,"},{"Start":"07:59.150 ","End":"08:06.745","Text":"72 over sine cubed Theta for this bit and so what I get well,"},{"Start":"08:06.745 ","End":"08:11.300","Text":"the 72 I can take completely in front so I\u0027ve got 72."},{"Start":"08:11.300 ","End":"08:20.435","Text":"Now I\u0027ve got the integral from Pi over 4 to Pi over 2 and I\u0027ve got"},{"Start":"08:20.435 ","End":"08:26.510","Text":"the cosine Theta and then here I have in"},{"Start":"08:26.510 ","End":"08:35.855","Text":"the denominator sine cubed Theta and along the left missing is the d Theta."},{"Start":"08:35.855 ","End":"08:40.745","Text":"This looks like it should be done by substitution."},{"Start":"08:40.745 ","End":"08:45.125","Text":"If I substitute sine Theta and I will be well,"},{"Start":"08:45.125 ","End":"08:49.715","Text":"because I already have its derivative cosine Theta here so at the side,"},{"Start":"08:49.715 ","End":"08:58.100","Text":"we\u0027ll do substitution t equals sine of Theta,"},{"Start":"08:58.100 ","End":"09:08.465","Text":"dt will be equal to cosine Theta d Theta and there\u0027s 2 general techniques."},{"Start":"09:08.465 ","End":"09:12.950","Text":"We either solve this into the indefinite integral in terms of t"},{"Start":"09:12.950 ","End":"09:17.510","Text":"and then substitute back Theta but we don\u0027t actually have to substitute back."},{"Start":"09:17.510 ","End":"09:19.460","Text":"The other way of doing it which I prefer,"},{"Start":"09:19.460 ","End":"09:22.675","Text":"is to substitute the limits of integration also."},{"Start":"09:22.675 ","End":"09:28.620","Text":"In that case, we get that when Theta is Pi over 4,"},{"Start":"09:28.620 ","End":"09:35.680","Text":"this gives me that t which is sine Theta is sine Pi over 4."},{"Start":"09:37.580 ","End":"09:42.390","Text":"I\u0027ll leave it for the moment of sine Pi over 4 and then we\u0027ll see what that is."},{"Start":"09:42.390 ","End":"09:45.375","Text":"When Theta is Pi over 2,"},{"Start":"09:45.375 ","End":"09:50.985","Text":"then t is sine of pi over 2."},{"Start":"09:50.985 ","End":"09:53.985","Text":"Now, these are famous angles."},{"Start":"09:53.985 ","End":"09:59.120","Text":"They are angles that you really should know is 45 degrees and 90 degrees."},{"Start":"09:59.120 ","End":"10:05.630","Text":"The sine of 45 degrees is known to be 1 over square root of 2 and the sine of"},{"Start":"10:05.630 ","End":"10:12.960","Text":"90 degrees is equal to 1 so after substituting all this stuff here,"},{"Start":"10:12.960 ","End":"10:21.745","Text":"what we will get will be the integral in t 72 times the integral from"},{"Start":"10:21.745 ","End":"10:29.580","Text":"1 over root 2 to 1 cosine Theta d Theta"},{"Start":"10:29.580 ","End":"10:38.420","Text":"is dt and sine cubed Theta will just be t cubed."},{"Start":"10:38.420 ","End":"10:44.400","Text":"This is a straight forward integral because instead of t _3,"},{"Start":"10:44.400 ","End":"10:47.060","Text":"I can write t_ minus 3."},{"Start":"10:47.060 ","End":"10:50.810","Text":"I forgot to write the equals throughout."},{"Start":"10:53.420 ","End":"10:56.355","Text":"I\u0027ll do this at the side."},{"Start":"10:56.355 ","End":"11:01.660","Text":"The indefinite integral of this would be the same as"},{"Start":"11:01.660 ","End":"11:07.075","Text":"t _minus 3 dt and I\u0027m just doing this part."},{"Start":"11:07.075 ","End":"11:11.140","Text":"This is equal to I raise the power by"},{"Start":"11:11.140 ","End":"11:19.150","Text":"1 and so it\u0027s t to the minus 2 and then we divide by that new power minus 2."},{"Start":"11:19.150 ","End":"11:20.470","Text":"If it was an indefinite integral,"},{"Start":"11:20.470 ","End":"11:25.670","Text":"I would add the C equals I don\u0027t need this here because I\u0027m going to substitute back."},{"Start":"11:28.370 ","End":"11:34.090","Text":"We\u0027ll get 72 times."},{"Start":"11:34.090 ","End":"11:36.950","Text":"The integral comes out."},{"Start":"11:36.950 ","End":"11:47.700","Text":"I\u0027ll put the 1 over minus 2 outside so I\u0027ve got minus 1/2 and the t to the minus 2,"},{"Start":"11:47.700 ","End":"11:53.770","Text":"I can write as 1 over t squared."},{"Start":"11:54.110 ","End":"11:57.345","Text":"This bit is 1 over t squared here,"},{"Start":"11:57.345 ","End":"12:05.310","Text":"1 over minus 2 is here and I need to substitute 1 over root 2 and 1."},{"Start":"12:05.310 ","End":"12:08.460","Text":"Let\u0027s see what we get."},{"Start":"12:08.460 ","End":"12:11.740","Text":"I\u0027ll continue over here."},{"Start":"12:11.830 ","End":"12:17.345","Text":"I\u0027ve got minus 36 from here."},{"Start":"12:17.345 ","End":"12:20.730","Text":"Maybe I\u0027ll go to a new line after all."},{"Start":"12:21.310 ","End":"12:26.395","Text":"Minus 36 and now,"},{"Start":"12:26.395 ","End":"12:33.659","Text":"the upper limit is 1 over 1 squared is 1 and the lower limit,"},{"Start":"12:33.659 ","End":"12:36.915","Text":"t squared is 1/2."},{"Start":"12:36.915 ","End":"12:42.840","Text":"1 over 1/2 is 2 so it\u0027s 1 minus 2."},{"Start":"12:42.840 ","End":"12:45.410","Text":"Now I\u0027ll continue here."},{"Start":"12:45.410 ","End":"12:48.905","Text":"1 minus 2 is minus 1,"},{"Start":"12:48.905 ","End":"12:53.540","Text":"minus with the minus cancels so I just get"},{"Start":"12:53.540 ","End":"12:59.010","Text":"36 and just highlight the answer and we\u0027re done."}],"ID":8701},{"Watched":false,"Name":"Exercise 17","Duration":"11m 10s","ChapterTopicVideoID":8506,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8506.jpeg","UploadDate":"2017-02-13T09:04:22.3730000","DurationForVideoObject":"PT11M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.880","Text":"In this exercise, we have to compute this integral by converting to polar coordinates."},{"Start":"00:05.880 ","End":"00:09.780","Text":"I might tell you straight away that if I had a choice,"},{"Start":"00:09.780 ","End":"00:11.945","Text":"I would not do it in polar coordinates,"},{"Start":"00:11.945 ","End":"00:15.495","Text":"but when you get a question on a test or an exam or a homework assignment,"},{"Start":"00:15.495 ","End":"00:19.395","Text":"the person who gave it you can insist that you do it with polar coordinates."},{"Start":"00:19.395 ","End":"00:24.210","Text":"Anyway, I have all the formulas for conversion on hand."},{"Start":"00:24.210 ","End":"00:28.140","Text":"The strategy will be to write this as a double"},{"Start":"00:28.140 ","End":"00:33.790","Text":"integral over some region D of the same thing, ydydx."},{"Start":"00:34.880 ","End":"00:39.165","Text":"Remember that dydx is also dA."},{"Start":"00:39.165 ","End":"00:43.140","Text":"This tells us when it\u0027s dydx that it\u0027s going to be"},{"Start":"00:43.140 ","End":"00:47.570","Text":"a type 1 region where we take vertical slices."},{"Start":"00:47.570 ","End":"00:53.225","Text":"Let me describe the region D from here and then we can sketch it."},{"Start":"00:53.225 ","End":"00:57.040","Text":"What we\u0027re saying is, from here, sorry,"},{"Start":"00:57.040 ","End":"01:02.300","Text":"the outer integral x goes from 0-2 and for each,"},{"Start":"01:02.300 ","End":"01:07.495","Text":"such x, the inner integral y goes from 0 to x."},{"Start":"01:07.495 ","End":"01:09.380","Text":"If I write this out,"},{"Start":"01:09.380 ","End":"01:13.265","Text":"I say 0 less than or equal to x,"},{"Start":"01:13.265 ","End":"01:15.275","Text":"less than or equal to 2."},{"Start":"01:15.275 ","End":"01:20.509","Text":"Given such, x, y goes between 0 and"},{"Start":"01:20.509 ","End":"01:23.900","Text":"x. I\u0027m going to bring in"},{"Start":"01:23.900 ","End":"01:28.330","Text":"a sketch and we need the first quadrant because everything is bigger or equal to 0."},{"Start":"01:28.330 ","End":"01:31.350","Text":"What happens here is x goes from 0-2,"},{"Start":"01:31.350 ","End":"01:36.190","Text":"so let\u0027s say this is 0 and this is 2 somewhere."},{"Start":"01:40.460 ","End":"01:43.790","Text":"Let\u0027s say this is a given typical x,"},{"Start":"01:43.790 ","End":"01:46.475","Text":"y goes from 0 up to x."},{"Start":"01:46.475 ","End":"01:51.030","Text":"In other words, we need to sketch where is y equals x."},{"Start":"01:51.030 ","End":"01:55.030","Text":"Well, we know this is a 45 degree line through the origin,"},{"Start":"01:55.030 ","End":"01:57.460","Text":"I\u0027ve seen it so many times."},{"Start":"01:57.460 ","End":"02:00.185","Text":"What we basically have,"},{"Start":"02:00.185 ","End":"02:02.455","Text":"this is y equals x,"},{"Start":"02:02.455 ","End":"02:07.185","Text":"so when this is 2, this is also 2 so we\u0027re going up to here."},{"Start":"02:07.185 ","End":"02:12.230","Text":"For each x, y goes vertically from this point here,"},{"Start":"02:12.230 ","End":"02:16.540","Text":"which is 0, to this point here, which is x."},{"Start":"02:16.540 ","End":"02:24.050","Text":"If I do these slices all the way from x equals 0 to x equals 2,"},{"Start":"02:24.050 ","End":"02:26.480","Text":"I can shade this triangle,"},{"Start":"02:26.480 ","End":"02:33.560","Text":"and this triangle will be my region d. Now I want this region in polar,"},{"Start":"02:33.560 ","End":"02:37.830","Text":"so I want to express D in terms of r and Theta."},{"Start":"02:37.900 ","End":"02:40.730","Text":"Theta is the easier one."},{"Start":"02:40.730 ","End":"02:44.630","Text":"Theta will go from this line,"},{"Start":"02:44.630 ","End":"02:46.370","Text":"the x-axis that was,"},{"Start":"02:46.370 ","End":"02:48.350","Text":"that\u0027s Theta equals 0."},{"Start":"02:48.350 ","End":"02:51.230","Text":"We said that y equals x as a slope of 1,"},{"Start":"02:51.230 ","End":"02:53.134","Text":"so an angle of 45 degrees,"},{"Start":"02:53.134 ","End":"02:57.050","Text":"in other words, Theta in radians is Pi over 4."},{"Start":"02:57.050 ","End":"02:59.270","Text":"That gives us Theta."},{"Start":"02:59.270 ","End":"03:02.720","Text":"I can already start writing my D in polar."},{"Start":"03:02.720 ","End":"03:06.320","Text":"We have 0 less than or equal to Theta,"},{"Start":"03:06.320 ","End":"03:10.115","Text":"less than or equal to Pi over 4,"},{"Start":"03:10.115 ","End":"03:17.330","Text":"the radius, well, you can easily see it starts from 0."},{"Start":"03:17.330 ","End":"03:19.280","Text":"Well, let us take a typical Theta."},{"Start":"03:19.280 ","End":"03:22.055","Text":"Let\u0027s say this is a general Theta,"},{"Start":"03:22.055 ","End":"03:25.370","Text":"radius starts from here and ends here."},{"Start":"03:25.370 ","End":"03:28.564","Text":"It starts at 0, but where does it end?"},{"Start":"03:28.564 ","End":"03:30.995","Text":"That\u0027s what we want to find out."},{"Start":"03:30.995 ","End":"03:32.780","Text":"I prefer to give it a name,"},{"Start":"03:32.780 ","End":"03:36.200","Text":"I\u0027ll call it r_1,"},{"Start":"03:36.200 ","End":"03:39.905","Text":"and then r goes from 0 to r_1,"},{"Start":"03:39.905 ","End":"03:42.520","Text":"but I wanted to see what r_1 is."},{"Start":"03:42.520 ","End":"03:45.605","Text":"Now, r_1 is on this vertical line,"},{"Start":"03:45.605 ","End":"03:49.190","Text":"and this vertical line has the equation x equals 2."},{"Start":"03:49.190 ","End":"03:52.620","Text":"In any event, I want to use this."},{"Start":"03:52.760 ","End":"04:01.080","Text":"My x here on this point is 2, r is r_1,"},{"Start":"04:01.080 ","End":"04:05.249","Text":"and this is my cosine Theta,"},{"Start":"04:05.249 ","End":"04:13.035","Text":"so I get that r_1 is equal to 2 over cosine Theta."},{"Start":"04:13.035 ","End":"04:15.060","Text":"R_1 depends on Theta."},{"Start":"04:15.060 ","End":"04:21.110","Text":"For different Thetas, I get different lengths here, but that\u0027s okay."},{"Start":"04:21.110 ","End":"04:27.350","Text":"Now I can write here 2 over cosine Theta."},{"Start":"04:27.350 ","End":"04:31.370","Text":"I\u0027ve changed the region from Cartesian to polar and now I can write"},{"Start":"04:31.370 ","End":"04:37.395","Text":"the integral in terms of the polar region."},{"Start":"04:37.395 ","End":"04:44.420","Text":"I\u0027ve got this time the integral and I do it as Theta from something to something."},{"Start":"04:44.420 ","End":"04:50.810","Text":"What will it be from 0 to Pi over 4,and r"},{"Start":"04:50.810 ","End":"04:57.830","Text":"goes from 0 to 2 over cosine Theta."},{"Start":"04:57.830 ","End":"05:00.905","Text":"Then I need y,"},{"Start":"05:00.905 ","End":"05:06.000","Text":"which I copy from here, r sine Theta."},{"Start":"05:07.820 ","End":"05:10.905","Text":"Then what else do I need?"},{"Start":"05:10.905 ","End":"05:15.430","Text":"I need dA, which is rdrd Theta."},{"Start":"05:19.580 ","End":"05:22.765","Text":"I just need to make a bit of room here."},{"Start":"05:22.765 ","End":"05:25.475","Text":"I\u0027m going to rewrite this, simplify it."},{"Start":"05:25.475 ","End":"05:31.170","Text":"The inner integral is dr and Theta is a constant,"},{"Start":"05:31.170 ","End":"05:40.880","Text":"so I can rewrite this as the integral from Theta equals 0 to Theta equals Pi over 4."},{"Start":"05:40.880 ","End":"05:44.900","Text":"Pull the sine Theta in front and then get the"},{"Start":"05:44.900 ","End":"05:53.205","Text":"integral from r equals 0 to 2 over cosine Theta."},{"Start":"05:53.205 ","End":"05:57.940","Text":"What are we left with? R squared drd Theta."},{"Start":"06:00.010 ","End":"06:03.050","Text":"We do the inner integral first,"},{"Start":"06:03.050 ","End":"06:05.365","Text":"which is this one."},{"Start":"06:05.365 ","End":"06:08.300","Text":"Like to compute this at the side or call it"},{"Start":"06:08.300 ","End":"06:11.600","Text":"asterisk and do it where I have some room, say here."},{"Start":"06:11.600 ","End":"06:17.180","Text":"What I have is the integral from 0 to 2"},{"Start":"06:17.180 ","End":"06:24.870","Text":"over cosine Theta of r squared dr."},{"Start":"06:27.350 ","End":"06:32.850","Text":"The last squared gives me 1/3 r cubed,"},{"Start":"06:32.850 ","End":"06:36.390","Text":"so I have 1/3 and the r cubed,"},{"Start":"06:36.390 ","End":"06:43.305","Text":"I\u0027ll take from 0 to 2 over cosine Theta."},{"Start":"06:43.305 ","End":"06:46.160","Text":"What will I get? Well, if I plug in r equals 0,"},{"Start":"06:46.160 ","End":"06:50.360","Text":"I just get 0, so I only need the 2 over cosine Theta."},{"Start":"06:50.360 ","End":"06:53.780","Text":"If I cube it and then divide by 3,"},{"Start":"06:53.780 ","End":"07:02.640","Text":"what I end up with is 2 cubed is 8 over cosine cubed Theta."},{"Start":"07:02.640 ","End":"07:07.700","Text":"I\u0027ll also stick the 3 in here, cosine cubed Theta."},{"Start":"07:07.700 ","End":"07:09.680","Text":"Now that I\u0027ve got this,"},{"Start":"07:09.680 ","End":"07:12.660","Text":"I\u0027m going to put it back here."},{"Start":"07:14.120 ","End":"07:16.320","Text":"Let me just write it again."},{"Start":"07:16.320 ","End":"07:17.510","Text":"Have it close on hand,"},{"Start":"07:17.510 ","End":"07:24.770","Text":"we computed that this was 8 over 3 cosine cubed Theta."},{"Start":"07:24.770 ","End":"07:28.470","Text":"Rewriting this, I\u0027m going to do it as follows."},{"Start":"07:28.470 ","End":"07:30.770","Text":"The 8/3 is a constant,"},{"Start":"07:30.770 ","End":"07:33.230","Text":"so I\u0027ll take completely in front."},{"Start":"07:33.230 ","End":"07:40.975","Text":"Then we have Theta from 0 to Pi over 4."},{"Start":"07:40.975 ","End":"07:45.960","Text":"I have the cosine cubed Theta in the denominator."},{"Start":"07:46.510 ","End":"07:54.970","Text":"I still have a sine Theta from here and this is all d Theta."},{"Start":"07:54.970 ","End":"07:57.470","Text":"Now how do we do this integral?"},{"Start":"07:57.470 ","End":"08:02.615","Text":"We\u0027ve had a similar one before and we do it by substitution."},{"Start":"08:02.615 ","End":"08:06.935","Text":"We\u0027ll substitute here t equals"},{"Start":"08:06.935 ","End":"08:13.895","Text":"cosine Theta and then dt is equal to the derivative of this,"},{"Start":"08:13.895 ","End":"08:18.985","Text":"which is minus sine Theta d Theta."},{"Start":"08:18.985 ","End":"08:23.630","Text":"If you don\u0027t want to return to Theta and just stay in the land of t,"},{"Start":"08:23.630 ","End":"08:27.320","Text":"then you can just substitute the limits also."},{"Start":"08:27.320 ","End":"08:30.470","Text":"When Theta equals 0,"},{"Start":"08:30.470 ","End":"08:35.470","Text":"we get that t is cosine 0."},{"Start":"08:38.420 ","End":"08:42.205","Text":"What is cosine 0? Is 1."},{"Start":"08:42.205 ","End":"08:47.020","Text":"When Theta is Pi over 4,"},{"Start":"08:47.020 ","End":"08:51.000","Text":"then t is cosine Pi over 4."},{"Start":"08:51.000 ","End":"08:52.170","Text":"That\u0027s a well-known angle,"},{"Start":"08:52.170 ","End":"08:53.730","Text":"cosine of 45 degrees,"},{"Start":"08:53.730 ","End":"08:56.095","Text":"1 over root 2."},{"Start":"08:56.095 ","End":"09:00.050","Text":"If I substitute all these things in here then"},{"Start":"09:00.050 ","End":"09:07.920","Text":"we will get 8/3."},{"Start":"09:07.920 ","End":"09:15.615","Text":"Now the lower integral is 1 and the upper integral is 1 over"},{"Start":"09:15.615 ","End":"09:23.510","Text":"root 2 and sine Theta d Theta is minus dt."},{"Start":"09:23.510 ","End":"09:28.410","Text":"I can put the minus here and I have dt."},{"Start":"09:30.170 ","End":"09:37.140","Text":"Finally, the cosine Theta here is t over t cubed."},{"Start":"09:37.140 ","End":"09:40.560","Text":"I\u0027m going to write this with the negative exponents,"},{"Start":"09:40.560 ","End":"09:44.550","Text":"so what we have is minus"},{"Start":"09:44.550 ","End":"09:51.260","Text":"8/3 times the integral of t to the minus 3 dt."},{"Start":"09:51.260 ","End":"09:56.115","Text":"Same limits here, 1 to 1 over root 2."},{"Start":"09:56.115 ","End":"10:06.290","Text":"Now negative exponent, I raise the power by 1t to the minus 2 and divide by minus 2."},{"Start":"10:06.290 ","End":"10:09.035","Text":"I have minus 8/3,"},{"Start":"10:09.035 ","End":"10:11.920","Text":"1 over minus 2,"},{"Start":"10:11.920 ","End":"10:17.760","Text":"and this between 1 and 1 over root 2."},{"Start":"10:17.760 ","End":"10:23.280","Text":"Continuing, I can cancel some stuff here."},{"Start":"10:23.280 ","End":"10:26.040","Text":"Minus 2 into minus 8 goes 4 times,"},{"Start":"10:26.040 ","End":"10:28.500","Text":"so I just have 4/3."},{"Start":"10:28.500 ","End":"10:30.050","Text":"Then t to the minus 2,"},{"Start":"10:30.050 ","End":"10:32.750","Text":"I can return to 1 over t squared."},{"Start":"10:32.750 ","End":"10:39.080","Text":"Put that between 1 and 1 over root 2 and let\u0027s see what we get."},{"Start":"10:39.080 ","End":"10:46.710","Text":"It\u0027s 4/3 times something minus something."},{"Start":"10:46.710 ","End":"10:50.100","Text":"The top limit, if t is 1 over root 2,"},{"Start":"10:50.100 ","End":"10:53.085","Text":"t squared is 1/2, 1 over 1/2 Is 2."},{"Start":"10:53.085 ","End":"10:56.435","Text":"If t is 1, this thing just comes out to be 1."},{"Start":"10:56.435 ","End":"10:58.580","Text":"2 minus 1 is 1,"},{"Start":"10:58.580 ","End":"11:03.130","Text":"1 times 4/3, so the answer is 4/3."},{"Start":"11:03.730 ","End":"11:10.470","Text":"I\u0027ll just highlight this and we are done."}],"ID":8702},{"Watched":false,"Name":"Exercise 18","Duration":"9m 48s","ChapterTopicVideoID":8479,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8479.jpeg","UploadDate":"2017-02-13T08:24:39.2970000","DurationForVideoObject":"PT9M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"In this exercise, we have a double integral to"},{"Start":"00:02.520 ","End":"00:05.580","Text":"compute and we have to do it using polar coordinates."},{"Start":"00:05.580 ","End":"00:09.510","Text":"What we have now is a type 1 region,"},{"Start":"00:09.510 ","End":"00:16.090","Text":"where x goes from something to something and then we have vertical slices for y."},{"Start":"00:17.360 ","End":"00:20.880","Text":"Let me just write it in the D form,"},{"Start":"00:20.880 ","End":"00:24.345","Text":"domain, of the same thing."},{"Start":"00:24.345 ","End":"00:31.455","Text":"2 over 1 plus the square root of x squared plus y squared."},{"Start":"00:31.455 ","End":"00:37.740","Text":"The dy dx tells me that it\u0027s a Type 1 region, vertical slices."},{"Start":"00:37.740 ","End":"00:43.795","Text":"The region D is described by an outer loop x going from minus 1-0."},{"Start":"00:43.795 ","End":"00:49.905","Text":"Then an inner loop for y that for each x goes from this also to 0."},{"Start":"00:49.905 ","End":"00:53.675","Text":"Whenever we\u0027re working all in the third quadrant, everything\u0027s negative."},{"Start":"00:53.675 ","End":"01:03.044","Text":"So I can describe D as x going from minus 1 to 0,"},{"Start":"01:03.044 ","End":"01:05.355","Text":"and for each such x,"},{"Start":"01:05.355 ","End":"01:14.330","Text":"y goes from minus the square root of 1 minus x squared to 0."},{"Start":"01:14.330 ","End":"01:21.915","Text":"Let\u0027s draw a quick sketch of this D. Here is the sketch I prepared,"},{"Start":"01:21.915 ","End":"01:25.445","Text":"I\u0027ll just justify it so that we\u0027ve seen this kind of thing before."},{"Start":"01:25.445 ","End":"01:32.650","Text":"This last equation is part of x squared plus y squared equals 1."},{"Start":"01:32.650 ","End":"01:35.480","Text":"Because if I, I mean not last equation,"},{"Start":"01:35.480 ","End":"01:39.125","Text":"the lower limit, this one,"},{"Start":"01:39.125 ","End":"01:41.930","Text":"if I let this equal to this, I square, both sides,"},{"Start":"01:41.930 ","End":"01:44.045","Text":"I get 1 minus x squared equals y squared,"},{"Start":"01:44.045 ","End":"01:48.420","Text":"bring everything over, so it\u0027s part of the circle of radius 1."},{"Start":"01:48.420 ","End":"01:54.515","Text":"But because x goes from minus 1 to 0 and y goes from 0 down to the minus the square root,"},{"Start":"01:54.515 ","End":"01:56.480","Text":"we only get a quarter of a circle,"},{"Start":"01:56.480 ","End":"02:00.825","Text":"so this is our D. I\u0027ll even illustrate further."},{"Start":"02:00.825 ","End":"02:05.280","Text":"It\u0027s a Type 1 region for each x in this,"},{"Start":"02:05.280 ","End":"02:07.730","Text":"if I take the vertical slice,"},{"Start":"02:07.730 ","End":"02:11.915","Text":"it enters the region or exits here,"},{"Start":"02:11.915 ","End":"02:14.750","Text":"depending on which way, actually we always go upward."},{"Start":"02:14.750 ","End":"02:19.175","Text":"So it enters here and exits here."},{"Start":"02:19.175 ","End":"02:21.515","Text":"The upper limit is 0."},{"Start":"02:21.515 ","End":"02:23.885","Text":"The lower limit is just this equation."},{"Start":"02:23.885 ","End":"02:29.760","Text":"That\u0027s the y equals minus the square root of 1 minus x squared."},{"Start":"02:30.220 ","End":"02:33.905","Text":"That shows how this works in Cartesian."},{"Start":"02:33.905 ","End":"02:38.285","Text":"Now, we want to convert the same region D to polar,"},{"Start":"02:38.285 ","End":"02:42.545","Text":"give a polar description and then write the integrals in polar form."},{"Start":"02:42.545 ","End":"02:45.530","Text":"All the equations that we need are here."},{"Start":"02:45.530 ","End":"02:47.630","Text":"I cleaned the sketch up,"},{"Start":"02:47.630 ","End":"02:50.690","Text":"what I don\u0027t need throughout the minus 1 and only"},{"Start":"02:50.690 ","End":"02:54.215","Text":"the minus it just to remind me that the radius is 1."},{"Start":"02:54.215 ","End":"02:56.470","Text":"What about Theta?"},{"Start":"02:56.470 ","End":"03:00.680","Text":"Well, Theta goes from here to here?"},{"Start":"03:00.680 ","End":"03:02.645","Text":"It starts from this line,"},{"Start":"03:02.645 ","End":"03:05.745","Text":"which is, just write Theta,"},{"Start":"03:05.745 ","End":"03:10.750","Text":"which is Theta equals a 180 degrees or Pi,"},{"Start":"03:10.750 ","End":"03:13.670","Text":"and it ends up at 270 degrees,"},{"Start":"03:13.670 ","End":"03:18.605","Text":"which I could write as Theta equals 3 Pi over 2."},{"Start":"03:18.605 ","End":"03:27.285","Text":"Now, for any particular Theta in this domain,"},{"Start":"03:27.285 ","End":"03:30.470","Text":"then r goes from 0 to 1."},{"Start":"03:30.470 ","End":"03:36.425","Text":"That\u0027s the limits for the r. So I can now rewrite the region in polar description"},{"Start":"03:36.425 ","End":"03:45.525","Text":"as Theta goes from pi to 3Pi over 2,"},{"Start":"03:45.525 ","End":"03:49.645","Text":"and r goes between 0 and 1."},{"Start":"03:49.645 ","End":"03:51.795","Text":"With this polar description,"},{"Start":"03:51.795 ","End":"03:57.830","Text":"I can now go back here and write this in polar form."},{"Start":"03:57.830 ","End":"04:00.245","Text":"So I have the double integral."},{"Start":"04:00.245 ","End":"04:03.410","Text":"Now this was the Cartesian form as a Type 1 region."},{"Start":"04:03.410 ","End":"04:10.720","Text":"Now I want a polar the Theta\u0027s always the outer one in this case."},{"Start":"04:15.770 ","End":"04:20.370","Text":"Pi to 3Pi over 2,"},{"Start":"04:20.370 ","End":"04:23.640","Text":"r goes from 0 to 1,"},{"Start":"04:23.640 ","End":"04:27.820","Text":"the dy dx is our dA."},{"Start":"04:27.820 ","End":"04:33.750","Text":"That is according to the formula rd, rd Theta."},{"Start":"04:33.940 ","End":"04:37.640","Text":"I don\u0027t know why I am doing the end first, never mind."},{"Start":"04:37.640 ","End":"04:45.740","Text":"Then 2 over 1 plus now x-squared plus y squared from this formula is r squared."},{"Start":"04:45.740 ","End":"04:48.960","Text":"So I have the square root of r squared."},{"Start":"04:49.910 ","End":"04:54.020","Text":"I\u0027ll just rewrite it a little bit."},{"Start":"04:54.020 ","End":"04:56.855","Text":"So what do I have?"},{"Start":"04:56.855 ","End":"05:05.175","Text":"I have that this is equal to the integral from Pi to 3, Pi over 2."},{"Start":"05:05.175 ","End":"05:15.290","Text":"I\u0027ll put, I\u0027ll take the 2 out front and make this integral"},{"Start":"05:15.290 ","End":"05:21.590","Text":"from 0 to 1 of r over"},{"Start":"05:21.590 ","End":"05:29.550","Text":"1 plus r. Something got wrote wrong here."},{"Start":"05:29.550 ","End":"05:31.710","Text":"Sorry, I forgot the d before the Theta."},{"Start":"05:31.710 ","End":"05:35.980","Text":"Yeah, dr and then d Theta."},{"Start":"05:35.980 ","End":"05:39.545","Text":"Now these integrals are worked from the inside out."},{"Start":"05:39.545 ","End":"05:43.140","Text":"First of all, the dr,"},{"Start":"05:44.440 ","End":"05:50.030","Text":"I\u0027d like to do this integral separately at the side or call it asterisk."},{"Start":"05:50.030 ","End":"05:59.760","Text":"What I want is the integral from 0 to 1 of r over 1 plus r dr."},{"Start":"05:59.760 ","End":"06:02.745","Text":"I can think of 2 main ways of doing this."},{"Start":"06:02.745 ","End":"06:06.065","Text":"one of them involve algebraic manipulation,"},{"Start":"06:06.065 ","End":"06:09.320","Text":"in which I would rewrite the numerator as"},{"Start":"06:09.320 ","End":"06:15.320","Text":"1 plus r minus 1 and then split it up into 2 separate bits."},{"Start":"06:15.320 ","End":"06:17.120","Text":"That\u0027s one way to go."},{"Start":"06:17.120 ","End":"06:22.595","Text":"The other way to go is a substitution and I\u0027ll go with substitution."},{"Start":"06:22.595 ","End":"06:25.070","Text":"But on your own later if you want the practice,"},{"Start":"06:25.070 ","End":"06:27.410","Text":"you might try doing it this way also."},{"Start":"06:27.410 ","End":"06:33.235","Text":"What I\u0027m going to substitute is t equals 1 plus r."},{"Start":"06:33.235 ","End":"06:43.205","Text":"Then r is easy because r is equal to t minus 1 if I just bring it over."},{"Start":"06:43.205 ","End":"06:48.770","Text":"Also, dt is the same as dr,"},{"Start":"06:48.770 ","End":"06:50.810","Text":"because the derivative of both is 1,"},{"Start":"06:50.810 ","End":"06:55.160","Text":"this with respect to t and this with respect to r. I also"},{"Start":"06:55.160 ","End":"07:00.930","Text":"want to substitute the limits is that when t equals 0,"},{"Start":"07:00.930 ","End":"07:11.290","Text":"then r is equal to, oops."},{"Start":"07:11.290 ","End":"07:16.655","Text":"Perhaps it\u0027s worth emphasizing that these are limits for r. Yeah, sorry,"},{"Start":"07:16.655 ","End":"07:20.660","Text":"when r equals 0 and when r equals 1,"},{"Start":"07:20.660 ","End":"07:24.830","Text":"then t is 1 plus r so here t is 1,"},{"Start":"07:24.830 ","End":"07:26.765","Text":"and here t is 2."},{"Start":"07:26.765 ","End":"07:29.665","Text":"If I pull, put all this into here,"},{"Start":"07:29.665 ","End":"07:35.190","Text":"I will get the integral from t equals 1"},{"Start":"07:35.190 ","End":"07:41.790","Text":"to t equals 2 of r is T minus 1."},{"Start":"07:41.790 ","End":"07:47.170","Text":"1 plus r is t and it is dt."},{"Start":"07:47.540 ","End":"07:51.105","Text":"Let\u0027s see what this equals."},{"Start":"07:51.105 ","End":"07:54.515","Text":"Break this up into 2 bits."},{"Start":"07:54.515 ","End":"07:59.975","Text":"I\u0027ve got the integral from 1 to 2. t over t is 1,"},{"Start":"07:59.975 ","End":"08:05.130","Text":"1 over t, 1 over t dt."},{"Start":"08:05.240 ","End":"08:07.380","Text":"This is equal too,"},{"Start":"08:07.380 ","End":"08:12.770","Text":"that\u0027s an easy integral because the integral of 1 is just t. The integral of 1 over t is"},{"Start":"08:12.770 ","End":"08:19.860","Text":"natural log of t. I\u0027m certainly in the positive range for t,"},{"Start":"08:19.860 ","End":"08:23.295","Text":"so no problems and we\u0027re far away from 0."},{"Start":"08:23.295 ","End":"08:27.915","Text":"So this between 1 and 2."},{"Start":"08:27.915 ","End":"08:31.715","Text":"Now let\u0027s make the substitution."},{"Start":"08:31.715 ","End":"08:33.890","Text":"When t is 2,"},{"Start":"08:33.890 ","End":"08:38.655","Text":"I\u0027ve got 2 minus natural log of 2."},{"Start":"08:38.655 ","End":"08:40.700","Text":"When t is 1,"},{"Start":"08:40.700 ","End":"08:48.060","Text":"I have 1 minus natural log of 1 is 0."},{"Start":"08:48.060 ","End":"08:49.955","Text":"So what I get over here,"},{"Start":"08:49.955 ","End":"08:53.670","Text":"if I simplify it as 2 minus 1 is 1,"},{"Start":"08:53.670 ","End":"09:00.530","Text":"this comes out to be 1 minus natural log of 2."},{"Start":"09:00.530 ","End":"09:04.775","Text":"That\u0027s my asterisk that I have over here."},{"Start":"09:04.775 ","End":"09:06.650","Text":"I\u0027m back here now."},{"Start":"09:06.650 ","End":"09:11.330","Text":"Now this 1 minus natural log of 2 does not depend on the Theta."},{"Start":"09:11.330 ","End":"09:16.775","Text":"It\u0027s a constant, so I can pull it out in front and what I get from here now"},{"Start":"09:16.775 ","End":"09:22.730","Text":"is twice this constant 1 minus natural log of 2."},{"Start":"09:22.730 ","End":"09:27.635","Text":"All I\u0027m left with is the integral from 0 to 1 of d Theta."},{"Start":"09:27.635 ","End":"09:32.360","Text":"Let\u0027s write it as 1 d Theta and we already know that when we have the integral of 1,"},{"Start":"09:32.360 ","End":"09:34.850","Text":"it\u0027s just the upper limit minus the lower limit,"},{"Start":"09:34.850 ","End":"09:36.800","Text":"1 minus 0 is 1."},{"Start":"09:36.800 ","End":"09:44.090","Text":"So altogether all we\u0027re left with is twice 1 minus natural log of 2."},{"Start":"09:44.090 ","End":"09:48.130","Text":"This is the answer. We are done."}],"ID":8703},{"Watched":false,"Name":"Exercise 19","Duration":"7m 19s","ChapterTopicVideoID":8480,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8480.jpeg","UploadDate":"2017-02-13T08:25:32.4500000","DurationForVideoObject":"PT7M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.355","Text":"In this exercise, we have to compute this double integral by converting to polar."},{"Start":"00:05.355 ","End":"00:07.890","Text":"What I\u0027m going to do first is write this as a double"},{"Start":"00:07.890 ","End":"00:11.970","Text":"integral over a region D of the same thing."},{"Start":"00:11.970 ","End":"00:16.080","Text":"I just copied it and it\u0027s dxdy,"},{"Start":"00:16.080 ","End":"00:20.790","Text":"which means that it\u0027s a type 2 region,"},{"Start":"00:20.790 ","End":"00:25.710","Text":"horizontal slices, and this is also dA."},{"Start":"00:25.710 ","End":"00:29.145","Text":"I want to find out what the region D is."},{"Start":"00:29.145 ","End":"00:36.150","Text":"Note that the outer loop y goes from minus 1 to 1 and then for each y,"},{"Start":"00:36.150 ","End":"00:42.574","Text":"the inner loop on x goes from minus this thing to 0."},{"Start":"00:42.574 ","End":"00:45.950","Text":"So it stays in the negative area."},{"Start":"00:45.950 ","End":"00:49.280","Text":"We\u0027re going to be in the left half plane."},{"Start":"00:49.280 ","End":"00:51.115","Text":"Let me bring a sketch."},{"Start":"00:51.115 ","End":"00:59.364","Text":"We see that y goes from minus 1 to 1 and for each particular y in this range,"},{"Start":"00:59.364 ","End":"01:05.185","Text":"we can take a horizontal slice and see where it cuts."},{"Start":"01:05.185 ","End":"01:08.885","Text":"On the right side, the upper limit,"},{"Start":"01:08.885 ","End":"01:12.880","Text":"this part is 0 because"},{"Start":"01:12.880 ","End":"01:20.130","Text":"the y-axis is x equals 0 and the left semicircle is just this,"},{"Start":"01:20.130 ","End":"01:23.405","Text":"x equals minus root of 1 minus y squared."},{"Start":"01:23.405 ","End":"01:24.710","Text":"We\u0027ve seen this before."},{"Start":"01:24.710 ","End":"01:29.570","Text":"This is part of the x squared plus y squared equals 1."},{"Start":"01:29.570 ","End":"01:36.095","Text":"If we isolate x and we get plus or minus the square root on the left side is the minus."},{"Start":"01:36.095 ","End":"01:38.060","Text":"This is what this x is,"},{"Start":"01:38.060 ","End":"01:39.740","Text":"and we travel from here to here,"},{"Start":"01:39.740 ","End":"01:42.205","Text":"and slice the region horizontally."},{"Start":"01:42.205 ","End":"01:44.325","Text":"This is the region D,"},{"Start":"01:44.325 ","End":"01:53.240","Text":"and the Cartesian description of D is from the outer loop that y goes from minus"},{"Start":"01:53.240 ","End":"01:57.440","Text":"1 to 1 and x goes from"},{"Start":"01:57.440 ","End":"02:02.990","Text":"something to 0 from minus the square root of 1 minus y squared to 0."},{"Start":"02:02.990 ","End":"02:08.560","Text":"I want this now in polar form."},{"Start":"02:08.560 ","End":"02:17.060","Text":"In polar terms, this region starts from here and continues all the way around to here."},{"Start":"02:17.060 ","End":"02:22.745","Text":"This vertical line, the positive y-axis is where Theta is equal to 90 degrees,"},{"Start":"02:22.745 ","End":"02:24.805","Text":"that\u0027s Pi over 2."},{"Start":"02:24.805 ","End":"02:29.340","Text":"Here, Theta is 270 degrees,"},{"Start":"02:29.340 ","End":"02:36.315","Text":"which is 3 Pi over 2 and for each such Theta,"},{"Start":"02:36.315 ","End":"02:40.245","Text":"r just always goes from 0 to 1."},{"Start":"02:40.245 ","End":"02:46.535","Text":"We can describe the region in polar terms as Theta"},{"Start":"02:46.535 ","End":"02:53.820","Text":"from Pi over 2 to 3 Pi over 2,"},{"Start":"02:53.820 ","End":"02:58.055","Text":"and r goes from 0 to 1."},{"Start":"02:58.055 ","End":"03:02.300","Text":"Now I need to convert the integral."},{"Start":"03:02.300 ","End":"03:05.590","Text":"There, I brought in all the conversion formulas."},{"Start":"03:05.590 ","End":"03:10.000","Text":"Now, what we have is over this region,"},{"Start":"03:10.000 ","End":"03:14.810","Text":"we get an iterated integral with the drd Theta."},{"Start":"03:15.360 ","End":"03:21.460","Text":"We have by this description that Theta goes on"},{"Start":"03:21.460 ","End":"03:27.475","Text":"the outer loop from Pi over 2 to 3 Pi over 2,"},{"Start":"03:27.475 ","End":"03:31.385","Text":"r goes from 0 to 1."},{"Start":"03:31.385 ","End":"03:34.545","Text":"Now, we going to use the conversion."},{"Start":"03:34.545 ","End":"03:43.200","Text":"What we have is x squared plus y squared and here it is, it\u0027s r squared."},{"Start":"03:43.200 ","End":"03:49.280","Text":"The square root of x squared plus y squared is just r. On the denominator,"},{"Start":"03:49.280 ","End":"03:59.460","Text":"we have 1 plus x squared plus y squared is r squared and dA is given by rdrd Theta."},{"Start":"04:03.010 ","End":"04:05.480","Text":"We can slightly simplify this."},{"Start":"04:05.480 ","End":"04:07.685","Text":"If I pull the 4 all the way up front,"},{"Start":"04:07.685 ","End":"04:14.175","Text":"I\u0027ve got 4 integral"},{"Start":"04:14.175 ","End":"04:22.715","Text":"from Pi over 2 to 3 Pi over 2 and then the integral from 0 to 1,"},{"Start":"04:22.715 ","End":"04:24.845","Text":"this r with this r combines."},{"Start":"04:24.845 ","End":"04:32.255","Text":"So I get r squared over 1 plus r squared drd Theta."},{"Start":"04:32.255 ","End":"04:37.855","Text":"As usual, we have to do these integrals from the inside out."},{"Start":"04:37.855 ","End":"04:40.550","Text":"Let me do this in an integral at the side."},{"Start":"04:40.550 ","End":"04:41.840","Text":"I\u0027ll call it asterisk."},{"Start":"04:41.840 ","End":"04:44.780","Text":"I\u0027ll do it over here where I have some room."},{"Start":"04:44.780 ","End":"04:50.820","Text":"I can rewrite this using standard algebraic tricks, 0 to 1."},{"Start":"04:50.820 ","End":"04:52.440","Text":"Instead of r squared,"},{"Start":"04:52.440 ","End":"04:57.450","Text":"I can write 1 plus r squared minus"},{"Start":"04:57.450 ","End":"05:02.945","Text":"1 because then the denominator is 1 plus r squared,"},{"Start":"05:02.945 ","End":"05:09.210","Text":"and this bit with this bit will cancel, and of course, it\u0027s dr."},{"Start":"05:09.210 ","End":"05:14.850","Text":"What we get after this algebra is the integral from 0 to 1."},{"Start":"05:14.850 ","End":"05:18.060","Text":"1 plus r squared over 1 plus r squared is just 1."},{"Start":"05:18.060 ","End":"05:23.930","Text":"Here, I have minus 1 over 1 plus r squared."},{"Start":"05:23.930 ","End":"05:31.855","Text":"All this, dr. Now these are all well-known integrals."},{"Start":"05:31.855 ","End":"05:35.940","Text":"The integral of 1 is just r,"},{"Start":"05:35.940 ","End":"05:38.190","Text":"the integral of 1 over 1 plus r squared,"},{"Start":"05:38.190 ","End":"05:44.670","Text":"you should recognize it\u0027s the arctangent of r."},{"Start":"05:44.670 ","End":"05:50.370","Text":"I have to take this between the limits of 0 and 1."},{"Start":"05:50.370 ","End":"05:55.890","Text":"What do we get? If I plug in r equals 1,"},{"Start":"05:58.040 ","End":"06:02.720","Text":"the arctangent of 1 is the angle whose tangent is 1."},{"Start":"06:02.720 ","End":"06:09.375","Text":"It\u0027s 45 degrees or Pi over 4 and then if I plug in 0,"},{"Start":"06:09.375 ","End":"06:14.830","Text":"I\u0027ve got 0 and the arctangent of 0 is also 0."},{"Start":"06:14.830 ","End":"06:20.360","Text":"Altogether, we\u0027re left with just the 1 minus Pi over 4."},{"Start":"06:20.360 ","End":"06:21.650","Text":"I\u0027ll just write that here, this is"},{"Start":"06:21.650 ","End":"06:27.030","Text":"1 minus Pi over 4 is our answer and now we can continue."},{"Start":"06:27.260 ","End":"06:30.590","Text":"Now, 1 minus Pi over 4 is a constant."},{"Start":"06:30.590 ","End":"06:32.810","Text":"I can also bring it out of the integral sign."},{"Start":"06:32.810 ","End":"06:38.975","Text":"So I\u0027ve got 4 times 1 minus Pi over 4 times the"},{"Start":"06:38.975 ","End":"06:46.670","Text":"integral from Pi over 2 to 3 Pi over 2 of just d Theta,"},{"Start":"06:46.670 ","End":"06:48.990","Text":"which we\u0027ll write as 1d Theta."},{"Start":"06:49.700 ","End":"06:52.610","Text":"Remember, when we had the integral of 1,"},{"Start":"06:52.610 ","End":"06:54.920","Text":"we just have to take away the top minus the bottom."},{"Start":"06:54.920 ","End":"06:58.250","Text":"3 Pi over 2 minus Pi over 2 is just Pi."},{"Start":"06:58.250 ","End":"07:05.445","Text":"So we get 4 times 1 minus Pi over 4 times Pi."},{"Start":"07:05.445 ","End":"07:08.820","Text":"I really should multiply this out, it will look simpler."},{"Start":"07:08.820 ","End":"07:19.540","Text":"We get 4 minus Pi times Pi and that\u0027s our answer. Done."}],"ID":8704},{"Watched":false,"Name":"Exercise 20","Duration":"8m 44s","ChapterTopicVideoID":8481,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8481.jpeg","UploadDate":"2017-02-13T08:26:36.5100000","DurationForVideoObject":"PT8M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this exercise, we got to compute the following double"},{"Start":"00:03.840 ","End":"00:07.065","Text":"integral by converting to polar coordinates."},{"Start":"00:07.065 ","End":"00:09.450","Text":"I\u0027ve already got all the formulas handy."},{"Start":"00:09.450 ","End":"00:13.290","Text":"What we\u0027re going to do is rewrite this as the double"},{"Start":"00:13.290 ","End":"00:17.010","Text":"integral over a region D and then figure out what D looks like."},{"Start":"00:17.010 ","End":"00:18.900","Text":"I\u0027ll just copy from here."},{"Start":"00:18.900 ","End":"00:20.700","Text":"Don\u0027t forget that dx,"},{"Start":"00:20.700 ","End":"00:23.430","Text":"dy is also dA."},{"Start":"00:23.430 ","End":"00:26.745","Text":"I want to see what this looks like."},{"Start":"00:26.745 ","End":"00:29.350","Text":"I could describe it first."},{"Start":"00:29.870 ","End":"00:32.220","Text":"Let\u0027s see which type is it."},{"Start":"00:32.220 ","End":"00:33.600","Text":"Well, y is the outer loop,"},{"Start":"00:33.600 ","End":"00:35.445","Text":"so it\u0027s horizontal slices."},{"Start":"00:35.445 ","End":"00:37.815","Text":"It\u0027s a type 2 region."},{"Start":"00:37.815 ","End":"00:43.905","Text":"Outer loop y from 0 to some constant here."},{"Start":"00:43.905 ","End":"00:49.525","Text":"In the inner loop, x goes from 0 and it depends on what y is this function here."},{"Start":"00:49.525 ","End":"00:52.250","Text":"We\u0027ll soon see what it looks like."},{"Start":"00:54.050 ","End":"00:58.845","Text":"I like to have the D written out."},{"Start":"00:58.845 ","End":"01:06.970","Text":"The outer loop is where y goes from 0 to this constant natural log of 2."},{"Start":"01:06.970 ","End":"01:10.420","Text":"We\u0027ve already see we\u0027re going to be in the first quadrant only because everything\u0027s from"},{"Start":"01:10.420 ","End":"01:15.440","Text":"0 upwards and x depends on what y is."},{"Start":"01:15.440 ","End":"01:19.985","Text":"It always starts from 0 but it goes up to the square root of this thing here."},{"Start":"01:19.985 ","End":"01:24.870","Text":"Natural log squared 2 minus y squared."},{"Start":"01:24.870 ","End":"01:28.180","Text":"I\u0027m going to give you the picture and then explain."},{"Start":"01:28.180 ","End":"01:34.490","Text":"We see that y goes from 0 to natural log of 2."},{"Start":"01:34.490 ","End":"01:37.520","Text":"For each typical y,"},{"Start":"01:37.520 ","End":"01:41.610","Text":"which like a horizontal slice like so."},{"Start":"01:41.610 ","End":"01:45.560","Text":"We enter the region here we\u0027re always going in this direction,"},{"Start":"01:45.560 ","End":"01:48.320","Text":"and we exit the region here."},{"Start":"01:48.320 ","End":"01:52.935","Text":"This of course is where x equals 0."},{"Start":"01:52.935 ","End":"01:54.350","Text":"On the other side,"},{"Start":"01:54.350 ","End":"02:02.420","Text":"x equals the square root of this expression minus y squared."},{"Start":"02:02.420 ","End":"02:06.980","Text":"Now, I\u0027m planning this what we get is a quarter of a circle."},{"Start":"02:06.980 ","End":"02:11.330","Text":"This is just of the form 0 less than or equal to x,"},{"Start":"02:11.330 ","End":"02:12.515","Text":"less than or equal to."},{"Start":"02:12.515 ","End":"02:16.070","Text":"If I write it as a squared minus y squared,"},{"Start":"02:16.070 ","End":"02:21.450","Text":"that will look a lot better where of course our a is equal to natural log of 2."},{"Start":"02:21.450 ","End":"02:24.665","Text":"This we already know is part of a circle of radius a,"},{"Start":"02:24.665 ","End":"02:28.310","Text":"because if you square both sides and bring the y squared over,"},{"Start":"02:28.310 ","End":"02:31.145","Text":"it\u0027s x squared plus y squared equals a squared."},{"Start":"02:31.145 ","End":"02:36.835","Text":"But it\u0027s only a quarter circle because both of these are non-negative,"},{"Start":"02:36.835 ","End":"02:39.180","Text":"we\u0027re taking the positive square root."},{"Start":"02:39.180 ","End":"02:45.320","Text":"What we have here is quarter of a circle of radius natural log 2."},{"Start":"02:45.320 ","End":"02:47.885","Text":"This is also natural log of 2,"},{"Start":"02:47.885 ","End":"02:55.260","Text":"and that\u0027s the region in rectangular in Cartesian coordinates."},{"Start":"02:55.260 ","End":"02:59.000","Text":"Now I want to write it in polar coordinates."},{"Start":"02:59.000 ","End":"03:01.255","Text":"Let me clear what I don\u0027t need."},{"Start":"03:01.255 ","End":"03:05.880","Text":"In polar coordinates, we can see that theta goes from"},{"Start":"03:05.880 ","End":"03:10.320","Text":"here to here in the positive direction counterclockwise."},{"Start":"03:10.320 ","End":"03:12.809","Text":"Here theta is equal to 0,"},{"Start":"03:12.809 ","End":"03:15.210","Text":"here theta equals 90 degrees,"},{"Start":"03:15.210 ","End":"03:17.145","Text":"that\u0027s Pi over 2."},{"Start":"03:17.145 ","End":"03:22.545","Text":"As for us for any given theta, wherever it is,"},{"Start":"03:22.545 ","End":"03:25.499","Text":"r goes from 0_ radius,"},{"Start":"03:25.499 ","End":"03:29.080","Text":"which is natural log of 2."},{"Start":"03:29.080 ","End":"03:34.265","Text":"I can rewrite the Cartesian region D as a polar region."},{"Start":"03:34.265 ","End":"03:36.340","Text":"In the polar form,"},{"Start":"03:36.340 ","End":"03:46.490","Text":"theta goes from 0 to Pi over 2 and the radius goes from 0 to natural log of 2."},{"Start":"03:46.490 ","End":"03:53.160","Text":"Now I can rewrite this integral in polar form as an iterated integral."},{"Start":"03:53.160 ","End":"03:56.850","Text":"Here theta, here r, let\u0027s see again."},{"Start":"03:56.850 ","End":"04:01.650","Text":"Theta from 0 to Pi over 2,"},{"Start":"04:01.650 ","End":"04:06.915","Text":"r from 0 to natural log of 2."},{"Start":"04:06.915 ","End":"04:14.805","Text":"Next, convert this, e_x squared plus y squared is r squared."},{"Start":"04:14.805 ","End":"04:23.230","Text":"This square root is just r and dA is rdrd theta."},{"Start":"04:25.570 ","End":"04:29.495","Text":"Here we have the polar form."},{"Start":"04:29.495 ","End":"04:35.285","Text":"We want to do the inner integral first which is this."},{"Start":"04:35.285 ","End":"04:38.440","Text":"Let me do this inner integral separately at the sides."},{"Start":"04:38.440 ","End":"04:48.270","Text":"What we have to compute is the integral from 0 to natural log of 2 of re_rdr."},{"Start":"04:49.810 ","End":"04:52.615","Text":"I Just switched the order."},{"Start":"04:52.615 ","End":"04:55.750","Text":"This is a classic case of integration by parts."},{"Start":"04:55.750 ","End":"05:01.285","Text":"Let this be u and this part be dv."},{"Start":"05:01.285 ","End":"05:08.685","Text":"What I want to get is uv minus the integral."},{"Start":"05:08.685 ","End":"05:13.590","Text":"I\u0027m just writing at the bottom here to remind myself vdu."},{"Start":"05:13.590 ","End":"05:16.410","Text":"There is an integral here."},{"Start":"05:16.410 ","End":"05:22.299","Text":"Let\u0027s see, u is r,"},{"Start":"05:22.790 ","End":"05:27.850","Text":"v is the integral of this is e_r."},{"Start":"05:28.960 ","End":"05:32.780","Text":"Then I have minus,"},{"Start":"05:32.780 ","End":"05:38.285","Text":"and then the integral of v we already said was e_r."},{"Start":"05:38.285 ","End":"05:40.810","Text":"It\u0027s the anti-derivative of e_r."},{"Start":"05:40.810 ","End":"05:48.420","Text":"We want a du, du is just the same as dr. 1du equals 1dr,"},{"Start":"05:48.420 ","End":"05:56.245","Text":"so this is dr. We can actually do the definite integral."},{"Start":"05:56.245 ","End":"05:58.630","Text":"If it\u0027s an indefinite integral form,"},{"Start":"05:58.630 ","End":"06:08.215","Text":"then I\u0027ll take this between 0 and natural log of 2."},{"Start":"06:08.215 ","End":"06:15.850","Text":"This is the integral from 0 to natural log of 2."},{"Start":"06:15.850 ","End":"06:18.070","Text":"Sorry, it\u0027s a bit crowded,"},{"Start":"06:18.070 ","End":"06:20.005","Text":"and the equal sign."},{"Start":"06:20.005 ","End":"06:23.805","Text":"Continuing, this is equal to."},{"Start":"06:23.805 ","End":"06:27.685","Text":"Now here I need to substitute the upper and the lower limits."},{"Start":"06:27.685 ","End":"06:31.450","Text":"If I put in natural log of 2,"},{"Start":"06:31.450 ","End":"06:35.110","Text":"e_ natural log, that canceled each other out."},{"Start":"06:35.110 ","End":"06:36.670","Text":"This is just 2."},{"Start":"06:36.670 ","End":"06:43.270","Text":"I\u0027ve got here 2 natural log of 2 from the upper limit."},{"Start":"06:43.270 ","End":"06:46.135","Text":"I substitute 0, I\u0027ve got nothing."},{"Start":"06:46.135 ","End":"06:50.740","Text":"This thing is just 2 natural log of 2 minus,"},{"Start":"06:50.740 ","End":"06:55.510","Text":"now the integral of e_r is e_r."},{"Start":"06:55.510 ","End":"07:04.810","Text":"I\u0027ve got e_r between 0 and natural log of 2."},{"Start":"07:05.720 ","End":"07:10.770","Text":"What do we get?"},{"Start":"07:10.770 ","End":"07:14.440","Text":"If I plug in natural log of 2,"},{"Start":"07:14.440 ","End":"07:18.030","Text":"I\u0027ve just got 2."},{"Start":"07:18.030 ","End":"07:20.895","Text":"If I plug in 0, I get 1."},{"Start":"07:20.895 ","End":"07:24.285","Text":"This thing here is 2 minus 1,"},{"Start":"07:24.285 ","End":"07:30.015","Text":"which is 1, so I get 2 natural log of 2 minus 1."},{"Start":"07:30.015 ","End":"07:33.165","Text":"Again, e_ natural log of 2 is just 2,"},{"Start":"07:33.165 ","End":"07:36.210","Text":"e_0 is 1, 2 minus 1 is 1."},{"Start":"07:36.210 ","End":"07:41.180","Text":"Now back here, perhaps I should have mentioned one of"},{"Start":"07:41.180 ","End":"07:47.465","Text":"the formulas I was using here is that e_natural log of something say a,"},{"Start":"07:47.465 ","End":"07:49.010","Text":"is just a itself."},{"Start":"07:49.010 ","End":"07:54.190","Text":"I was using this in the case where a was 2."},{"Start":"07:54.230 ","End":"07:59.690","Text":"Back here, since this is a constant,"},{"Start":"07:59.690 ","End":"08:03.125","Text":"I can pull it out in front of the integral sign,"},{"Start":"08:03.125 ","End":"08:11.790","Text":"and so I get 2 natural log 2 minus 1,"},{"Start":"08:14.660 ","End":"08:25.425","Text":"times the integral of theta from 0 to Pi over 2 of 1d theta."},{"Start":"08:25.425 ","End":"08:28.890","Text":"We have the integral of 1, we just subtract the lower from the upper."},{"Start":"08:28.890 ","End":"08:32.565","Text":"Pi over 2 minus 0 is Pi over 2."},{"Start":"08:32.565 ","End":"08:36.735","Text":"Our answer is Pi over 2,"},{"Start":"08:36.735 ","End":"08:41.070","Text":"2 natural log of 2 minus 1."},{"Start":"08:41.070 ","End":"08:44.770","Text":"That is our answer."}],"ID":8705},{"Watched":false,"Name":"Exercise 21","Duration":"7m 59s","ChapterTopicVideoID":8482,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8482.jpeg","UploadDate":"2017-02-13T08:27:34.7930000","DurationForVideoObject":"PT7M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this exercise, we have a double integral to do,"},{"Start":"00:02.730 ","End":"00:06.135","Text":"but we want to convert it to polar coordinates first."},{"Start":"00:06.135 ","End":"00:07.890","Text":"We see it\u0027s dy dx,"},{"Start":"00:07.890 ","End":"00:12.510","Text":"which is a type 1 region."},{"Start":"00:12.510 ","End":"00:18.330","Text":"Let me write that as integral over a region, integral over D,"},{"Start":"00:18.330 ","End":"00:22.185","Text":"and then we want to sketch D of the same thing,"},{"Start":"00:22.185 ","End":"00:27.044","Text":"e to the minus x squared plus y squared,"},{"Start":"00:27.044 ","End":"00:29.655","Text":"and as I said, dy, dx,"},{"Start":"00:29.655 ","End":"00:32.560","Text":"which we also can call dA."},{"Start":"00:34.820 ","End":"00:43.085","Text":"I can describe the region D by noticing that the outer loop x goes from 0-1,"},{"Start":"00:43.085 ","End":"00:45.825","Text":"so that\u0027s what I write."},{"Start":"00:45.825 ","End":"00:49.230","Text":"For each such x, y runs"},{"Start":"00:49.230 ","End":"00:57.900","Text":"from 0 to square root of 1 minus x squared."},{"Start":"00:57.900 ","End":"00:59.640","Text":"If I sketch this,"},{"Start":"00:59.640 ","End":"01:03.195","Text":"I could probably tell straight away it\u0027s 1/4 of a circle,"},{"Start":"01:03.195 ","End":"01:06.630","Text":"at least we\u0027re in the first quadrant."},{"Start":"01:06.630 ","End":"01:11.415","Text":"X on the outside goes from 0-1,"},{"Start":"01:11.415 ","End":"01:15.204","Text":"and for each particular x in this range,"},{"Start":"01:15.204 ","End":"01:21.785","Text":"the vertical slice y goes from this point to this point."},{"Start":"01:21.785 ","End":"01:25.190","Text":"Obviously on the x-axis, y is 0,"},{"Start":"01:25.190 ","End":"01:30.080","Text":"and this is y equals the square root of 1 minus x squared,"},{"Start":"01:30.080 ","End":"01:32.225","Text":"which we recognize as part of the circle."},{"Start":"01:32.225 ","End":"01:33.470","Text":"We\u0027ve done this so many times."},{"Start":"01:33.470 ","End":"01:35.120","Text":"If you square it,"},{"Start":"01:35.120 ","End":"01:36.810","Text":"you get y squared is 1 minus x squared,"},{"Start":"01:36.810 ","End":"01:38.750","Text":"so x squared plus y squared is 1."},{"Start":"01:38.750 ","End":"01:40.340","Text":"It\u0027s part of a circle,"},{"Start":"01:40.340 ","End":"01:42.710","Text":"and in each case we\u0027re going from 0 to something."},{"Start":"01:42.710 ","End":"01:50.060","Text":"It\u0027s 1/4 of a circle and that\u0027s what it is in Cartesian coordinates."},{"Start":"01:50.060 ","End":"01:56.000","Text":"Now we want to convert this region into polar coordinates and write"},{"Start":"01:56.000 ","End":"02:02.920","Text":"it as Theta something and r something."},{"Start":"02:03.320 ","End":"02:06.570","Text":"Let me simplify the picture."},{"Start":"02:06.570 ","End":"02:09.720","Text":"We have this quarter circle radius 1,"},{"Start":"02:09.720 ","End":"02:13.300","Text":"Theta goes from here to here."},{"Start":"02:13.610 ","End":"02:17.025","Text":"This is our typical Theta,"},{"Start":"02:17.025 ","End":"02:22.590","Text":"then it goes from Theta equals 0 to Theta equals 90 degrees,"},{"Start":"02:22.590 ","End":"02:26.010","Text":"but we use radians Pi over 2."},{"Start":"02:26.010 ","End":"02:28.609","Text":"For each Theta also,"},{"Start":"02:28.609 ","End":"02:33.480","Text":"radius goes from here to here, from 0-1."},{"Start":"02:34.430 ","End":"02:37.140","Text":"We can fill in the missing bits here,"},{"Start":"02:37.140 ","End":"02:41.650","Text":"Theta from 0 to Pi over 2, from 0-1."},{"Start":"02:41.650 ","End":"02:46.635","Text":"Now we rewrite this integral in polar."},{"Start":"02:46.635 ","End":"02:50.450","Text":"We have Theta something here and r something here."},{"Start":"02:50.450 ","End":"02:56.535","Text":"Let\u0027s see, outer loop Theta from 0 to Pi over 2,"},{"Start":"02:56.535 ","End":"03:00.340","Text":"inner loop r from 0-1."},{"Start":"03:00.340 ","End":"03:03.530","Text":"Now we use the conversion formulas,"},{"Start":"03:03.530 ","End":"03:09.045","Text":"e to the minus, x squared plus y squared, here it is,"},{"Start":"03:09.045 ","End":"03:11.985","Text":"is r squared, dA,"},{"Start":"03:11.985 ","End":"03:16.120","Text":"here it is, rdrd Theta."},{"Start":"03:18.620 ","End":"03:23.245","Text":"This is our integral in polar form."},{"Start":"03:23.245 ","End":"03:26.395","Text":"First we do the inner loop,"},{"Start":"03:26.395 ","End":"03:32.455","Text":"the r. This inner integral,"},{"Start":"03:32.455 ","End":"03:36.310","Text":"I\u0027d like to compute it at the side and then return here."},{"Start":"03:36.310 ","End":"03:41.905","Text":"What we have to do is the integral of,"},{"Start":"03:41.905 ","End":"03:50.650","Text":"let me just rewrite it slightly as r e to the minus r squared dr,"},{"Start":"03:50.650 ","End":"03:54.510","Text":"which is a definite integral from 0-1."},{"Start":"03:54.510 ","End":"03:57.865","Text":"Now, 1 way to do this is substitution."},{"Start":"03:57.865 ","End":"04:02.810","Text":"We could let t equals minus r-squared."},{"Start":"04:02.810 ","End":"04:06.140","Text":"I think substitution is too heavy here."},{"Start":"04:06.140 ","End":"04:10.715","Text":"What I\u0027m thinking is more on the lines of the,"},{"Start":"04:10.715 ","End":"04:19.700","Text":"we have this rule formula that if I have the integral of e to the power of something,"},{"Start":"04:19.700 ","End":"04:21.950","Text":"call it box some function of x,"},{"Start":"04:21.950 ","End":"04:30.185","Text":"but I also have the derivative of that box alongside d whatever,"},{"Start":"04:30.185 ","End":"04:35.375","Text":"then this is just equal to e to the box,"},{"Start":"04:35.375 ","End":"04:38.795","Text":"well, plus C. I\u0027m not even going to write that."},{"Start":"04:38.795 ","End":"04:41.960","Text":"Because if you differentiate this by the chain rule,"},{"Start":"04:41.960 ","End":"04:44.690","Text":"you get e to the something gives you e to the something,"},{"Start":"04:44.690 ","End":"04:47.430","Text":"but the inner derivative is that."},{"Start":"04:48.470 ","End":"04:54.095","Text":"My box is going to be this here."},{"Start":"04:54.095 ","End":"05:00.300","Text":"But I don\u0027t have box prime because box prime is minus 2r."},{"Start":"05:00.300 ","End":"05:02.795","Text":"I\u0027m going to just fix it up a bit."},{"Start":"05:02.795 ","End":"05:06.435","Text":"If I put a minus 2 here,"},{"Start":"05:06.435 ","End":"05:07.950","Text":"then I have it but,"},{"Start":"05:07.950 ","End":"05:09.915","Text":"I can\u0027t just go ahead and change it,"},{"Start":"05:09.915 ","End":"05:13.230","Text":"so I\u0027ll also put a minus 1/2 here."},{"Start":"05:13.230 ","End":"05:15.155","Text":"If I do all that,"},{"Start":"05:15.155 ","End":"05:22.670","Text":"then the part after the integral looks very much like this and so what we get is,"},{"Start":"05:22.670 ","End":"05:24.995","Text":"the minus 1/2 stays here,"},{"Start":"05:24.995 ","End":"05:28.300","Text":"and then by this formula, like I said,"},{"Start":"05:28.300 ","End":"05:34.570","Text":"this bit here is this thing prime,"},{"Start":"05:34.570 ","End":"05:44.500","Text":"so we get just e to this bit here, minus r-squared."},{"Start":"05:44.500 ","End":"05:47.369","Text":"But it\u0027s a definite integral,"},{"Start":"05:47.369 ","End":"05:50.715","Text":"that\u0027s why we didn\u0027t need the plus C here,"},{"Start":"05:50.715 ","End":"05:55.950","Text":"because we\u0027re going to substitute 0 and 1,"},{"Start":"05:55.950 ","End":"05:58.335","Text":"and what will that give us?"},{"Start":"05:58.335 ","End":"06:02.985","Text":"Well, if r is 1,"},{"Start":"06:02.985 ","End":"06:11.920","Text":"we get minus 1/2 e to the minus 1."},{"Start":"06:12.020 ","End":"06:16.260","Text":"Well, I\u0027ll take the minus 1/2 outside."},{"Start":"06:16.260 ","End":"06:19.920","Text":"Yeah, e to the minus 1,"},{"Start":"06:19.920 ","End":"06:21.670","Text":"and if I let r equals 0,"},{"Start":"06:21.670 ","End":"06:24.384","Text":"I\u0027ve got e to the minus 0,"},{"Start":"06:24.384 ","End":"06:27.020","Text":"which is just 1."},{"Start":"06:27.020 ","End":"06:30.250","Text":"That solves this part here,"},{"Start":"06:30.250 ","End":"06:32.230","Text":"what we call the asterisk."},{"Start":"06:32.230 ","End":"06:37.180","Text":"Back here, because this is a constant,"},{"Start":"06:37.180 ","End":"06:43.640","Text":"I can pull it in front of the integral and maybe slightly rewrite it."},{"Start":"06:45.300 ","End":"06:49.420","Text":"Well, I could leave it as is, doesn\u0027t really matter,"},{"Start":"06:49.420 ","End":"06:56.495","Text":"but I\u0027d like to write the e to the minus 1 as 1 over e minus 1,"},{"Start":"06:56.495 ","End":"07:05.700","Text":"and then the integral from 0 to Pi over 2 of just d Theta."},{"Start":"07:05.700 ","End":"07:08.505","Text":"I put a 1 there."},{"Start":"07:08.505 ","End":"07:12.575","Text":"Now we know how to do this."},{"Start":"07:12.575 ","End":"07:15.320","Text":"Integral of 1 is just the upper minus the lower,"},{"Start":"07:15.320 ","End":"07:18.845","Text":"so it\u0027s Pi over 2 minus 0."},{"Start":"07:18.845 ","End":"07:25.620","Text":"I\u0027ve got Pi minus 1/2,"},{"Start":"07:25.790 ","End":"07:32.560","Text":"1 over e minus 1 times Pi over 2."},{"Start":"07:34.060 ","End":"07:37.115","Text":"Nothing much to simplify."},{"Start":"07:37.115 ","End":"07:42.830","Text":"Personally, I like to get rid of the minus by reversing the order of the subtraction,"},{"Start":"07:42.830 ","End":"07:49.250","Text":"and also combining the Pi over 2 with 1/2 so I can get Pi over 4."},{"Start":"07:49.250 ","End":"07:55.175","Text":"That\u0027s this with this and the minus makes this subtraction the reverse order,"},{"Start":"07:55.175 ","End":"07:59.070","Text":"and this is the answer."}],"ID":8706},{"Watched":false,"Name":"Exercise 22","Duration":"15m 41s","ChapterTopicVideoID":8483,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8483.jpeg","UploadDate":"2017-02-13T08:29:32.1100000","DurationForVideoObject":"PT15M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.135","Text":"In this exercise we are given a double integral,"},{"Start":"00:03.135 ","End":"00:06.900","Text":"but we have to compute it by converting to polar coordinates."},{"Start":"00:06.900 ","End":"00:10.920","Text":"This one is a bit different from the ones that have been so far."},{"Start":"00:10.920 ","End":"00:17.505","Text":"The big difference is that we don\u0027t have x squared here we have x minus 1,"},{"Start":"00:17.505 ","End":"00:20.715","Text":"square then you\u0027ll see later on this will make all the difference."},{"Start":"00:20.715 ","End":"00:22.905","Text":"We\u0027re not going to have circular symmetry,"},{"Start":"00:22.905 ","End":"00:25.560","Text":"but I\u0027m getting ahead of myself."},{"Start":"00:25.560 ","End":"00:34.040","Text":"As usual I like to write this generally as the double integral over a region D. In fact,"},{"Start":"00:34.040 ","End":"00:40.760","Text":"this is going to be a Type 1 region because it\u0027s dydx of the same thing,"},{"Start":"00:40.760 ","End":"00:48.760","Text":"x plus y over x squared plus y squared and then dydx,"},{"Start":"00:48.760 ","End":"00:51.610","Text":"which we also call dA,"},{"Start":"00:51.610 ","End":"00:54.874","Text":"and because of the outer loop is on x,"},{"Start":"00:54.874 ","End":"01:00.290","Text":"it\u0027s a Type 1 vertical slice region."},{"Start":"01:00.290 ","End":"01:07.565","Text":"I could describe D with inequalities saying that the outer loop, which is dx,"},{"Start":"01:07.565 ","End":"01:12.840","Text":"so x goes from here, 0-2,"},{"Start":"01:12.840 ","End":"01:17.465","Text":"that\u0027s straightforward enough, and then for each such x,"},{"Start":"01:17.465 ","End":"01:22.460","Text":"y also goes from 0 to something."},{"Start":"01:22.460 ","End":"01:24.230","Text":"Obviously we\u0027re going to be all in"},{"Start":"01:24.230 ","End":"01:31.345","Text":"the first quadrant and then I\u0027m just copying this expression,"},{"Start":"01:31.345 ","End":"01:36.690","Text":"1 minus x minus 1 squared positive square root."},{"Start":"01:36.690 ","End":"01:44.180","Text":"Let me spend a few moments looking at the upper limit of y."},{"Start":"01:44.180 ","End":"01:47.495","Text":"What is this function here?"},{"Start":"01:47.495 ","End":"01:55.445","Text":"If y was equal to square root of 1 minus x minus 1 squared,"},{"Start":"01:55.445 ","End":"01:56.855","Text":"what does this look like?"},{"Start":"01:56.855 ","End":"01:59.750","Text":"Well, we square both sides of the equation."},{"Start":"01:59.750 ","End":"02:05.150","Text":"We\u0027ll get y squared equals 1 minus x minus 1 squared,"},{"Start":"02:05.150 ","End":"02:12.560","Text":"and then bring this over to the other side we get x minus 1 squared plus y squared."},{"Start":"02:12.560 ","End":"02:15.809","Text":"I\u0027ll emphasize it by,"},{"Start":"02:16.000 ","End":"02:20.720","Text":"y squared is y minus 0 squared equals 1,"},{"Start":"02:20.720 ","End":"02:24.560","Text":"which I\u0027ll write as 1 squared because I\u0027m trying to get it to look like the equation of"},{"Start":"02:24.560 ","End":"02:29.255","Text":"a circle where x minus a squared plus y minus b squared is r squared."},{"Start":"02:29.255 ","End":"02:34.790","Text":"In this case we see that the center is 1,"},{"Start":"02:34.790 ","End":"02:38.705","Text":"0, I\u0027ll write that."},{"Start":"02:38.705 ","End":"02:42.035","Text":"Center is 1, 0,"},{"Start":"02:42.035 ","End":"02:46.315","Text":"and the radius of the circle is 1,"},{"Start":"02:46.315 ","End":"02:50.660","Text":"but y is bigger or equal to 0,"},{"Start":"02:50.660 ","End":"02:52.730","Text":"because it\u0027s the square root, so it\u0027s actually"},{"Start":"02:52.730 ","End":"02:56.960","Text":"the upper semicircle and I\u0027ll bring in the sketch."},{"Start":"02:56.960 ","End":"03:01.380","Text":"Just made room for it and here it is."},{"Start":"03:01.380 ","End":"03:07.035","Text":"Circle with the center 1, 0 that\u0027s 1,"},{"Start":"03:07.035 ","End":"03:08.700","Text":"and radius is 1,"},{"Start":"03:08.700 ","End":"03:10.440","Text":"which means it goes from 0-2,"},{"Start":"03:10.440 ","End":"03:14.380","Text":"and it\u0027s the upper semicircle."},{"Start":"03:14.960 ","End":"03:18.430","Text":"We take it as a Type 1 region,"},{"Start":"03:18.430 ","End":"03:23.740","Text":"meaning that the outer loop is x from 0-2,"},{"Start":"03:23.740 ","End":"03:28.045","Text":"and for each typical x from 0-2,"},{"Start":"03:28.045 ","End":"03:31.510","Text":"we take a vertical slice and we"},{"Start":"03:31.510 ","End":"03:40.020","Text":"go in at this point and out at this point."},{"Start":"03:40.020 ","End":"03:47.250","Text":"This line of course is the x-axis where y equals 0,"},{"Start":"03:47.250 ","End":"03:51.855","Text":"that\u0027s this 0 here or say here,"},{"Start":"03:51.855 ","End":"03:57.130","Text":"and the upper one is the y equals the square root of 1"},{"Start":"03:57.130 ","End":"04:03.615","Text":"minus x minus 1 squared and so we sweep across in vertical strips."},{"Start":"04:03.615 ","End":"04:07.850","Text":"That\u0027s Cartesian, x and y. I want to describe"},{"Start":"04:07.850 ","End":"04:12.710","Text":"this D in terms of polar coordinates in r and Theta."},{"Start":"04:12.710 ","End":"04:17.480","Text":"Just clean up a bit and now comes the tricky bit of"},{"Start":"04:17.480 ","End":"04:22.535","Text":"describing this region D in polar terms."},{"Start":"04:22.535 ","End":"04:28.230","Text":"I want to say where Theta goes from and to,"},{"Start":"04:28.230 ","End":"04:32.610","Text":"and I want to say where r goes from and to."},{"Start":"04:32.610 ","End":"04:35.335","Text":"The Theta part is not so bad."},{"Start":"04:35.335 ","End":"04:46.750","Text":"Let\u0027s take any point on the circumference here and let\u0027s connect it to the origin."},{"Start":"04:46.910 ","End":"04:50.655","Text":"This is our Theta."},{"Start":"04:50.655 ","End":"04:53.400","Text":"Let\u0027s see, where does Theta go from and to?"},{"Start":"04:53.400 ","End":"04:56.710","Text":"Well, clearly it goes from Theta equals 0."},{"Start":"04:56.710 ","End":"04:58.460","Text":"Where does it stop?"},{"Start":"04:58.460 ","End":"05:02.650","Text":"If you think about it because this is the tangent to the circle."},{"Start":"05:02.650 ","End":"05:07.415","Text":"As long as we\u0027re less than 90 degrees we\u0027re going to cut the circle."},{"Start":"05:07.415 ","End":"05:10.795","Text":"Even at 90 degrees we just cut it at this one point."},{"Start":"05:10.795 ","End":"05:13.030","Text":"So we\u0027re going from 0-90 degrees."},{"Start":"05:13.030 ","End":"05:17.375","Text":"Of course, we use radians and that\u0027s Pi over 2."},{"Start":"05:17.375 ","End":"05:21.180","Text":"This part is the straightforward part."},{"Start":"05:21.180 ","End":"05:24.465","Text":"As for r it starts from 0,"},{"Start":"05:24.465 ","End":"05:26.075","Text":"but the question is,"},{"Start":"05:26.075 ","End":"05:28.620","Text":"where does it end?"},{"Start":"05:28.670 ","End":"05:33.665","Text":"This is 0, but this length here,"},{"Start":"05:33.665 ","End":"05:39.320","Text":"I don\u0027t know what it is, I\u0027ll call it r1."},{"Start":"05:39.320 ","End":"05:42.755","Text":"Just don\u0027t use the same letter r again."},{"Start":"05:42.755 ","End":"05:46.075","Text":"R goes from 0 up to,"},{"Start":"05:46.075 ","End":"05:48.390","Text":"we\u0027ve given it a name, r1."},{"Start":"05:48.390 ","End":"05:50.420","Text":"We have to find out what this is and it\u0027s"},{"Start":"05:50.420 ","End":"05:53.180","Text":"certainly going to depend on Theta because as we"},{"Start":"05:53.180 ","End":"05:59.230","Text":"cut different angles we have different lengths for r or r1."},{"Start":"05:59.230 ","End":"06:03.020","Text":"The part we\u0027re missing is the semicircle and we"},{"Start":"06:03.020 ","End":"06:06.020","Text":"need to know the equation of this in polar coordinates."},{"Start":"06:06.020 ","End":"06:08.420","Text":"We\u0027re going to use these formulas to help us."},{"Start":"06:08.420 ","End":"06:11.255","Text":"We\u0027ll start off with Cartesian."},{"Start":"06:11.255 ","End":"06:15.965","Text":"In Cartesian we have that this is given by"},{"Start":"06:15.965 ","End":"06:22.700","Text":"y equals the square root of 1 minus x minus 1 squared."},{"Start":"06:22.700 ","End":"06:23.720","Text":"Just copying it."},{"Start":"06:23.720 ","End":"06:28.250","Text":"I want to get this as r in terms of Theta."},{"Start":"06:28.250 ","End":"06:31.895","Text":"Let\u0027s raise both sides to the power of 2, square them."},{"Start":"06:31.895 ","End":"06:39.170","Text":"We\u0027ve got y squared equals 1 minus x minus 1 squared."},{"Start":"06:39.170 ","End":"06:44.885","Text":"Now bring this x minus 1 squared to the other side,"},{"Start":"06:44.885 ","End":"06:50.135","Text":"but I\u0027ll also open it up and we\u0027ve got x squared minus 2x"},{"Start":"06:50.135 ","End":"06:58.420","Text":"plus 1 plus y squared is equal to 1."},{"Start":"06:58.460 ","End":"07:02.350","Text":"The 1s cancel I\u0027ll bring the 2x over."},{"Start":"07:02.350 ","End":"07:09.380","Text":"I have x squared plus y squared equals 2x."},{"Start":"07:09.380 ","End":"07:13.985","Text":"Now I can use these formulas here,"},{"Start":"07:13.985 ","End":"07:24.790","Text":"x squared plus y squared is r squared and 2x is 2r cosine Theta."},{"Start":"07:27.800 ","End":"07:36.375","Text":"Now I\u0027m going to divide both sides by r and say r equals 2 cosine Theta."},{"Start":"07:36.375 ","End":"07:39.060","Text":"You might say, hey, wait a minute, what if r is 0?"},{"Start":"07:39.060 ","End":"07:41.760","Text":"Well, there is one point where r is 0,"},{"Start":"07:41.760 ","End":"07:44.670","Text":"that\u0027s exactly at 90 degrees."},{"Start":"07:44.670 ","End":"07:49.230","Text":"But if you substitute 90 degrees since cosine of 90,"},{"Start":"07:49.230 ","End":"07:51.630","Text":"or should I say Pi over 2 is 0,"},{"Start":"07:51.630 ","End":"07:53.445","Text":"it works there also."},{"Start":"07:53.445 ","End":"07:57.630","Text":"We\u0027re fine even if r is 0."},{"Start":"07:57.630 ","End":"08:01.365","Text":"Let me say r equals 0 is okay."},{"Start":"08:01.365 ","End":"08:03.840","Text":"Works also."},{"Start":"08:03.840 ","End":"08:09.075","Text":"Now we have this formula for the semicircle."},{"Start":"08:09.075 ","End":"08:11.980","Text":"I can now replace it here."},{"Start":"08:12.560 ","End":"08:22.875","Text":"I\u0027ll just put a line through that and say this is 2 cosine Theta."},{"Start":"08:22.875 ","End":"08:30.250","Text":"I actually should have used r_1 here because that\u0027s the particular r_1."},{"Start":"08:30.920 ","End":"08:34.995","Text":"It\u0027 s being technical."},{"Start":"08:34.995 ","End":"08:43.500","Text":"Now we can write this as a double iterated integral in polar form."},{"Start":"08:43.500 ","End":"08:46.485","Text":"We have the outer integral,"},{"Start":"08:46.485 ","End":"08:52.110","Text":"Theta from 0 to Pi over 2,"},{"Start":"08:52.110 ","End":"08:53.355","Text":"does that look like a Theta?"},{"Start":"08:53.355 ","End":"08:57.600","Text":"Yeah. Then we have the inner integral,"},{"Start":"08:57.600 ","End":"09:06.390","Text":"which is r which goes from 0 to cosine Theta."},{"Start":"09:06.390 ","End":"09:07.920","Text":"It depends on Theta,"},{"Start":"09:07.920 ","End":"09:10.604","Text":"as most of our previous examples."},{"Start":"09:10.604 ","End":"09:14.085","Text":"Then we look at the integrant,"},{"Start":"09:14.085 ","End":"09:16.020","Text":"the thing that we integrate."},{"Start":"09:16.020 ","End":"09:19.545","Text":"We\u0027re also going to have to use the formulas here."},{"Start":"09:19.545 ","End":"09:22.530","Text":"X is r cosine Theta,"},{"Start":"09:22.530 ","End":"09:24.465","Text":"y is r sine Theta."},{"Start":"09:24.465 ","End":"09:26.805","Text":"I could take r outside the brackets,"},{"Start":"09:26.805 ","End":"09:34.630","Text":"so I have r cosine Theta plus sine Theta."},{"Start":"09:36.950 ","End":"09:41.895","Text":"Then on the denominator I have x squared plus y squared,"},{"Start":"09:41.895 ","End":"09:44.865","Text":"which is r squared."},{"Start":"09:44.865 ","End":"09:47.475","Text":"Getting a bit cramped here."},{"Start":"09:47.475 ","End":"09:49.590","Text":"Here I can squeeze in a dA,"},{"Start":"09:49.590 ","End":"09:55.915","Text":"which is r dr d Theta that\u0027s here."},{"Start":"09:55.915 ","End":"10:00.335","Text":"Now I can simplify this a bit because I have this r,"},{"Start":"10:00.335 ","End":"10:04.415","Text":"with this r, will cancel with this r squared."},{"Start":"10:04.415 ","End":"10:10.205","Text":"So I get the integral from 0 to Pi over 2."},{"Start":"10:10.205 ","End":"10:19.110","Text":"That\u0027s for Theta integral from 0-2 cosine Theta"},{"Start":"10:19.880 ","End":"10:25.950","Text":"of cosine Theta plus sine"},{"Start":"10:25.950 ","End":"10:32.775","Text":"Theta dr d Theta."},{"Start":"10:32.775 ","End":"10:37.020","Text":"Yes, technically this is not defined for r equals 0."},{"Start":"10:37.020 ","End":"10:38.340","Text":"It can be justified."},{"Start":"10:38.340 ","End":"10:43.709","Text":"Let\u0027s just let that 1 slide."},{"Start":"10:43.709 ","End":"10:46.290","Text":"I don\u0027t want to bore you with all the details."},{"Start":"10:46.290 ","End":"10:48.640","Text":"It\u0027s okay."},{"Start":"10:49.730 ","End":"10:55.110","Text":"Now this is actually a constant as far as r goes."},{"Start":"10:55.110 ","End":"10:58.545","Text":"I could even take this bit and put it in front of the integral."},{"Start":"10:58.545 ","End":"11:03.840","Text":"Let\u0027s do 1 more step and say that this is the integral from 0 to"},{"Start":"11:03.840 ","End":"11:10.830","Text":"Pi over 2 of cosine Theta plus sine Theta."},{"Start":"11:10.830 ","End":"11:18.030","Text":"Then the integral from 0-2 cosine Theta,"},{"Start":"11:18.030 ","End":"11:25.240","Text":"nothing left, just 1 dr and then d Theta."},{"Start":"11:25.280 ","End":"11:29.520","Text":"Now this inner integral is very simple."},{"Start":"11:29.520 ","End":"11:34.800","Text":"It\u0027s the integral of 1, so it\u0027s just the upper limit minus the lower limit,"},{"Start":"11:34.800 ","End":"11:38.910","Text":"which is 2 cosine Theta minus nothing."},{"Start":"11:38.910 ","End":"11:49.155","Text":"We end up with an integral for Theta from 0 to Pi over 2"},{"Start":"11:49.155 ","End":"11:54.840","Text":"of 2 cosine Theta times"},{"Start":"11:54.840 ","End":"12:02.280","Text":"cosine Theta plus sine Theta d Theta."},{"Start":"12:02.280 ","End":"12:06.240","Text":"We\u0027re getting close, just hold on a bit more."},{"Start":"12:06.240 ","End":"12:10.064","Text":"We can multiply out and get the integral."},{"Start":"12:10.064 ","End":"12:14.220","Text":"The first bit is 2 cosine squared Theta."},{"Start":"12:14.220 ","End":"12:23.530","Text":"The second bit is 2 cosine Theta sine Theta d Theta."},{"Start":"12:23.630 ","End":"12:30.750","Text":"Then I can use some trigonometrical identities 2 cosine Theta,"},{"Start":"12:30.750 ","End":"12:35.520","Text":"sine Theta is the same as sine 2 Theta."},{"Start":"12:35.520 ","End":"12:38.490","Text":"There look up your trigonometrical identities."},{"Start":"12:38.490 ","End":"12:43.110","Text":"2 cosine squared Theta comes out to"},{"Start":"12:43.110 ","End":"12:50.370","Text":"be 1 plus cosine 2 Theta actually is not a formula for that."},{"Start":"12:50.370 ","End":"12:53.910","Text":"There\u0027s a formula that cosine squared Theta is Theta over 2,"},{"Start":"12:53.910 ","End":"12:57.430","Text":"and I just brought the 2 over to the other side."},{"Start":"12:58.460 ","End":"13:02.865","Text":"These 2 together give us,"},{"Start":"13:02.865 ","End":"13:06.090","Text":"I\u0027ll just write it again,"},{"Start":"13:06.090 ","End":"13:10.065","Text":"integral of this plus this."},{"Start":"13:10.065 ","End":"13:14.040","Text":"We have, let\u0027s see,"},{"Start":"13:14.040 ","End":"13:21.780","Text":"this part first was 1 plus cosine 2 Theta and"},{"Start":"13:21.780 ","End":"13:31.410","Text":"then plus sine of 2 Theta and all this d Theta."},{"Start":"13:31.410 ","End":"13:35.920","Text":"A bit lengthy, but we\u0027re getting there."},{"Start":"13:36.800 ","End":"13:40.365","Text":"It\u0027s 0 to Pi over 2."},{"Start":"13:40.365 ","End":"13:44.325","Text":"Integral of 1 is just Theta."},{"Start":"13:44.325 ","End":"13:47.805","Text":"The integral of cosine 2 Theta,"},{"Start":"13:47.805 ","End":"13:51.030","Text":"it\u0027s almost sine of 2 Theta,"},{"Start":"13:51.030 ","End":"13:53.550","Text":"but because of the 2 Theta,"},{"Start":"13:53.550 ","End":"13:55.515","Text":"we need to put 1/2 here."},{"Start":"13:55.515 ","End":"13:58.170","Text":"Similarly, the integral of sine is minus cosine,"},{"Start":"13:58.170 ","End":"14:03.690","Text":"so we have minus 1/2 cosine of 2 Theta."},{"Start":"14:03.690 ","End":"14:12.675","Text":"Now we have to evaluate this between 0 and Pi over 2."},{"Start":"14:12.675 ","End":"14:15.045","Text":"Let me continue over here."},{"Start":"14:15.045 ","End":"14:17.685","Text":"Let\u0027s put in the Pi over 2 first."},{"Start":"14:17.685 ","End":"14:21.010","Text":"We have Pi over 2,"},{"Start":"14:22.190 ","End":"14:27.675","Text":"where am I? Plus 1/2."},{"Start":"14:27.675 ","End":"14:29.580","Text":"Now if Theta\u0027s Pi over 2,"},{"Start":"14:29.580 ","End":"14:30.990","Text":"2 Theta is Pi,"},{"Start":"14:30.990 ","End":"14:34.990","Text":"sine of Pi is 0."},{"Start":"14:35.150 ","End":"14:43.965","Text":"Similarly here, cosine of Pi is minus 1."},{"Start":"14:43.965 ","End":"14:47.295","Text":"This is the upper limit."},{"Start":"14:47.295 ","End":"14:53.745","Text":"Now I need to subtract what I get when I plug in 0. That\u0027s just 0."},{"Start":"14:53.745 ","End":"14:58.110","Text":"Theta 0, 2 Theta is 0 sine of 0 is 0."},{"Start":"14:58.110 ","End":"15:04.995","Text":"We have a 0 here minus 1/2,"},{"Start":"15:04.995 ","End":"15:06.960","Text":"I\u0027ve cosine twice 0,"},{"Start":"15:06.960 ","End":"15:11.110","Text":"cosine 0 is 1."},{"Start":"15:11.540 ","End":"15:14.849","Text":"What have we got?"},{"Start":"15:14.849 ","End":"15:23.295","Text":"We\u0027ve got Pi over 2 minus a 1/2, plus 1/2."},{"Start":"15:23.295 ","End":"15:28.380","Text":"I make it just Pi over 2. Wait a minute."},{"Start":"15:28.380 ","End":"15:30.225","Text":"This is plus 1/2,"},{"Start":"15:30.225 ","End":"15:34.304","Text":"this is plus 1/2, so it\u0027s plus 1."},{"Start":"15:34.304 ","End":"15:38.550","Text":"Finally we have reached the solution."},{"Start":"15:38.550 ","End":"15:40.690","Text":"I\u0027m done."}],"ID":8707},{"Watched":false,"Name":"Exercise 23","Duration":"15m 18s","ChapterTopicVideoID":8484,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8484.jpeg","UploadDate":"2017-02-13T08:31:29.3500000","DurationForVideoObject":"PT15M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.415","Text":"In this exercise, we\u0027re given this double integral in Cartesian form as dxdy,"},{"Start":"00:08.415 ","End":"00:15.990","Text":"which means that it\u0027s a type 2 region, horizontal slices."},{"Start":"00:15.990 ","End":"00:19.320","Text":"My first goal is usually to sketch the region."},{"Start":"00:19.320 ","End":"00:23.580","Text":"I\u0027m just rewriting this in region form."},{"Start":"00:23.580 ","End":"00:29.625","Text":"It\u0027s the same integrant x y squared and dxdy,"},{"Start":"00:29.625 ","End":"00:31.950","Text":"which is also dA."},{"Start":"00:31.950 ","End":"00:35.640","Text":"This just tells me that this is, as I said,"},{"Start":"00:35.640 ","End":"00:39.720","Text":"a type 2, is that y is the outer loop."},{"Start":"00:39.720 ","End":"00:42.755","Text":"In fact, let\u0027s see how this works here,"},{"Start":"00:42.755 ","End":"00:45.260","Text":"how I can describe this region."},{"Start":"00:45.260 ","End":"00:51.000","Text":"The outer loop is y and it goes from 0-2,"},{"Start":"00:52.640 ","End":"00:56.285","Text":"that\u0027s in the y\u0027s positive."},{"Start":"00:56.285 ","End":"00:58.940","Text":"I noticed that x is negative,"},{"Start":"00:58.940 ","End":"01:01.069","Text":"so we\u0027re going to be in the second quadrant."},{"Start":"01:01.069 ","End":"01:10.245","Text":"For each such y, x goes between 0 above and below,"},{"Start":"01:10.245 ","End":"01:11.730","Text":"we have this expression,"},{"Start":"01:11.730 ","End":"01:14.450","Text":"I\u0027ll just copy it, but it\u0027s negative,"},{"Start":"01:14.450 ","End":"01:21.590","Text":"minus the square root of 1 minus y minus 1 squared."},{"Start":"01:21.590 ","End":"01:28.940","Text":"The question is, what is this x equals minus the square root and all that?"},{"Start":"01:28.940 ","End":"01:31.350","Text":"I want to do that at the side."},{"Start":"01:31.780 ","End":"01:40.429","Text":"If x is minus the square root of 1 minus y minus 1 squared and I square both sides,"},{"Start":"01:40.429 ","End":"01:43.940","Text":"then I get x squared is equal,"},{"Start":"01:43.940 ","End":"01:48.470","Text":"the minus disappears, and I have 1 minus y minus 1 squared."},{"Start":"01:48.470 ","End":"01:52.340","Text":"But we have to remember that x is less than or equal to 0 because when you square,"},{"Start":"01:52.340 ","End":"01:54.140","Text":"you lose that information."},{"Start":"01:54.140 ","End":"01:56.435","Text":"Now bring to the other side,"},{"Start":"01:56.435 ","End":"01:58.429","Text":"and I\u0027ve got x squared."},{"Start":"01:58.429 ","End":"02:05.900","Text":"Let me write it as x minus 0 squared plus y minus 1 squared equals 1,"},{"Start":"02:05.900 ","End":"02:07.850","Text":"and let me write 1 as 1 squared."},{"Start":"02:07.850 ","End":"02:11.180","Text":"I\u0027m doing this so it\u0027s in the form of the circle x minus a squared"},{"Start":"02:11.180 ","End":"02:14.595","Text":"plus y minus b squared equals r squared,"},{"Start":"02:14.595 ","End":"02:16.905","Text":"and then we know the center is a, b,"},{"Start":"02:16.905 ","End":"02:20.190","Text":"so that\u0027s the center and that\u0027s the radius 1."},{"Start":"02:20.190 ","End":"02:25.990","Text":"Just write that center and radius."},{"Start":"02:27.050 ","End":"02:29.400","Text":"This is a circle,"},{"Start":"02:29.400 ","End":"02:32.100","Text":"then now we can do the sketch,"},{"Start":"02:32.100 ","End":"02:34.920","Text":"and here it is in the second quadrant."},{"Start":"02:34.920 ","End":"02:40.920","Text":"Like I said, we can see that the outer loop y goes from 0-2."},{"Start":"02:42.470 ","End":"02:47.890","Text":"For each typical y in this 0-2,"},{"Start":"02:47.890 ","End":"02:53.030","Text":"we take horizontal slices like so,"},{"Start":"02:53.030 ","End":"02:58.670","Text":"which enter the region here and leave the region here."},{"Start":"02:58.670 ","End":"03:01.945","Text":"These 2 points, we know what they are now,"},{"Start":"03:01.945 ","End":"03:03.980","Text":"these are the limits for x."},{"Start":"03:03.980 ","End":"03:07.700","Text":"This is where x equals 0, that\u0027s the y-axis,"},{"Start":"03:07.700 ","End":"03:17.250","Text":"and this is where x is equal to minus the square root of 1 minus y minus 1 squared."},{"Start":"03:17.250 ","End":"03:18.680","Text":"So we go from here to here,"},{"Start":"03:18.680 ","End":"03:22.830","Text":"so we sweep it as a type 2 region."},{"Start":"03:22.830 ","End":"03:28.120","Text":"Now, I want to get away from Cartesian and I want to describe the same region."},{"Start":"03:28.120 ","End":"03:34.835","Text":"Let\u0027s call it region D. I want to describe this in polar terms."},{"Start":"03:34.835 ","End":"03:39.695","Text":"Let me just get rid of some stuff I don\u0027t need."},{"Start":"03:39.695 ","End":"03:42.145","Text":"Here we are."},{"Start":"03:42.145 ","End":"03:44.310","Text":"Let\u0027s take care of Theta first."},{"Start":"03:44.310 ","End":"03:48.890","Text":"Theta starts at the positive x-axis and goes counterclockwise."},{"Start":"03:48.890 ","End":"03:51.080","Text":"That\u0027s how it is in mathematics."},{"Start":"03:51.080 ","End":"03:56.365","Text":"What we have, the range of Theta we need it from here up to"},{"Start":"03:56.365 ","End":"04:04.040","Text":"the negative x-axis because this is tangent to this circle."},{"Start":"04:04.040 ","End":"04:10.835","Text":"The angle goes all the way up to the full 180 degrees from 90 degrees."},{"Start":"04:10.835 ","End":"04:15.680","Text":"In radians, this is Theta equals Pi over 2,"},{"Start":"04:15.680 ","End":"04:20.355","Text":"and this is where Theta equals Pi."},{"Start":"04:20.355 ","End":"04:23.055","Text":"The thing is that r is variable."},{"Start":"04:23.055 ","End":"04:24.540","Text":"For each particular Theta,"},{"Start":"04:24.540 ","End":"04:29.970","Text":"let\u0027s say this is a typical Theta in our semicircle,"},{"Start":"04:29.970 ","End":"04:34.895","Text":"in our domain, we know that r goes from here to here."},{"Start":"04:34.895 ","End":"04:36.980","Text":"It\u0027s always from 0,"},{"Start":"04:36.980 ","End":"04:41.810","Text":"but where r goes up to is variable."},{"Start":"04:41.810 ","End":"04:44.285","Text":"I\u0027m going to call it, say, r_1,"},{"Start":"04:44.285 ","End":"04:46.655","Text":"use a different symbol than r,"},{"Start":"04:46.655 ","End":"04:49.175","Text":"and r_1 will depend on Theta."},{"Start":"04:49.175 ","End":"04:52.860","Text":"We know that when Theta is Pi over 2, it\u0027ll be 2."},{"Start":"04:52.860 ","End":"04:54.575","Text":"We saw that this is the point 2."},{"Start":"04:54.575 ","End":"04:57.335","Text":"When we get to 180 degrees or Pi,"},{"Start":"04:57.335 ","End":"04:59.005","Text":"we\u0027ll get to this point."},{"Start":"04:59.005 ","End":"05:01.860","Text":"We slowly get shorter and shorter."},{"Start":"05:01.860 ","End":"05:06.495","Text":"We have to describe this r_1 in terms of Theta."},{"Start":"05:06.495 ","End":"05:09.660","Text":"I\u0027ve got all the formulas ready here."},{"Start":"05:09.660 ","End":"05:14.280","Text":"Let me first write what I I do know about D and polar."},{"Start":"05:14.280 ","End":"05:20.659","Text":"I know that Theta goes from 90 degrees to 180 degrees."},{"Start":"05:20.659 ","End":"05:23.300","Text":"I often say degrees and write radians."},{"Start":"05:23.300 ","End":"05:25.610","Text":"That\u0027s just the way I do it."},{"Start":"05:25.610 ","End":"05:34.589","Text":"Then we know that r goes from 0 up to this r_1,"},{"Start":"05:34.589 ","End":"05:39.800","Text":"and this is what I have to replace by something more explicit."},{"Start":"05:39.800 ","End":"05:42.680","Text":"What I\u0027m going to do is use the equation of"},{"Start":"05:42.680 ","End":"05:46.670","Text":"this semicircle and use the formulas that I already brought"},{"Start":"05:46.670 ","End":"05:54.190","Text":"with me and see if we can write this r_1 as a function of Theta."},{"Start":"05:54.190 ","End":"05:57.410","Text":"I want to do some algebra and then some conversion."},{"Start":"05:57.410 ","End":"06:00.755","Text":"The semicircle, we have the equation for it."},{"Start":"06:00.755 ","End":"06:05.885","Text":"It\u0027s from here, we have that x equals minus"},{"Start":"06:05.885 ","End":"06:11.690","Text":"the square root of 1 minus y minus 1 squared."},{"Start":"06:11.690 ","End":"06:21.450","Text":"We square this and we get that x squared equals 1 minus y minus 1 squared."},{"Start":"06:21.450 ","End":"06:27.600","Text":"I can bring this on the other side and say plus y minus 1 squared equals 1."},{"Start":"06:27.600 ","End":"06:29.535","Text":"In fact, we did this before."},{"Start":"06:29.535 ","End":"06:31.970","Text":"Now, if I open the brackets,"},{"Start":"06:31.970 ","End":"06:34.250","Text":"I\u0027ve got x squared,"},{"Start":"06:34.250 ","End":"06:37.510","Text":"and here I have plus y squared,"},{"Start":"06:37.510 ","End":"06:40.565","Text":"and everything else I\u0027ll bring to the right-hand side,"},{"Start":"06:40.565 ","End":"06:47.285","Text":"so the minus 2y becomes 2y on the right-hand side."},{"Start":"06:47.285 ","End":"06:50.040","Text":"Then I\u0027ll get plus 1,"},{"Start":"06:50.040 ","End":"06:51.270","Text":"which becomes minus 1,"},{"Start":"06:51.270 ","End":"06:52.845","Text":"which cancels with this 1."},{"Start":"06:52.845 ","End":"06:55.370","Text":"Basically, this is what we get."},{"Start":"06:55.370 ","End":"07:00.650","Text":"Now, I already brought with me from the beginning these substitution equations."},{"Start":"07:00.650 ","End":"07:08.560","Text":"In polar, I will get x squared plus y squared is r squared."},{"Start":"07:08.990 ","End":"07:15.585","Text":"Well, it\u0027s r_1 squared if we\u0027re talking about a point on the circumference."},{"Start":"07:15.585 ","End":"07:21.040","Text":"I\u0027m talking about this particular point and this particular Theta."},{"Start":"07:21.040 ","End":"07:28.975","Text":"R_1 squared is 2 and y is r sine Theta."},{"Start":"07:28.975 ","End":"07:32.125","Text":"In our case r_1 sine Theta."},{"Start":"07:32.125 ","End":"07:33.580","Text":"You don\u0027t really have to use r_1."},{"Start":"07:33.580 ","End":"07:36.940","Text":"You can use r as long as you\u0027re not getting confused between the concept"},{"Start":"07:36.940 ","End":"07:40.900","Text":"of the variable r and this particular r. But I like to separate."},{"Start":"07:40.900 ","End":"07:46.630","Text":"Now, it works also when r_1 is 0,"},{"Start":"07:46.630 ","End":"07:55.630","Text":"when I divide both sides by r_1,and get r_1 equals 2 sine Theta."},{"Start":"07:55.630 ","End":"07:59.530","Text":"The only place that r_1 is 0 is when we go all the"},{"Start":"07:59.530 ","End":"08:04.015","Text":"way to 180 degrees and we get back to the origin."},{"Start":"08:04.015 ","End":"08:05.695","Text":"But it works there too,"},{"Start":"08:05.695 ","End":"08:11.395","Text":"because the sine of a 180 degrees is 0 and this is 0, so it\u0027s fine."},{"Start":"08:11.395 ","End":"08:13.600","Text":"The division by r_1,"},{"Start":"08:13.600 ","End":"08:15.700","Text":"we didn\u0027t have to worry about whether it\u0027s 0 or not."},{"Start":"08:15.700 ","End":"08:20.725","Text":"This thing works. This gives me the equation here that I need."},{"Start":"08:20.725 ","End":"08:23.749","Text":"Now, I can plug that in here."},{"Start":"08:23.749 ","End":"08:28.635","Text":"I\u0027ll just write what it is that is set at this."},{"Start":"08:28.635 ","End":"08:33.915","Text":"I could write 2 sine Theta."},{"Start":"08:33.915 ","End":"08:37.760","Text":"Just doing a quick mental check to see this is reasonable."},{"Start":"08:37.760 ","End":"08:39.925","Text":"If you know the function sine Theta,"},{"Start":"08:39.925 ","End":"08:42.670","Text":"then from 90 to 180 degrees,"},{"Start":"08:42.670 ","End":"08:45.055","Text":"it goes down from 1 to 0."},{"Start":"08:45.055 ","End":"08:47.260","Text":"This thing goes from 2 down to 0."},{"Start":"08:47.260 ","End":"08:49.690","Text":"That makes sense. At 90 degrees,"},{"Start":"08:49.690 ","End":"08:51.520","Text":"the distance is 2."},{"Start":"08:51.520 ","End":"08:53.590","Text":"Then the distance from the origin gets shorter and"},{"Start":"08:53.590 ","End":"08:56.470","Text":"shorter and shorter until it goes down to 0."},{"Start":"08:56.470 ","End":"08:58.075","Text":"That\u0027s makes sense."},{"Start":"08:58.075 ","End":"09:01.809","Text":"Now, let\u0027s write the integral in polar."},{"Start":"09:01.809 ","End":"09:07.585","Text":"We have all the conversions that we need and we have the domain described."},{"Start":"09:07.585 ","End":"09:11.425","Text":"In polar, we get the integral,"},{"Start":"09:11.425 ","End":"09:16.345","Text":"first of all, Theta from Pi over 2 to Pi."},{"Start":"09:16.345 ","End":"09:21.955","Text":"Then R from, where is it"},{"Start":"09:21.955 ","End":"09:28.210","Text":"now from 0 to 2 sine Theta."},{"Start":"09:28.210 ","End":"09:32.005","Text":"Next, I have to convert all these."},{"Start":"09:32.005 ","End":"09:39.610","Text":"Xy squared is, x is r cosine Theta."},{"Start":"09:40.650 ","End":"09:48.295","Text":"Then I have y, which is r sine Theta squared."},{"Start":"09:48.295 ","End":"09:55.750","Text":"DA is rdrd Theta."},{"Start":"09:55.750 ","End":"09:58.465","Text":"I want to rewrite this a bit."},{"Start":"09:58.465 ","End":"10:05.845","Text":"This is equal to the integral from Pi over 2 to Pi."},{"Start":"10:05.845 ","End":"10:12.355","Text":"Now, here I have cosine Theta,"},{"Start":"10:12.355 ","End":"10:15.025","Text":"and here I have sine squared Theta."},{"Start":"10:15.025 ","End":"10:17.470","Text":"But all of these don\u0027t depend on r,"},{"Start":"10:17.470 ","End":"10:18.880","Text":"so I can put them here."},{"Start":"10:18.880 ","End":"10:20.710","Text":"I have cosine Theta,"},{"Start":"10:20.710 ","End":"10:25.720","Text":"that\u0027s this, sine squared Theta."},{"Start":"10:25.720 ","End":"10:29.500","Text":"Now, I can put the stuff with r. What do I have for r?"},{"Start":"10:29.500 ","End":"10:30.730","Text":"I have r here,"},{"Start":"10:30.730 ","End":"10:32.860","Text":"I have r squared from here,"},{"Start":"10:32.860 ","End":"10:34.720","Text":"and I have another r here."},{"Start":"10:34.720 ","End":"10:41.185","Text":"I have r to the 4th dr. Then finally d Theta,"},{"Start":"10:41.185 ","End":"10:47.870","Text":"and the limits of r goes from 0 to 2 sine Theta."},{"Start":"10:48.390 ","End":"10:51.100","Text":"Let\u0027s start with the inner integral,"},{"Start":"10:51.100 ","End":"10:54.890","Text":"that\u0027s this 1, the dr integral."},{"Start":"10:55.290 ","End":"10:58.630","Text":"I\u0027d like to do this as a side exercise."},{"Start":"10:58.630 ","End":"11:01.030","Text":"Let me call it asterisk."},{"Start":"11:01.030 ","End":"11:10.465","Text":"What I have to compute is the integral from 0 to 2 sine Theta r to the 4th dr."},{"Start":"11:10.465 ","End":"11:15.175","Text":"This is equal to the integral of this is 1/5,"},{"Start":"11:15.175 ","End":"11:16.660","Text":"r to the 5th."},{"Start":"11:16.660 ","End":"11:20.155","Text":"I have 1/5 and I can take outside,"},{"Start":"11:20.155 ","End":"11:21.850","Text":"and I have r to the 5th,"},{"Start":"11:21.850 ","End":"11:28.975","Text":"which I have to evaluate between 0 and 2 sine Theta."},{"Start":"11:28.975 ","End":"11:32.080","Text":"This is equal to when r is 0,"},{"Start":"11:32.080 ","End":"11:33.370","Text":"I don\u0027t get anything."},{"Start":"11:33.370 ","End":"11:38.964","Text":"I just have to substitute r equals 2 sine Theta."},{"Start":"11:38.964 ","End":"11:41.410","Text":"Now, if I raise 2 sine Theta to the 5th,"},{"Start":"11:41.410 ","End":"11:46.900","Text":"I get 2 to the 5th sine to the 5th Theta, 2 to the 5th is 32."},{"Start":"11:46.900 ","End":"11:53.905","Text":"It\u0027s 32/5 sine of Theta to the 5th,"},{"Start":"11:53.905 ","End":"11:55.675","Text":"write the 5 here."},{"Start":"11:55.675 ","End":"12:02.180","Text":"Now, we\u0027re ready to plug that back into here."},{"Start":"12:02.430 ","End":"12:05.845","Text":"The constant 32/ 5,"},{"Start":"12:05.845 ","End":"12:09.065","Text":"I\u0027ll take in front of the integral sign."},{"Start":"12:09.065 ","End":"12:14.520","Text":"We get the integral from Pi over 2 to Pi."},{"Start":"12:14.520 ","End":"12:19.665","Text":"I have to throw in sine 5th Theta into here."},{"Start":"12:19.665 ","End":"12:23.565","Text":"We already have sine squared,"},{"Start":"12:23.565 ","End":"12:27.850","Text":"so it looks like we\u0027re going to get sine to the 7th."},{"Start":"12:28.110 ","End":"12:30.310","Text":"Write the sines first,"},{"Start":"12:30.310 ","End":"12:34.450","Text":"sine squared sine 5th is sine to the 7th Theta."},{"Start":"12:34.450 ","End":"12:40.480","Text":"I still have a cosine Theta and it\u0027s d Theta."},{"Start":"12:40.480 ","End":"12:43.375","Text":"How do we do this integral?"},{"Start":"12:43.375 ","End":"12:49.225","Text":"I suggest doing this with a substitution of t equals sine Theta."},{"Start":"12:49.225 ","End":"12:53.260","Text":"I know this will work because I already have the derivative of sine Theta,"},{"Start":"12:53.260 ","End":"12:55.465","Text":"which is cosine Theta alongside."},{"Start":"12:55.465 ","End":"12:57.970","Text":"That will guarantee success."},{"Start":"12:57.970 ","End":"13:01.180","Text":"Let us see then if I set,"},{"Start":"13:01.180 ","End":"13:07.300","Text":"I\u0027ll do this at the side t equals sine Theta."},{"Start":"13:07.300 ","End":"13:14.590","Text":"Then I have that dt is cosine Theta d Theta."},{"Start":"13:14.590 ","End":"13:16.195","Text":"But that\u0027s not all."},{"Start":"13:16.195 ","End":"13:18.100","Text":"If I don\u0027t want to come back to the Theta,"},{"Start":"13:18.100 ","End":"13:24.530","Text":"I want to stay in the land of t. I have to substitute the limits also."},{"Start":"13:26.640 ","End":"13:30.490","Text":"Do the lower 1 first Pi over 2,"},{"Start":"13:30.490 ","End":"13:36.955","Text":"then t equals sine of Pi over 2 is 1."},{"Start":"13:36.955 ","End":"13:40.270","Text":"When Theta equals Pi,"},{"Start":"13:40.270 ","End":"13:46.435","Text":"then I get that t equals sine of Pi, which is 0."},{"Start":"13:46.435 ","End":"13:49.420","Text":"After all this substituting,"},{"Start":"13:49.420 ","End":"13:55.990","Text":"what we get is 32/5 still."},{"Start":"13:55.990 ","End":"13:59.365","Text":"The integral now this time it\u0027s with respect to t,"},{"Start":"13:59.365 ","End":"14:03.430","Text":"so it\u0027s from Pi over 2, which is 1."},{"Start":"14:03.430 ","End":"14:11.050","Text":"The Pi gives 0 there backwards the bigger ones on the bottom, but that\u0027s okay."},{"Start":"14:11.050 ","End":"14:16.150","Text":"Then I have t to the 7th,"},{"Start":"14:16.150 ","End":"14:20.000","Text":"and cosine d to the d Theta is just dt."},{"Start":"14:21.510 ","End":"14:25.585","Text":"Another integral of t to the 7th,"},{"Start":"14:25.585 ","End":"14:29.320","Text":"that\u0027s 1/8, t to the 8."},{"Start":"14:29.320 ","End":"14:34.285","Text":"The 1/8 will get swallowed up in the 32 and leave 4,"},{"Start":"14:34.285 ","End":"14:39.340","Text":"so I\u0027ve got 4/5 of t to"},{"Start":"14:39.340 ","End":"14:45.550","Text":"the 8th between 1 and 0."},{"Start":"14:45.550 ","End":"14:48.100","Text":"Did you follow that thing with the 8?"},{"Start":"14:48.100 ","End":"14:49.395","Text":"When I divide it by 8,"},{"Start":"14:49.395 ","End":"14:53.460","Text":"I just canceled partially with the 32."},{"Start":"14:53.530 ","End":"15:01.390","Text":"If we plug in here 0, we get 0."},{"Start":"15:01.390 ","End":"15:03.175","Text":"If we plug in 1,"},{"Start":"15:03.175 ","End":"15:06.640","Text":"we get 1 to the 8th, which is 1."},{"Start":"15:06.640 ","End":"15:14.345","Text":"Altogether, our answer is equal to minus 4/5."},{"Start":"15:14.345 ","End":"15:18.690","Text":"I\u0027ll just highlight it and we are done."}],"ID":8708},{"Watched":false,"Name":"Exercise 24","Duration":"10m 36s","ChapterTopicVideoID":8485,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8485.jpeg","UploadDate":"2017-02-13T08:32:54.4930000","DurationForVideoObject":"PT10M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this exercise, we have a double integral."},{"Start":"00:03.180 ","End":"00:06.570","Text":"It\u0027s written in x, y coordinates and we want to"},{"Start":"00:06.570 ","End":"00:10.065","Text":"change to our Theta from Cartesian to polar."},{"Start":"00:10.065 ","End":"00:12.290","Text":"This is a dxdy, in other words,"},{"Start":"00:12.290 ","End":"00:15.870","Text":"type 2 region with horizontal slices."},{"Start":"00:16.970 ","End":"00:25.425","Text":"Let me first write this as a double integral over a region D as the same thing,"},{"Start":"00:25.425 ","End":"00:30.465","Text":"natural log of x squared plus y squared plus 1,"},{"Start":"00:30.465 ","End":"00:34.785","Text":"and it\u0027s dxdy, which is also dA."},{"Start":"00:34.785 ","End":"00:37.800","Text":"I want to describe this region,"},{"Start":"00:37.800 ","End":"00:42.790","Text":"what we can see is that the region D,"},{"Start":"00:44.090 ","End":"00:46.815","Text":"the outer loop is dy,"},{"Start":"00:46.815 ","End":"00:52.919","Text":"and y goes from minus 1-1, inclusive."},{"Start":"00:52.919 ","End":"00:55.290","Text":"Now, as y goes from minus 1-1,"},{"Start":"00:55.290 ","End":"00:58.965","Text":"x goes between 2 things,"},{"Start":"00:58.965 ","End":"01:00.390","Text":"and they\u0027re dependent on y."},{"Start":"01:00.390 ","End":"01:04.030","Text":"It goes from minus the square root of 1 minus y"},{"Start":"01:04.030 ","End":"01:09.605","Text":"squared to plus square root of 1 minus y squared."},{"Start":"01:09.605 ","End":"01:12.715","Text":"I think we\u0027ve seen this many times before."},{"Start":"01:12.715 ","End":"01:15.730","Text":"It\u0027s actually just part of the,"},{"Start":"01:15.730 ","End":"01:23.035","Text":"or maybe all of the circle x squared plus y squared equals 1."},{"Start":"01:23.035 ","End":"01:24.970","Text":"The insides of it,"},{"Start":"01:24.970 ","End":"01:26.965","Text":"so really less than or equal to 1."},{"Start":"01:26.965 ","End":"01:28.315","Text":"We\u0027ve seen this before,"},{"Start":"01:28.315 ","End":"01:34.310","Text":"I\u0027ll bring in the sketch and it is the full circle."},{"Start":"01:34.310 ","End":"01:36.050","Text":"This is what D is,"},{"Start":"01:36.050 ","End":"01:43.925","Text":"because you see y goes from minus 1-1 and for each particular y,"},{"Start":"01:43.925 ","End":"01:49.669","Text":"if I take the horizontal slice through the region,"},{"Start":"01:49.669 ","End":"01:57.640","Text":"it enters here and leaves here and we know these two points."},{"Start":"01:57.640 ","End":"02:01.450","Text":"These equations on the circle, this is plus,"},{"Start":"02:01.450 ","End":"02:08.210","Text":"this is where x is the square root of 1 minus y squared."},{"Start":"02:08.210 ","End":"02:11.840","Text":"Here x is minus the square root of 1 minus y"},{"Start":"02:11.840 ","End":"02:17.510","Text":"squared and next I wanted to describe this same region in polar terms,"},{"Start":"02:17.510 ","End":"02:20.135","Text":"not x, y, but r Theta."},{"Start":"02:20.135 ","End":"02:22.940","Text":"We\u0027ve seen the circle many times before,"},{"Start":"02:22.940 ","End":"02:26.780","Text":"the unit circle we know that Theta goes from,"},{"Start":"02:26.780 ","End":"02:31.280","Text":"figure all the way around from 0-2 Pi here."},{"Start":"02:31.280 ","End":"02:37.020","Text":"It\u0027s 0 and it ends up at 2 Pi and asked for r,"},{"Start":"02:37.020 ","End":"02:38.595","Text":"for any given Theta,"},{"Start":"02:38.595 ","End":"02:41.910","Text":"r always goes from 0-1 it\u0027s the same."},{"Start":"02:41.910 ","End":"02:50.265","Text":"Our domain in polar terms for Theta C and for r,"},{"Start":"02:50.265 ","End":"02:55.050","Text":"here we\u0027re from 0-2 Pi here from 0-1."},{"Start":"02:55.050 ","End":"02:59.525","Text":"We can rewrite this integral in polar form."},{"Start":"02:59.525 ","End":"03:03.840","Text":"Theta goes from 0-2 Pi,"},{"Start":"03:03.840 ","End":"03:07.665","Text":"r goes from 0-1,"},{"Start":"03:07.665 ","End":"03:10.105","Text":"we have all the conversion formulas."},{"Start":"03:10.105 ","End":"03:16.300","Text":"Let\u0027s see, natural log of x squared plus y squared is r squared,"},{"Start":"03:16.300 ","End":"03:18.895","Text":"so it\u0027s r squared plus 1,"},{"Start":"03:18.895 ","End":"03:22.910","Text":"dA is rdd Theta."},{"Start":"03:25.530 ","End":"03:28.930","Text":"We always do the inner integral first,"},{"Start":"03:28.930 ","End":"03:32.990","Text":"the one that\u0027s in this case, dr."},{"Start":"03:33.290 ","End":"03:36.860","Text":"I like doing it at the side I call it asterisk"},{"Start":"03:36.860 ","End":"03:40.265","Text":"and then I go over to the side where I have more room."},{"Start":"03:40.265 ","End":"03:45.350","Text":"We have to compute the integral from 0-1 of"},{"Start":"03:45.350 ","End":"03:54.600","Text":"natural log of r squared plus 1, rdr."},{"Start":"03:54.620 ","End":"03:58.710","Text":"Fixed. Going to do it with the substitution"},{"Start":"03:58.710 ","End":"04:03.380","Text":"because we have the derivative of r squared plus 1 outside,"},{"Start":"04:03.380 ","End":"04:06.910","Text":"we don\u0027t have 2r, we have r, but it\u0027s close enough."},{"Start":"04:06.910 ","End":"04:13.960","Text":"The substitution I propose is to let t equals r squared plus 1"},{"Start":"04:13.960 ","End":"04:21.990","Text":"and then dt is equal to the derivative of this dr, 2rdr."},{"Start":"04:25.090 ","End":"04:28.670","Text":"You can either go back to r or you can stay in"},{"Start":"04:28.670 ","End":"04:31.340","Text":"the land of t if you also substitute the limits."},{"Start":"04:31.340 ","End":"04:33.035","Text":"I\u0027m going to do that,"},{"Start":"04:33.035 ","End":"04:37.790","Text":"that when r equals 0,"},{"Start":"04:37.790 ","End":"04:44.345","Text":"we get that t equals 0 squared plus 1 is 1."},{"Start":"04:44.345 ","End":"04:47.405","Text":"When r equals 1 the upper limit,"},{"Start":"04:47.405 ","End":"04:53.480","Text":"then t is equal to 1 plus 1 squared is 2 and"},{"Start":"04:53.480 ","End":"04:59.870","Text":"so we can substitute the whole thing and then we get the integral."},{"Start":"04:59.870 ","End":"05:03.730","Text":"But with respect to t it\u0027s from 1-2."},{"Start":"05:03.730 ","End":"05:12.830","Text":"Natural log, r squared plus 1 is what we let to be t and rdr,"},{"Start":"05:12.830 ","End":"05:16.910","Text":"well, it\u0027s not dt, it\u0027s 1/2dt,"},{"Start":"05:16.910 ","End":"05:21.630","Text":"so I\u0027ll put dt and I\u0027ll put the 1/2 up front."},{"Start":"05:22.190 ","End":"05:30.425","Text":"Continuing, now this integral we can either use a formula sheet or do it by parts."},{"Start":"05:30.425 ","End":"05:33.320","Text":"Let me bring it with the formula sheet and maybe at"},{"Start":"05:33.320 ","End":"05:36.620","Text":"the end I\u0027ll show you how to do it with integration by parts."},{"Start":"05:36.620 ","End":"05:45.310","Text":"The formula sheet says that this integral is t natural log of t minus t,"},{"Start":"05:45.310 ","End":"05:49.940","Text":"certainly you could differentiate this and see that you get back to this."},{"Start":"05:49.940 ","End":"05:58.540","Text":"Anyway, between 1 and 2 and this is equal to 1.5,"},{"Start":"05:58.540 ","End":"06:02.860","Text":"maybe I\u0027ll just remind you that IOU an"},{"Start":"06:02.860 ","End":"06:07.100","Text":"explanation of how to get from here to here if you want to do it by parts."},{"Start":"06:07.100 ","End":"06:10.950","Text":"As I was saying 1.5."},{"Start":"06:10.950 ","End":"06:16.694","Text":"Now, let\u0027s plug in 2 so we get 2 natural log"},{"Start":"06:16.694 ","End":"06:29.670","Text":"2 minus 2 and then if t is 1,"},{"Start":"06:29.670 ","End":"06:32.190","Text":"natural log of 1 is 0."},{"Start":"06:32.190 ","End":"06:37.934","Text":"We just get minus 1,"},{"Start":"06:37.934 ","End":"06:42.840","Text":"but we\u0027re subtracting so it\u0027s minus minus 1."},{"Start":"06:42.840 ","End":"06:46.670","Text":"Well, I\u0027ll write it as minus minus 1 may be able to show"},{"Start":"06:46.670 ","End":"06:50.680","Text":"you that this is the part with the 2 part with the 1."},{"Start":"06:50.680 ","End":"06:53.445","Text":"Let\u0027s see if I can simplify this."},{"Start":"06:53.445 ","End":"06:59.270","Text":"This is 2 natural log of 2 minus 2 plus"},{"Start":"06:59.270 ","End":"07:07.475","Text":"1 is minus 1 so I can get it to simplify to natural log of 2."},{"Start":"07:07.475 ","End":"07:12.040","Text":"Was it minus 1, so it\u0027s minus 1/2."},{"Start":"07:12.040 ","End":"07:17.210","Text":"Now if you remember, this was the asterisks that we took from here."},{"Start":"07:17.210 ","End":"07:22.820","Text":"Let me just to copy it again just so it\u0027ll be handy."},{"Start":"07:22.820 ","End":"07:28.250","Text":"This turned out to be natural log of 2 minus 1/2."},{"Start":"07:28.250 ","End":"07:30.590","Text":"Now we do the integral d Theta."},{"Start":"07:30.590 ","End":"07:33.640","Text":"But this is a constant so I can bring it out front."},{"Start":"07:33.640 ","End":"07:41.865","Text":"I get natural log of 2 minus 1/2 integral from 0-2 Pi,"},{"Start":"07:41.865 ","End":"07:44.370","Text":"why I\u0027m I in this color?"},{"Start":"07:44.370 ","End":"07:48.370","Text":"Here, we have just d Theta or 1d Theta."},{"Start":"07:48.370 ","End":"07:50.060","Text":"As we know, when we have the integral of 1,"},{"Start":"07:50.060 ","End":"07:54.690","Text":"it\u0027s just the upper limit minus the lower limit so it\u0027s 2 Pi."},{"Start":"07:54.890 ","End":"08:06.480","Text":"What we get is 2 Pi times this natural log of 2 minus 1/2."},{"Start":"08:06.560 ","End":"08:08.625","Text":"I could leave it like that."},{"Start":"08:08.625 ","End":"08:13.520","Text":"I\u0027d like to multiply the 2 in and put the Pi at the end so I"},{"Start":"08:13.520 ","End":"08:20.470","Text":"have 2 natural log 2 minus 1 times Pi."},{"Start":"08:20.470 ","End":"08:23.920","Text":"I think that we are done."},{"Start":"08:24.860 ","End":"08:26.910","Text":"Oops, no, we\u0027re not,"},{"Start":"08:26.910 ","End":"08:29.600","Text":"if you want to see the integration by parts,"},{"Start":"08:29.600 ","End":"08:34.010","Text":"then you can stay otherwise you can finish now."},{"Start":"08:34.010 ","End":"08:39.010","Text":"Let\u0027s see the integral of natural log of t,"},{"Start":"08:39.010 ","End":"08:40.940","Text":"where shall I do it?"},{"Start":"08:40.940 ","End":"08:43.235","Text":"Maybe over here, squeeze it in."},{"Start":"08:43.235 ","End":"08:52.260","Text":"The integral of natural log of t, dt."},{"Start":"08:52.400 ","End":"08:57.920","Text":"I\u0027ll do the indefinite integral just want to get to this answer."},{"Start":"08:57.920 ","End":"09:04.705","Text":"We want to do it, and I\u0027ll call this part u and this part dv."},{"Start":"09:04.705 ","End":"09:09.585","Text":"So this is u and this is dv,"},{"Start":"09:09.585 ","End":"09:18.770","Text":"I\u0027m I going to use the formula that the integral of udv is uv minus the integral of vdu."},{"Start":"09:19.700 ","End":"09:28.630","Text":"This is equal to u is natural log of t. Now if dt is dv,"},{"Start":"09:28.630 ","End":"09:37.205","Text":"then t equals v so we have V is,"},{"Start":"09:37.205 ","End":"09:42.815","Text":"put the brackets here is t and then minus the integral,"},{"Start":"09:42.815 ","End":"09:47.310","Text":"v again is t,"},{"Start":"09:51.520 ","End":"09:55.535","Text":"and du is 1 over t,"},{"Start":"09:55.535 ","End":"10:01.650","Text":"but dt, 1 over t because derivative of natural log is 1 over tdt."},{"Start":"10:02.380 ","End":"10:05.330","Text":"It\u0027s a bit cramped here."},{"Start":"10:05.330 ","End":"10:07.250","Text":"I\u0027ll continue over here."},{"Start":"10:07.250 ","End":"10:09.695","Text":"What I have is this part,"},{"Start":"10:09.695 ","End":"10:15.710","Text":"t natural log of t minus the integral,"},{"Start":"10:15.710 ","End":"10:19.080","Text":"t times 1 over t is just 1dt,"},{"Start":"10:20.800 ","End":"10:24.810","Text":"and the integral of 1dt is just t,"},{"Start":"10:24.810 ","End":"10:30.470","Text":"so we get t natural log of t minus t and that\u0027s what I said here."},{"Start":"10:30.470 ","End":"10:37.140","Text":"So we\u0027re okay, so that repays that IOU debt. Now we\u0027re done."}],"ID":8709},{"Watched":false,"Name":"Exercise 25","Duration":"6m 32s","ChapterTopicVideoID":8486,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8486.jpeg","UploadDate":"2017-02-13T08:33:45.2630000","DurationForVideoObject":"PT6M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.300","Text":"In this exercise, you have to compute the double integral by converting to polar,"},{"Start":"00:06.300 ","End":"00:09.180","Text":"and I decided to speed up things a bit."},{"Start":"00:09.180 ","End":"00:12.420","Text":"Already, have all the formulas ready and you\u0027ve seen enough of these to"},{"Start":"00:12.420 ","End":"00:16.140","Text":"know that this is a unit circle, because it\u0027s dy/dx."},{"Start":"00:16.140 ","End":"00:17.790","Text":"It\u0027s vertical slices."},{"Start":"00:17.790 ","End":"00:19.635","Text":"I\u0027ll just show you what I mean."},{"Start":"00:19.635 ","End":"00:21.750","Text":"The outer loop is dx,"},{"Start":"00:21.750 ","End":"00:25.664","Text":"which goes from minus 1 to 1,"},{"Start":"00:25.664 ","End":"00:30.375","Text":"and the inner loop for y."},{"Start":"00:30.375 ","End":"00:34.780","Text":"Let\u0027s say this is a specific y."},{"Start":"00:34.780 ","End":"00:37.430","Text":"Then we get a vertical slice."},{"Start":"00:37.430 ","End":"00:40.175","Text":"We enter the region here,"},{"Start":"00:40.175 ","End":"00:43.980","Text":"and we exit the region here."},{"Start":"00:43.980 ","End":"00:46.370","Text":"We already seen this many times before."},{"Start":"00:46.370 ","End":"00:49.805","Text":"This is the upper semicircle, the lower semicircle."},{"Start":"00:49.805 ","End":"00:52.760","Text":"All this is part of the equation,"},{"Start":"00:52.760 ","End":"00:55.295","Text":"x squared plus y squared equals 1."},{"Start":"00:55.295 ","End":"00:59.660","Text":"I\u0027m not going to repeat it at all."},{"Start":"00:59.660 ","End":"01:06.455","Text":"All we have to do now is get this region described in polar terms."},{"Start":"01:06.455 ","End":"01:08.735","Text":"I forgot to give it a name."},{"Start":"01:08.735 ","End":"01:10.535","Text":"D is our usual name."},{"Start":"01:10.535 ","End":"01:17.585","Text":"So this thing could be written as double integral over d of the same thing."},{"Start":"01:17.585 ","End":"01:21.200","Text":"This dy/dx, remember, is also"},{"Start":"01:21.200 ","End":"01:28.559","Text":"da type 1 region because x goes on the outside,"},{"Start":"01:28.559 ","End":"01:31.185","Text":"and for each xy goes vertical."},{"Start":"01:31.185 ","End":"01:33.050","Text":"We don\u0027t get that in polar."},{"Start":"01:33.050 ","End":"01:34.850","Text":"It\u0027s always dr d Theta."},{"Start":"01:34.850 ","End":"01:38.425","Text":"It\u0027s never d Theta, dr. If haven\u0027t seen it."},{"Start":"01:38.425 ","End":"01:44.280","Text":"Okay, how do we get this to polar?"},{"Start":"01:44.280 ","End":"01:47.690","Text":"I\u0027d like to just formally write down what the region is,"},{"Start":"01:47.690 ","End":"01:48.755","Text":"to describe it in."},{"Start":"01:48.755 ","End":"01:50.450","Text":"First of all, in Cartesian,"},{"Start":"01:50.450 ","End":"01:53.060","Text":"as a type one region,"},{"Start":"01:53.060 ","End":"01:56.580","Text":"we have the outer loop,"},{"Start":"01:56.580 ","End":"01:57.900","Text":"x, the inner loop,"},{"Start":"01:57.900 ","End":"02:02.840","Text":"y. X on the outside goes from minus 1 to 1,"},{"Start":"02:02.840 ","End":"02:07.530","Text":"and for each such xy goes from square root of 1 minus x squared,"},{"Start":"02:07.530 ","End":"02:11.475","Text":"to minus the square root of 1 minus x squared."},{"Start":"02:11.475 ","End":"02:13.955","Text":"Now we\u0027re going to have to write something else."},{"Start":"02:13.955 ","End":"02:16.295","Text":"We\u0027re going to describe d in polar terms."},{"Start":"02:16.295 ","End":"02:17.975","Text":"Something with a Theta here,"},{"Start":"02:17.975 ","End":"02:20.060","Text":"and something with an r here."},{"Start":"02:20.060 ","End":"02:22.070","Text":"I\u0027m just giving you the general scheme,"},{"Start":"02:22.070 ","End":"02:24.050","Text":"and then we\u0027re going to have an integral"},{"Start":"02:24.050 ","End":"02:27.785","Text":"where here Theta goes from something to something in here,"},{"Start":"02:27.785 ","End":"02:29.600","Text":"r goes from something to something."},{"Start":"02:29.600 ","End":"02:33.630","Text":"Let me just describe the region in polar term."},{"Start":"02:33.630 ","End":"02:42.655","Text":"Need I even say it Theta goes all the way around from 0 up to Pi,"},{"Start":"02:42.655 ","End":"02:44.600","Text":"and for each Theta,"},{"Start":"02:44.600 ","End":"02:47.100","Text":"wherever Theta is r,"},{"Start":"02:47.100 ","End":"02:49.260","Text":"goes from 0 to 1 always."},{"Start":"02:49.260 ","End":"02:51.350","Text":"So to be complete,"},{"Start":"02:51.350 ","End":"02:56.660","Text":"describe the region that c Thetas between 0 and 2 Pi,"},{"Start":"02:56.660 ","End":"03:00.010","Text":"and between 0 and 1."},{"Start":"03:00.010 ","End":"03:02.040","Text":"Copy that in here,"},{"Start":"03:02.040 ","End":"03:06.415","Text":"0 to 2 Pi, 0 to 1."},{"Start":"03:06.415 ","End":"03:09.440","Text":"Now use these formulas to convert this."},{"Start":"03:09.440 ","End":"03:12.405","Text":"I get 2 over,"},{"Start":"03:12.405 ","End":"03:14.990","Text":"let\u0027s see, x squared plus y squared is r squared."},{"Start":"03:14.990 ","End":"03:19.335","Text":"1 plus r squared here,"},{"Start":"03:19.335 ","End":"03:22.625","Text":"da is r dr d theta."},{"Start":"03:22.625 ","End":"03:24.650","Text":"Can I, with your permission,"},{"Start":"03:24.650 ","End":"03:29.540","Text":"put the r up here and then dr d theta here."},{"Start":"03:29.540 ","End":"03:31.949","Text":"I don\u0027t think anyone will object."},{"Start":"03:31.990 ","End":"03:37.445","Text":"As always, the inner integral is the one we do first."},{"Start":"03:37.445 ","End":"03:41.780","Text":"That\u0027s the dr and you probably know me by"},{"Start":"03:41.780 ","End":"03:45.649","Text":"now I like to do this inner integral as a side exercise,"},{"Start":"03:45.649 ","End":"03:47.255","Text":"and then return here."},{"Start":"03:47.255 ","End":"03:53.045","Text":"What I want is the integral from 0 to 1 of"},{"Start":"03:53.045 ","End":"03:59.905","Text":"2 over 1 plus r squared squared dr."},{"Start":"03:59.905 ","End":"04:02.710","Text":"I\u0027ll do this with a substitution,"},{"Start":"04:02.710 ","End":"04:10.385","Text":"because I see that I have exactly the derivative of 1 plus r squared here as 2r."},{"Start":"04:10.385 ","End":"04:14.390","Text":"I\u0027m going to substitute and see where do I have room."},{"Start":"04:14.390 ","End":"04:16.560","Text":"I\u0027ll write it over here."},{"Start":"04:17.140 ","End":"04:22.160","Text":"I\u0027m substituting t equals"},{"Start":"04:22.160 ","End":"04:29.610","Text":"1 plus r squared and then dt is 2r dr."},{"Start":"04:29.900 ","End":"04:38.280","Text":"I don\u0027t want us to come back to r. I want to stay with t. When r equals 0,"},{"Start":"04:38.280 ","End":"04:39.900","Text":"then t equals 1 plus r squared."},{"Start":"04:39.900 ","End":"04:42.300","Text":"Then t equals 1\u0027s lower limit."},{"Start":"04:42.300 ","End":"04:48.640","Text":"Upper limit r equals 1 gives me t equals 1 plus 1 squared is 2."},{"Start":"04:48.640 ","End":"04:50.620","Text":"I got everything I need."},{"Start":"04:50.620 ","End":"04:56.530","Text":"Now back here, we get the integral."},{"Start":"04:56.530 ","End":"04:58.360","Text":"I am in the land of t,"},{"Start":"04:58.360 ","End":"05:01.210","Text":"so it\u0027s from 1 to 2."},{"Start":"05:01.210 ","End":"05:05.900","Text":"Now, 2r dr is just dt,"},{"Start":"05:06.950 ","End":"05:13.510","Text":"and 1 plus r squared is t. So it\u0027s d t over t squared."},{"Start":"05:13.940 ","End":"05:17.260","Text":"This is equal to, well,"},{"Start":"05:17.260 ","End":"05:25.970","Text":"the integral of 1 over t squared is minus 1 over t. This is minus 1 over t,"},{"Start":"05:25.970 ","End":"05:30.885","Text":"from 1 to 2."},{"Start":"05:30.885 ","End":"05:34.540","Text":"This is equal to, let\u0027s see,"},{"Start":"05:34.540 ","End":"05:36.910","Text":"if I plug in 2,"},{"Start":"05:36.910 ","End":"05:42.285","Text":"I get minus 1 1/2."},{"Start":"05:42.285 ","End":"05:45.245","Text":"If I plug in 1,"},{"Start":"05:45.245 ","End":"05:50.525","Text":"I get minus 1 over 1 is minus 1."},{"Start":"05:50.525 ","End":"05:55.095","Text":"What we get is minus 1/2 plus 1, one minus 1/2."},{"Start":"05:55.095 ","End":"05:57.435","Text":"This is just 1/2."},{"Start":"05:57.435 ","End":"05:59.990","Text":"So we\u0027ve computed this asterisk, I\u0027ll just make a note of it."},{"Start":"05:59.990 ","End":"06:02.635","Text":"This came out to be 1 1/2."},{"Start":"06:02.635 ","End":"06:06.105","Text":"Now we\u0027re back in the main stream things."},{"Start":"06:06.105 ","End":"06:09.965","Text":"1/2 is a constant, I can bring it right in front of the integral,"},{"Start":"06:09.965 ","End":"06:12.785","Text":"which is from 0 to 2 Pi."},{"Start":"06:12.785 ","End":"06:14.345","Text":"There\u0027s nothing left."},{"Start":"06:14.345 ","End":"06:17.600","Text":"Just one d Theta."},{"Start":"06:17.600 ","End":"06:20.915","Text":"The integral of 1 is the upper limit minus the lower limit,"},{"Start":"06:20.915 ","End":"06:23.980","Text":"2 Pi minus 0 is 2 Pi."},{"Start":"06:23.980 ","End":"06:27.974","Text":"So we get 1 1/2 times 2 Pi,"},{"Start":"06:27.974 ","End":"06:32.890","Text":"which is Pi, and that\u0027s the answer."}],"ID":8710},{"Watched":false,"Name":"Exercise 26","Duration":"13m 33s","ChapterTopicVideoID":8487,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8487.jpeg","UploadDate":"2017-02-13T08:35:25.3830000","DurationForVideoObject":"PT13M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.915","Text":"In this exercise, we have to compute the volume of the solid bounded by these 2 shapes."},{"Start":"00:06.915 ","End":"00:15.520","Text":"This is a sphere of radius 3 and this is a cylinder centered on the z-axis."},{"Start":"00:16.180 ","End":"00:19.885","Text":"You\u0027re wondering which kind of problem this is,"},{"Start":"00:19.885 ","End":"00:23.820","Text":"the volume of a solid bounded by 2 surfaces."},{"Start":"00:23.820 ","End":"00:27.420","Text":"The chapter on the applications of double integrals."},{"Start":"00:27.420 ","End":"00:34.140","Text":"When we have a solid that\u0027s bounded above and below by 2 surfaces,"},{"Start":"00:34.140 ","End":"00:37.520","Text":"it\u0027s projected onto a domain D,"},{"Start":"00:37.520 ","End":"00:41.370","Text":"then there\u0027s this formula that we use which says that the volume of this"},{"Start":"00:41.370 ","End":"00:45.304","Text":"solid is the upper surface minus the lower surface,"},{"Start":"00:45.304 ","End":"00:52.230","Text":"double integral of that over the projection D. Now,"},{"Start":"00:52.230 ","End":"00:56.885","Text":"we don\u0027t even see 2 surfaces here and what\u0027s going on anyway?"},{"Start":"00:56.885 ","End":"00:59.675","Text":"Let me try to explain with another diagram."},{"Start":"00:59.675 ","End":"01:02.600","Text":"Imagine that we\u0027re looking at the side that this is"},{"Start":"01:02.600 ","End":"01:07.834","Text":"the z-axis but that this is the xy plane."},{"Start":"01:07.834 ","End":"01:11.690","Text":"This, for example, would be our sphere and if this is"},{"Start":"01:11.690 ","End":"01:19.170","Text":"3 then and this is 3 also and this is our sphere of radius 3."},{"Start":"01:19.170 ","End":"01:20.650","Text":"It\u0027s in 3 dimensions,"},{"Start":"01:20.650 ","End":"01:22.940","Text":"but we\u0027re just looking from the side."},{"Start":"01:22.940 ","End":"01:26.945","Text":"Now, the cylinder x squared plus y squared equals 1."},{"Start":"01:26.945 ","End":"01:29.810","Text":"If we look from the side,"},{"Start":"01:29.810 ","End":"01:33.590","Text":"let\u0027s say this is 1."},{"Start":"01:33.590 ","End":"01:39.790","Text":"The cylinder is just 2 lines."},{"Start":"01:40.220 ","End":"01:44.670","Text":"I shaded the solid that\u0027s bounded and,"},{"Start":"01:44.670 ","End":"01:45.980","Text":"of course, it looks flat,"},{"Start":"01:45.980 ","End":"01:47.630","Text":"but it\u0027s meant to be a solid."},{"Start":"01:47.630 ","End":"01:50.255","Text":"This is 1 I should have marked it."},{"Start":"01:50.255 ","End":"01:51.980","Text":"It looks a bit like this."},{"Start":"01:51.980 ","End":"01:59.660","Text":"If this was continued to be a sphere and this D would be this bit here and,"},{"Start":"01:59.660 ","End":"02:01.115","Text":"in fact, this will be,"},{"Start":"02:01.115 ","End":"02:04.700","Text":"if I look from above at the xy plane,"},{"Start":"02:04.700 ","End":"02:07.040","Text":"this would be a circle of radius 1."},{"Start":"02:07.040 ","End":"02:08.885","Text":"Maybe I\u0027ll bring an extra picture."},{"Start":"02:08.885 ","End":"02:11.420","Text":"So this is the view from above,"},{"Start":"02:11.420 ","End":"02:15.530","Text":"I\u0027m looking from the direction of the z-axis onto the x, y plane."},{"Start":"02:15.530 ","End":"02:21.960","Text":"This will be the D like in this picture and this is a circle of radius 1."},{"Start":"02:21.960 ","End":"02:25.625","Text":"The equation of this is just this 1 here,"},{"Start":"02:25.625 ","End":"02:30.265","Text":"x squared plus y squared equals 1."},{"Start":"02:30.265 ","End":"02:33.680","Text":"Now, what about the upper and lower surfaces?"},{"Start":"02:33.680 ","End":"02:36.500","Text":"Well, the sphere is 1 equation,"},{"Start":"02:36.500 ","End":"02:39.160","Text":"but it\u0027s really 2 surfaces if you think about it."},{"Start":"02:39.160 ","End":"02:41.900","Text":"There\u0027s the upper hemisphere and the lower hemisphere."},{"Start":"02:41.900 ","End":"02:47.225","Text":"If I take this equation of a sphere and just rearrange it, what will I get?"},{"Start":"02:47.225 ","End":"02:57.470","Text":"Z squared equals 9 minus x squared minus y squared."},{"Start":"02:57.470 ","End":"02:59.585","Text":"If I take the square root,"},{"Start":"02:59.585 ","End":"03:03.665","Text":"I get that z is equal to plus or minus"},{"Start":"03:03.665 ","End":"03:09.010","Text":"the square root of 9 minus x squared minus y squared."},{"Start":"03:09.010 ","End":"03:11.450","Text":"That\u0027s the upper hemisphere."},{"Start":"03:11.450 ","End":"03:14.089","Text":"This whole top hemisphere,"},{"Start":"03:14.089 ","End":"03:16.355","Text":"but I\u0027m just going to sketch part of it."},{"Start":"03:16.355 ","End":"03:22.430","Text":"The lower hemisphere, and here\u0027s the part that we\u0027re interested in is"},{"Start":"03:22.430 ","End":"03:29.480","Text":"the same thing here with the minus the square root of 9 minus x squared minus y squared."},{"Start":"03:29.480 ","End":"03:32.105","Text":"In this scheme of things like in this picture,"},{"Start":"03:32.105 ","End":"03:34.230","Text":"this would be the upper one,"},{"Start":"03:34.230 ","End":"03:36.090","Text":"I\u0027ll call it f of x, y,"},{"Start":"03:36.090 ","End":"03:37.695","Text":"and the lower one,"},{"Start":"03:37.695 ","End":"03:40.185","Text":"g of x, y."},{"Start":"03:40.185 ","End":"03:45.470","Text":"What the theorem says is that the volume is just the"},{"Start":"03:45.470 ","End":"03:51.635","Text":"integral over this region of the upper minus the lower."},{"Start":"03:51.635 ","End":"03:59.180","Text":"That this volume v is equal to the"},{"Start":"03:59.180 ","End":"04:05.960","Text":"double integral over the region D of the upper minus the lower."},{"Start":"04:05.960 ","End":"04:08.800","Text":"But look they\u0027re the same except for the sign,"},{"Start":"04:08.800 ","End":"04:14.390","Text":"so I can just write a 2 here and write here,"},{"Start":"04:14.390 ","End":"04:21.605","Text":"the square root of 9 minus x squared minus y squared dA."},{"Start":"04:21.605 ","End":"04:25.489","Text":"I\u0027ve got the square root of something minus,"},{"Start":"04:25.489 ","End":"04:27.785","Text":"minus the square root of something."},{"Start":"04:27.785 ","End":"04:32.015","Text":"The difference between the upper and the lower is just twice that square root."},{"Start":"04:32.015 ","End":"04:35.705","Text":"Nothing very clever going on here."},{"Start":"04:35.705 ","End":"04:40.865","Text":"But we suddenly remember that we\u0027re in the chapter on polar coordinates."},{"Start":"04:40.865 ","End":"04:43.715","Text":"I\u0027ll just get a bit more space here."},{"Start":"04:43.715 ","End":"04:49.165","Text":"Let me bring in my conversion formulas from Cartesian to polar."},{"Start":"04:49.165 ","End":"04:51.705","Text":"These are the formulas."},{"Start":"04:51.705 ","End":"04:56.750","Text":"Now, D, if I want to express it in polar,"},{"Start":"04:56.750 ","End":"04:57.860","Text":"it\u0027s the unit circle."},{"Start":"04:57.860 ","End":"04:59.825","Text":"You\u0027ve seen this many times before."},{"Start":"04:59.825 ","End":"05:01.820","Text":"I\u0027ll just write it out."},{"Start":"05:01.820 ","End":"05:05.530","Text":"Theta goes from 0-2Pi,"},{"Start":"05:05.530 ","End":"05:07.665","Text":"so this is what I write,"},{"Start":"05:07.665 ","End":"05:13.500","Text":"and the radius r goes from 0-1."},{"Start":"05:13.500 ","End":"05:21.140","Text":"That gives me the region D. That means I can now write the limits of the integration as"},{"Start":"05:21.140 ","End":"05:27.080","Text":"the integral Theta 0 up to"},{"Start":"05:27.080 ","End":"05:36.120","Text":"2Pi and then the integral r goes from 0-1."},{"Start":"05:36.610 ","End":"05:40.850","Text":"Maybe I\u0027ll sketch a little bit just to emphasize it,"},{"Start":"05:40.850 ","End":"05:42.815","Text":"but this is the polar."},{"Start":"05:42.815 ","End":"05:45.890","Text":"I\u0027m starting off with Theta equals 0,"},{"Start":"05:45.890 ","End":"05:51.075","Text":"working my way all the way around up to here and"},{"Start":"05:51.075 ","End":"05:56.820","Text":"there Theta is 2Pi and for each particular Theta on the way,"},{"Start":"05:56.820 ","End":"06:00.090","Text":"my r goes from 0-1,"},{"Start":"06:00.090 ","End":"06:03.430","Text":"and that covers the unit circle."},{"Start":"06:04.160 ","End":"06:11.040","Text":"Then we have to convert this r and I mustn\u0027t forget the"},{"Start":"06:11.040 ","End":"06:17.300","Text":"2 made a bit of room to write v equals and there\u0027s the 2."},{"Start":"06:17.300 ","End":"06:22.070","Text":"Then I have the square root of 9 minus."},{"Start":"06:22.070 ","End":"06:25.035","Text":"Now x squared plus y squared, where is it?"},{"Start":"06:25.035 ","End":"06:29.550","Text":"Here is r squared and dA over here"},{"Start":"06:29.550 ","End":"06:35.940","Text":"is rdrd Theta rdrd Theta."},{"Start":"06:35.940 ","End":"06:41.165","Text":"This is now a purely technical integration problem with r and Theta."},{"Start":"06:41.165 ","End":"06:44.605","Text":"As usual, we work from the inside out."},{"Start":"06:44.605 ","End":"06:49.560","Text":"We\u0027re going to first do the integral, the dr."},{"Start":"06:49.960 ","End":"06:53.825","Text":"I\u0027d like to do this bit here at the side."},{"Start":"06:53.825 ","End":"07:01.170","Text":"What we have is the integral from"},{"Start":"07:01.170 ","End":"07:06.200","Text":"0-1 of square root of"},{"Start":"07:06.200 ","End":"07:14.095","Text":"9 minus r squared times rdr."},{"Start":"07:14.095 ","End":"07:17.345","Text":"My suggestion is to do it with a substitution."},{"Start":"07:17.345 ","End":"07:24.054","Text":"If we take this 9 minus r squared and let that be t,"},{"Start":"07:24.054 ","End":"07:28.830","Text":"dt equals minus 2rdr."},{"Start":"07:32.420 ","End":"07:37.285","Text":"If I don\u0027t want to go back from t to r,"},{"Start":"07:37.285 ","End":"07:40.520","Text":"I can just substitute the limits of integration 2."},{"Start":"07:40.520 ","End":"07:45.255","Text":"When r equals 0,"},{"Start":"07:45.255 ","End":"07:49.910","Text":"we get that t equals 9 minus 0 squared is"},{"Start":"07:49.910 ","End":"07:57.125","Text":"9 and when r equals 1,"},{"Start":"07:57.125 ","End":"08:05.250","Text":"we get that t equals 9 minus 1 squared is 8."},{"Start":"08:05.840 ","End":"08:09.645","Text":"Now, let\u0027s make the substitution."},{"Start":"08:09.645 ","End":"08:13.835","Text":"This integral equals now I\u0027m continuing down here."},{"Start":"08:13.835 ","End":"08:18.245","Text":"The limits of the integration are from 9-8."},{"Start":"08:18.245 ","End":"08:23.360","Text":"It\u0027s okay that the top is smaller than the bottom not really a problem."},{"Start":"08:23.360 ","End":"08:32.300","Text":"The next thing I have is the square root of 9 minus r squared is t and rdr,"},{"Start":"08:32.300 ","End":"08:38.060","Text":"well, I could just extract that from here and say that it\u0027s dt over minus 2."},{"Start":"08:38.060 ","End":"08:40.800","Text":"So look, dividing by the minus 2,"},{"Start":"08:40.800 ","End":"08:46.410","Text":"I\u0027ll put a minus 1/2 in front and then a dt here."},{"Start":"08:46.410 ","End":"08:49.715","Text":"What is the integral of square root of t?"},{"Start":"08:49.715 ","End":"08:51.830","Text":"Well, instead of the square root of t,"},{"Start":"08:51.830 ","End":"08:56.150","Text":"I could think of it as t_1/2 and then when I get the integral,"},{"Start":"08:56.150 ","End":"09:03.575","Text":"you raise the power by 1 so it\u0027s t_3/2 and then you divide by the new exponent."},{"Start":"09:03.575 ","End":"09:06.140","Text":"Dividing by 3 over 2 is like multiplying"},{"Start":"09:06.140 ","End":"09:10.175","Text":"by 2/3 and we don\u0027t need the plus C, I\u0027ve just wrote it."},{"Start":"09:10.175 ","End":"09:12.675","Text":"We don\u0027t need it because it\u0027s a definite integral."},{"Start":"09:12.675 ","End":"09:19.550","Text":"What we get is minus 1/2 and then the 2/3,"},{"Start":"09:19.550 ","End":"09:21.830","Text":"I can write here,"},{"Start":"09:21.830 ","End":"09:24.300","Text":"and then I have"},{"Start":"09:24.700 ","End":"09:34.390","Text":"t_3/2 evaluated between 9 and 8."},{"Start":"09:34.390 ","End":"09:38.270","Text":"I\u0027d like to tidy up a bit."},{"Start":"09:38.270 ","End":"09:44.390","Text":"What I can do is first of all this 2 with this 2 will cancel."},{"Start":"09:45.320 ","End":"09:50.360","Text":"The other thing I like to do is when I have the limits that"},{"Start":"09:50.360 ","End":"09:55.175","Text":"are backwards and I also have a minus here,"},{"Start":"09:55.175 ","End":"10:00.350","Text":"I can actually switch the order of these 2 and that gets rid of the minus."},{"Start":"10:00.350 ","End":"10:05.720","Text":"What I get is 1/3 with"},{"Start":"10:05.720 ","End":"10:13.260","Text":"a plus t_3/2 but taken between,"},{"Start":"10:13.260 ","End":"10:16.960","Text":"and I\u0027ll reverse these, 9 and 8."},{"Start":"10:16.960 ","End":"10:22.940","Text":"Just a bit of computation with indices with powers."},{"Start":"10:22.940 ","End":"10:32.080","Text":"What I get is 1/3 of"},{"Start":"10:32.080 ","End":"10:43.540","Text":"9_3/2 minus 8_3/2 and each of these can be simplified or evaluated,"},{"Start":"10:44.060 ","End":"10:48.540","Text":"9_3/2 is the square root of 9_3,"},{"Start":"10:48.540 ","End":"10:58.250","Text":"which is 27 and 8 _3/2."},{"Start":"10:58.250 ","End":"11:02.975","Text":"Let me do just a little side exercise here 8 is 2_3."},{"Start":"11:02.975 ","End":"11:10.250","Text":"So I have 2_3_3/2"},{"Start":"11:10.250 ","End":"11:15.365","Text":"which is 2_9 over 2 is 4 1/2."},{"Start":"11:15.365 ","End":"11:18.335","Text":"Now 4 1/2 is 4 plus 1/2."},{"Start":"11:18.335 ","End":"11:24.890","Text":"2_4 is 16, 2_1/2 is square root of 2."},{"Start":"11:24.890 ","End":"11:31.740","Text":"So that is what I have here, 16 root 2."},{"Start":"11:34.970 ","End":"11:38.975","Text":"This is the bit I was doing at the side."},{"Start":"11:38.975 ","End":"11:43.285","Text":"It\u0027s time to go back and plug it in here."},{"Start":"11:43.285 ","End":"11:46.325","Text":"We get v equals twice."},{"Start":"11:46.325 ","End":"11:49.760","Text":"Now, this is a constant,"},{"Start":"11:49.760 ","End":"11:52.535","Text":"so I can also bring it in front."},{"Start":"11:52.535 ","End":"12:02.120","Text":"So 1/3 times 27 minus 16 root 2 and all we\u0027re left"},{"Start":"12:02.120 ","End":"12:07.020","Text":"with now is the integral from 0-2Pi of"},{"Start":"12:07.020 ","End":"12:11.990","Text":"d Theta or 1d Theta and we know that we have the integral of 1."},{"Start":"12:11.990 ","End":"12:14.585","Text":"It\u0027s just the top limit minus the bottom limit."},{"Start":"12:14.585 ","End":"12:16.920","Text":"So this bit is 2Pi."},{"Start":"12:16.970 ","End":"12:23.130","Text":"So altogether I get 2 times 1/3"},{"Start":"12:23.130 ","End":"12:29.830","Text":"times 27 minus 16 root 2 times 2Pi."},{"Start":"12:29.930 ","End":"12:32.690","Text":"There\u0027s a minus here."},{"Start":"12:32.690 ","End":"12:35.465","Text":"So let\u0027s do each of the pieces separately."},{"Start":"12:35.465 ","End":"12:37.475","Text":"First of all, the 27."},{"Start":"12:37.475 ","End":"12:41.195","Text":"27 times 1/3 is 9."},{"Start":"12:41.195 ","End":"12:46.790","Text":"9 times 2 is 18."},{"Start":"12:46.790 ","End":"12:53.130","Text":"18 times 2Pi is,"},{"Start":"12:53.130 ","End":"12:55.835","Text":"well, leave the Pi maybe at the end."},{"Start":"12:55.835 ","End":"12:58.925","Text":"So 18 times 2 is 36."},{"Start":"12:58.925 ","End":"13:02.195","Text":"Then we\u0027ll have something Pi."},{"Start":"13:02.195 ","End":"13:05.770","Text":"Let\u0027s see what the other bit comes out to be."},{"Start":"13:05.770 ","End":"13:08.160","Text":"Here, what do we have?"},{"Start":"13:08.160 ","End":"13:10.005","Text":"The Pi we were keeping."},{"Start":"13:10.005 ","End":"13:14.115","Text":"2 times 2 is 4."},{"Start":"13:14.115 ","End":"13:19.830","Text":"4 times 16 root 2 is 64 root 2,"},{"Start":"13:19.830 ","End":"13:24.940","Text":"and the only thing we haven\u0027t taken is the over 3."},{"Start":"13:25.900 ","End":"13:33.450","Text":"I\u0027m going to stop here and declare that this is our answer and we\u0027re finally done."}],"ID":8711},{"Watched":false,"Name":"Exercise 27","Duration":"14m 20s","ChapterTopicVideoID":8488,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8488.jpeg","UploadDate":"2017-02-13T08:37:11.7000000","DurationForVideoObject":"PT14M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this exercise, we have another one of those problems where we have a"},{"Start":"00:03.750 ","End":"00:10.080","Text":"solid bounded above and below by 2 surfaces,"},{"Start":"00:10.080 ","End":"00:15.705","Text":"and we have its projection onto the xy plane as in this picture."},{"Start":"00:15.705 ","End":"00:19.650","Text":"Then there\u0027s a formula that the volume is the"},{"Start":"00:19.650 ","End":"00:27.010","Text":"integral over the projection of the upper minus the lower."},{"Start":"00:27.110 ","End":"00:35.390","Text":"In our case, the upper surface is the cone."},{"Start":"00:35.390 ","End":"00:42.530","Text":"We can see it\u0027s always positive or at least non-negative so that our f of x,"},{"Start":"00:42.530 ","End":"00:49.605","Text":"y is the cone is the square root of x squared plus y squared, that\u0027s the upper."},{"Start":"00:49.605 ","End":"00:53.040","Text":"The lower, which here they call g of x,"},{"Start":"00:53.040 ","End":"00:58.440","Text":"y is the xy plane,"},{"Start":"00:58.440 ","End":"01:02.850","Text":"which is just 0."},{"Start":"01:02.850 ","End":"01:12.900","Text":"Z is 0, and the projection D in the xy plane is"},{"Start":"01:12.900 ","End":"01:18.480","Text":"what is inside including the boundary of"},{"Start":"01:18.480 ","End":"01:25.415","Text":"this x squared plus y squared equals 2y,"},{"Start":"01:25.415 ","End":"01:29.900","Text":"which we\u0027ll see will turn out to be a circle."},{"Start":"01:29.900 ","End":"01:33.640","Text":"I\u0027d like to start by sketching this,"},{"Start":"01:33.640 ","End":"01:35.590","Text":"which I claimed is a circle."},{"Start":"01:35.590 ","End":"01:38.590","Text":"That\u0027s the projection of the cylinder onto the xy plane is"},{"Start":"01:38.590 ","End":"01:41.830","Text":"just a curve in the xy plane. I\u0027ll copy it first."},{"Start":"01:41.830 ","End":"01:49.510","Text":"Now, bring the 2y over x squared plus y squared minus 2y."},{"Start":"01:49.510 ","End":"01:53.135","Text":"Looks like we want to complete the square."},{"Start":"01:53.135 ","End":"01:55.895","Text":"We want x squared plus something squared."},{"Start":"01:55.895 ","End":"01:58.740","Text":"If I add a 1 here,"},{"Start":"01:58.740 ","End":"02:01.290","Text":"and then I write a 1 here,"},{"Start":"02:01.290 ","End":"02:07.305","Text":"then I can write this as x squared or x minus 0 squared,"},{"Start":"02:07.305 ","End":"02:11.180","Text":"just so we want to emphasize the center of the circle, plus,"},{"Start":"02:11.180 ","End":"02:14.435","Text":"and this bit is y minus 1 squared,"},{"Start":"02:14.435 ","End":"02:16.745","Text":"and this is 1 squared."},{"Start":"02:16.745 ","End":"02:20.005","Text":"Now, we really see that it is a circle,"},{"Start":"02:20.005 ","End":"02:25.724","Text":"and the circle has the center at 0,"},{"Start":"02:25.724 ","End":"02:31.335","Text":"1, and radius is 1."},{"Start":"02:31.335 ","End":"02:37.275","Text":"That\u0027s the center, that\u0027s the radius."},{"Start":"02:37.275 ","End":"02:39.930","Text":"A little sketch."},{"Start":"02:39.930 ","End":"02:45.180","Text":"Here\u0027s the sketch of this circle center 0, 1,"},{"Start":"02:45.180 ","End":"02:48.210","Text":"radius 1, means it cuts at the origin,"},{"Start":"02:48.210 ","End":"02:50.145","Text":"and the point 2 here,"},{"Start":"02:50.145 ","End":"02:53.465","Text":"that\u0027s our D. According to this formula,"},{"Start":"02:53.465 ","End":"03:02.720","Text":"the volume is the double integral over D of the upper minus the lower,"},{"Start":"03:02.720 ","End":"03:12.345","Text":"the f minus the g. It\u0027s just the square root of x squared plus y squared,"},{"Start":"03:12.345 ","End":"03:15.700","Text":"and that is dA."},{"Start":"03:16.970 ","End":"03:20.090","Text":"I guess I forgot to say in the question,"},{"Start":"03:20.090 ","End":"03:23.150","Text":"because it is in the chapter on polar coordinates,"},{"Start":"03:23.150 ","End":"03:29.150","Text":"that I want this to be done with polar coordinates."},{"Start":"03:29.150 ","End":"03:36.185","Text":"I need to bring in the equations for conversion to polar over here."},{"Start":"03:36.185 ","End":"03:42.215","Text":"Now we want to describe D as a polar region,"},{"Start":"03:42.215 ","End":"03:47.940","Text":"so I want to give limits on theta going from something to something,"},{"Start":"03:47.940 ","End":"03:51.160","Text":"and then r also."},{"Start":"03:52.520 ","End":"03:55.035","Text":"This is a typical Theta,"},{"Start":"03:55.035 ","End":"04:00.190","Text":"but Theta goes all the way back to the x-axis,"},{"Start":"04:00.190 ","End":"04:04.490","Text":"which is Theta equals 0."},{"Start":"04:04.700 ","End":"04:07.375","Text":"As this point travels,"},{"Start":"04:07.375 ","End":"04:11.170","Text":"we\u0027re going to get up to 90 degrees and continue all the way down"},{"Start":"04:11.170 ","End":"04:16.280","Text":"here up to 180 degrees where Theta is equal to Pi."},{"Start":"04:16.280 ","End":"04:21.205","Text":"In the middle we pass through Theta equals Pi over 2, the 90 degrees."},{"Start":"04:21.205 ","End":"04:26.330","Text":"This point keeps traveling as Theta keeps traveling."},{"Start":"04:26.990 ","End":"04:35.970","Text":"So we go from 0 to 180 degrees or Pi."},{"Start":"04:35.970 ","End":"04:42.380","Text":"Now r it goes always from 0,"},{"Start":"04:42.380 ","End":"04:44.240","Text":"so this part\u0027s not the problem,"},{"Start":"04:44.240 ","End":"04:48.425","Text":"but this thing here is variable in length,"},{"Start":"04:48.425 ","End":"04:52.100","Text":"because as I change the Theta,"},{"Start":"04:52.100 ","End":"04:54.410","Text":"I get a different length each time."},{"Start":"04:54.410 ","End":"04:57.040","Text":"Let me give it a name."},{"Start":"04:57.040 ","End":"04:59.000","Text":"I could use r again,"},{"Start":"04:59.000 ","End":"05:01.310","Text":"but I really like to give it a different letter."},{"Start":"05:01.310 ","End":"05:02.825","Text":"Let\u0027s call it r_1,"},{"Start":"05:02.825 ","End":"05:07.340","Text":"which will be a function of Theta because different Thetas will give me different r_1."},{"Start":"05:07.340 ","End":"05:09.770","Text":"Lets say we just choose 1 for the moment."},{"Start":"05:09.770 ","End":"05:13.390","Text":"Now, how do we find out what this r_1 is?"},{"Start":"05:13.390 ","End":"05:16.170","Text":"R_1 is on the circumference of the circle,"},{"Start":"05:16.170 ","End":"05:20.990","Text":"and we know that the equation of the circle is this here."},{"Start":"05:20.990 ","End":"05:23.570","Text":"I have it here also I guess."},{"Start":"05:23.570 ","End":"05:26.455","Text":"On the left hand side,"},{"Start":"05:26.455 ","End":"05:34.260","Text":"I\u0027ve got x squared plus y squared is r squared."},{"Start":"05:34.260 ","End":"05:38.145","Text":"Here going over here just to do the conversion,"},{"Start":"05:38.145 ","End":"05:42.100","Text":"so r squared equals 2,"},{"Start":"05:42.100 ","End":"05:45.485","Text":"and then y, I have a formula for that."},{"Start":"05:45.485 ","End":"05:49.020","Text":"It\u0027s r sine Theta,"},{"Start":"05:49.820 ","End":"05:53.565","Text":"but we\u0027re using r_1 for this,"},{"Start":"05:53.565 ","End":"05:56.760","Text":"just to not double use the letter r,"},{"Start":"05:56.760 ","End":"06:05.115","Text":"divide by r_1, and I\u0027ve got that r_1 equals 2 sine Theta."},{"Start":"06:05.115 ","End":"06:09.905","Text":"You might say, \"Aha, but couldn\u0027t I have been dividing by 0?\""},{"Start":"06:09.905 ","End":"06:15.360","Text":"Well, when r_1 is 0,"},{"Start":"06:15.950 ","End":"06:18.635","Text":"where is r equal to 0?"},{"Start":"06:18.635 ","End":"06:20.320","Text":"It\u0027s at this point here."},{"Start":"06:20.320 ","End":"06:23.200","Text":"This point here is either the first or the last point,"},{"Start":"06:23.200 ","End":"06:26.615","Text":"so Theta is going to be the 0, or,"},{"Start":"06:26.615 ","End":"06:28.560","Text":"let me just write that down,"},{"Start":"06:28.560 ","End":"06:30.600","Text":"if r_1 is 0,"},{"Start":"06:30.600 ","End":"06:35.490","Text":"then we can either have Theta is 0 or Theta is Pi."},{"Start":"06:35.490 ","End":"06:40.485","Text":"But either way, sine Theta is 0,"},{"Start":"06:40.485 ","End":"06:44.655","Text":"so we get 0 equals 0, so we\u0027re okay."},{"Start":"06:44.655 ","End":"06:48.075","Text":"Now, I can write that here,"},{"Start":"06:48.075 ","End":"06:52.860","Text":"that this is 2 sine Theta,"},{"Start":"06:52.860 ","End":"06:57.150","Text":"and now we have a description of our domain."},{"Start":"06:57.150 ","End":"07:01.710","Text":"This integral can be written as,"},{"Start":"07:02.150 ","End":"07:06.030","Text":"first of all, the outer loop is Theta,"},{"Start":"07:06.030 ","End":"07:09.240","Text":"goes from 0 to Pi."},{"Start":"07:09.240 ","End":"07:12.385","Text":"I\u0027ll write that, Theta equals 0 to Pi."},{"Start":"07:12.385 ","End":"07:19.550","Text":"For each Theta, r goes from 0 up to this r_1,"},{"Start":"07:19.550 ","End":"07:22.800","Text":"which is 2 sine Theta."},{"Start":"07:22.990 ","End":"07:25.340","Text":"Now, what else do I have?"},{"Start":"07:25.340 ","End":"07:33.375","Text":"The integrant square root of x squared plus y squared from this formula is just r,"},{"Start":"07:33.375 ","End":"07:39.650","Text":"and dA is rdrdTheta."},{"Start":"07:39.970 ","End":"07:48.350","Text":"Let me just write it as r squared drdTheta."},{"Start":"07:48.350 ","End":"07:54.244","Text":"Once again, I\u0027ve got from the square root of x squared plus y squared,"},{"Start":"07:54.244 ","End":"07:57.650","Text":"I get an r, and from the dA,"},{"Start":"07:57.650 ","End":"08:00.950","Text":"I get this other r here."},{"Start":"08:00.950 ","End":"08:05.120","Text":"This r with this r give me this r squared."},{"Start":"08:05.120 ","End":"08:10.415","Text":"We do the inner integral first, that\u0027s this bit."},{"Start":"08:10.415 ","End":"08:12.530","Text":"I\u0027d like to do this bit,"},{"Start":"08:12.530 ","End":"08:15.380","Text":"the one that I highlighted at the side here."},{"Start":"08:15.380 ","End":"08:20.090","Text":"I want to compute the integral from 0 to"},{"Start":"08:20.090 ","End":"08:26.130","Text":"2 sine Theta of r squared dr."},{"Start":"08:26.130 ","End":"08:28.075","Text":"Fairly straightforward."},{"Start":"08:28.075 ","End":"08:32.590","Text":"This is equal to the integral of r squared is"},{"Start":"08:32.590 ","End":"08:41.585","Text":"1 third r cubed and then I have to take it between the limits."},{"Start":"08:41.585 ","End":"08:45.650","Text":"I can leave the third constant outside the brackets and put that"},{"Start":"08:45.650 ","End":"08:51.480","Text":"this goes from 0 to 2 sine Theta."},{"Start":"08:51.940 ","End":"09:00.200","Text":"Plug in the upper 2 sine Theta cubed is 2 cubed is 8,"},{"Start":"09:00.200 ","End":"09:02.750","Text":"so I have 8/3,"},{"Start":"09:02.750 ","End":"09:05.540","Text":"and then sine Theta cubed,"},{"Start":"09:05.540 ","End":"09:08.525","Text":"we write a sine cubed Theta."},{"Start":"09:08.525 ","End":"09:13.350","Text":"If I plug in 0, I just get nothing."},{"Start":"09:13.450 ","End":"09:21.020","Text":"Now I can go back here and write this as the constant,"},{"Start":"09:21.020 ","End":"09:23.865","Text":"it can come out in front."},{"Start":"09:23.865 ","End":"09:30.220","Text":"It\u0027s 8/3 and then the integral from 0 to Pi"},{"Start":"09:30.220 ","End":"09:37.455","Text":"of sine cubed Theta d Theta."},{"Start":"09:37.455 ","End":"09:41.645","Text":"First of all, I\u0027m going to look it up on a formula sheet,"},{"Start":"09:41.645 ","End":"09:43.670","Text":"but then at the end of the exercise,"},{"Start":"09:43.670 ","End":"09:46.280","Text":"I\u0027ll go back and show you how we arrive at it."},{"Start":"09:46.280 ","End":"09:52.955","Text":"If you look up the indefinite integral of sine cubed Theta,"},{"Start":"09:52.955 ","End":"09:56.720","Text":"you get from the formula sheet,"},{"Start":"09:56.720 ","End":"10:03.810","Text":"1/3 of cosine cubed Theta minus cosine Theta."},{"Start":"10:03.940 ","End":"10:09.035","Text":"Evaluating it as a definite integral from 0 to Pi,"},{"Start":"10:09.035 ","End":"10:12.965","Text":"but I also want to show you how I get from here to here."},{"Start":"10:12.965 ","End":"10:14.570","Text":"I\u0027ll do it at the end,"},{"Start":"10:14.570 ","End":"10:16.835","Text":"so I\u0027ll write an IOU,"},{"Start":"10:16.835 ","End":"10:19.655","Text":"an explanation of this at the end."},{"Start":"10:19.655 ","End":"10:22.895","Text":"Those who don\u0027t want the explanation can skip it."},{"Start":"10:22.895 ","End":"10:25.295","Text":"Meanwhile, we\u0027ll continue."},{"Start":"10:25.295 ","End":"10:30.270","Text":"First of all, plugin the Pi."},{"Start":"10:30.760 ","End":"10:35.030","Text":"We need to know what is cosine of Pi and cosine of 0."},{"Start":"10:35.030 ","End":"10:44.390","Text":"I\u0027ll just remind you that cosine of Pi is negative 1 and cosine of 0 is 1,"},{"Start":"10:44.390 ","End":"10:46.354","Text":"for those who forgot."},{"Start":"10:46.354 ","End":"10:48.169","Text":"If I plug in Pi,"},{"Start":"10:48.169 ","End":"10:54.365","Text":"I get 1/3 times minus 1 cubed,"},{"Start":"10:54.365 ","End":"10:57.215","Text":"which makes it minus a 1/3."},{"Start":"10:57.215 ","End":"11:01.980","Text":"Minus minus 1 is plus 1."},{"Start":"11:03.190 ","End":"11:07.519","Text":"Put it on the brackets here, so that\u0027s for Pi,"},{"Start":"11:07.519 ","End":"11:13.325","Text":"and now for 0, I\u0027ll get something minus something."},{"Start":"11:13.325 ","End":"11:15.290","Text":"With 0, it\u0027s 1,"},{"Start":"11:15.290 ","End":"11:22.040","Text":"so this is just 1/3 minus 1,"},{"Start":"11:22.040 ","End":"11:31.055","Text":"32/9 and this is the answer."},{"Start":"11:31.055 ","End":"11:34.065","Text":"We\u0027re not really done,"},{"Start":"11:34.065 ","End":"11:37.870","Text":"because I want to show you how I did this integral."},{"Start":"11:37.870 ","End":"11:40.525","Text":"I\u0027ll do this integral here at the side."},{"Start":"11:40.525 ","End":"11:43.000","Text":"With the indefinite integral,"},{"Start":"11:43.000 ","End":"11:50.270","Text":"we want of sine cubed Theta d Theta."},{"Start":"11:50.270 ","End":"11:56.975","Text":"Now there\u0027s a few standard tricks of the trade."},{"Start":"11:56.975 ","End":"12:05.749","Text":"What we do is we can write this first of all as the integral of"},{"Start":"12:05.749 ","End":"12:15.290","Text":"sine squared Theta times sine Theta d Theta."},{"Start":"12:15.290 ","End":"12:16.850","Text":"Why would I do that?"},{"Start":"12:16.850 ","End":"12:24.695","Text":"Because then I can use the formula that sine squared Theta is"},{"Start":"12:24.695 ","End":"12:32.855","Text":"1 minus cosine squared Theta and sine Theta d Theta."},{"Start":"12:32.855 ","End":"12:36.125","Text":"This is good for a substitution."},{"Start":"12:36.125 ","End":"12:41.480","Text":"T equals cosine Theta and then"},{"Start":"12:41.480 ","End":"12:49.295","Text":"dt derivative of cosine is minus sine Theta d Theta."},{"Start":"12:49.295 ","End":"12:56.450","Text":"We get the integral of 1 minus t squared,"},{"Start":"12:56.450 ","End":"13:06.095","Text":"because cosine Theta is t and sine Theta d Theta is minus dt."},{"Start":"13:06.095 ","End":"13:11.674","Text":"I can put a minus here and a dt here,"},{"Start":"13:11.674 ","End":"13:14.360","Text":"so that the minus,"},{"Start":"13:14.360 ","End":"13:18.260","Text":"I prefer to have different order of subtraction,"},{"Start":"13:18.260 ","End":"13:24.300","Text":"t squared minus 1 dt."},{"Start":"13:25.060 ","End":"13:35.300","Text":"Now we can get that the integral of t squared is 1/3 t cubed,"},{"Start":"13:35.300 ","End":"13:43.865","Text":"the integral of 1 is just t. Then those plus that constant for indefinite integrals."},{"Start":"13:43.865 ","End":"13:50.225","Text":"Finally, we substitute back from t to Theta."},{"Start":"13:50.225 ","End":"13:52.490","Text":"T is cosine Theta,"},{"Start":"13:52.490 ","End":"13:56.750","Text":"so what we get is 1/3 cosine Theta cubed."},{"Start":"13:56.750 ","End":"13:59.600","Text":"You add the 3 here, minus t,"},{"Start":"13:59.600 ","End":"14:03.155","Text":"which is cosine Theta,"},{"Start":"14:03.155 ","End":"14:07.280","Text":"and plus the constant for indefinite integrals."},{"Start":"14:07.280 ","End":"14:10.850","Text":"If you look at this and you look what we have here,"},{"Start":"14:10.850 ","End":"14:21.090","Text":"then that is fine and so that\u0027s the IOU that taken care of, and we\u0027re done."}],"ID":8712},{"Watched":false,"Name":"Exercise 28","Duration":"20m 57s","ChapterTopicVideoID":8489,"CourseChapterTopicPlaylistID":4969,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8489.jpeg","UploadDate":"2017-02-13T08:40:04.5230000","DurationForVideoObject":"PT20M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"Here we have another one of those questions"},{"Start":"00:03.300 ","End":"00:06.045","Text":"like in the applications of the double integral,"},{"Start":"00:06.045 ","End":"00:09.930","Text":"where we\u0027re given a surface above and a surface below,"},{"Start":"00:09.930 ","End":"00:13.430","Text":"and the projection of the solid onto the xy plane,"},{"Start":"00:13.430 ","End":"00:15.405","Text":"and we have to figure out its volume."},{"Start":"00:15.405 ","End":"00:20.690","Text":"We\u0027ve done these before and I used the diagram here."},{"Start":"00:20.690 ","End":"00:25.010","Text":"The upper surface, the lower surface, the projection,"},{"Start":"00:25.010 ","End":"00:30.179","Text":"and the formula that the volume of the solid is the upper minus lower"},{"Start":"00:30.179 ","End":"00:36.735","Text":"double integral of that over the projected domain."},{"Start":"00:36.735 ","End":"00:39.395","Text":"Let\u0027s see what\u0027s what in our case."},{"Start":"00:39.395 ","End":"00:41.660","Text":"Well, I say that D is,"},{"Start":"00:41.660 ","End":"00:44.810","Text":"if you ignore the z and just look at the xy plane,"},{"Start":"00:44.810 ","End":"00:49.554","Text":"this is the D. Our D is given by"},{"Start":"00:49.554 ","End":"00:55.740","Text":"x squared plus y squared equals x."},{"Start":"00:55.740 ","End":"01:03.560","Text":"Actually, it\u0027s less than or equal to x because it also includes the inside."},{"Start":"01:03.560 ","End":"01:06.500","Text":"But the border is,"},{"Start":"01:06.500 ","End":"01:09.115","Text":"I\u0027ll write it as equals,"},{"Start":"01:09.115 ","End":"01:13.195","Text":"but that\u0027s just the boundary of D. Now,"},{"Start":"01:13.195 ","End":"01:18.900","Text":"I want to transform this a bit and show you this is actually a circle."},{"Start":"01:19.250 ","End":"01:22.460","Text":"If I bring x to the other side,"},{"Start":"01:22.460 ","End":"01:25.460","Text":"I get x squared minus x,"},{"Start":"01:25.460 ","End":"01:28.610","Text":"and let me leave space here for completing the square,"},{"Start":"01:28.610 ","End":"01:33.270","Text":"plus y squared equals,"},{"Start":"01:33.270 ","End":"01:34.380","Text":"and then we\u0027ll see in a moment,"},{"Start":"01:34.380 ","End":"01:36.390","Text":"it would be 0, but now,"},{"Start":"01:36.390 ","End":"01:37.620","Text":"I want to complete the square."},{"Start":"01:37.620 ","End":"01:40.100","Text":"I take 1/2 this coefficient and square it."},{"Start":"01:40.100 ","End":"01:42.280","Text":"It\u0027s 1/2 squared is 1/4,"},{"Start":"01:42.280 ","End":"01:45.080","Text":"and I add 1/4 to the other side also."},{"Start":"01:45.080 ","End":"01:51.120","Text":"Now I can write this as x minus 1/2 squared plus y squared,"},{"Start":"01:51.120 ","End":"01:53.700","Text":"I\u0027ll write as y minus 0 squared,"},{"Start":"01:53.700 ","End":"01:56.735","Text":"and 1/4, I\u0027ll write as 1/2 squared."},{"Start":"01:56.735 ","End":"02:05.465","Text":"We see that it\u0027s a circle where the center of the circle is at 1/2,"},{"Start":"02:05.465 ","End":"02:12.810","Text":"0, and the radius of the circle is 1/2."},{"Start":"02:12.810 ","End":"02:16.960","Text":"That describes domain D. Now,"},{"Start":"02:16.960 ","End":"02:20.120","Text":"I want to make sure that in the domain D,"},{"Start":"02:20.120 ","End":"02:25.620","Text":"that the paraboloid really is above the xy plane."},{"Start":"02:26.090 ","End":"02:30.170","Text":"The paraboloid is f of x,"},{"Start":"02:30.170 ","End":"02:35.255","Text":"y equals, hopefully the upper one as we\u0027re going to check,"},{"Start":"02:35.255 ","End":"02:39.245","Text":"is 1 minus x squared minus y squared,"},{"Start":"02:39.245 ","End":"02:41.045","Text":"and g of x, y,"},{"Start":"02:41.045 ","End":"02:43.220","Text":"is just the xy plane,"},{"Start":"02:43.220 ","End":"02:48.170","Text":"which is the equation where z is our g,"},{"Start":"02:48.170 ","End":"02:49.910","Text":"so this is 0."},{"Start":"02:49.910 ","End":"02:53.030","Text":"F minus g is just this."},{"Start":"02:53.030 ","End":"02:58.040","Text":"Now, how do I know that this is bigger or equal to 0 on my domain?"},{"Start":"02:58.040 ","End":"03:00.860","Text":"Well, let\u0027s check. Where is it equal to 0?"},{"Start":"03:00.860 ","End":"03:08.040","Text":"Let me see where is 1 minus x squared minus y squared equals to 0."},{"Start":"03:08.560 ","End":"03:16.420","Text":"That will give me that x squared plus y squared equals 1."},{"Start":"03:17.240 ","End":"03:22.730","Text":"Inside the circle it\u0027s bigger or equal to 0 this thing because, for example,"},{"Start":"03:22.730 ","End":"03:25.040","Text":"I could take the center and put in x is 0,"},{"Start":"03:25.040 ","End":"03:29.615","Text":"y is 0, so it\u0027s bigger or equal to 0 inside the circle."},{"Start":"03:29.615 ","End":"03:33.870","Text":"I\u0027d actually like to plot both of these."},{"Start":"03:36.500 ","End":"03:39.510","Text":"Here we are, I squeezed the picture in."},{"Start":"03:39.510 ","End":"03:44.550","Text":"Our domain is the inside of the circle that\u0027s here,"},{"Start":"03:44.550 ","End":"03:46.815","Text":"which has center 1/2,"},{"Start":"03:46.815 ","End":"03:50.875","Text":"0, that\u0027s here, and radius of a 1/2."},{"Start":"03:50.875 ","End":"03:56.520","Text":"But the place where f is bigger or equal to g, well,"},{"Start":"03:56.520 ","End":"04:02.225","Text":"I should have had a bigger or equal to 0 here and a less than or equal to 1 here,"},{"Start":"04:02.225 ","End":"04:04.250","Text":"so it\u0027s the interior of this circle."},{"Start":"04:04.250 ","End":"04:06.920","Text":"This is the circle x squared plus y squared equals"},{"Start":"04:06.920 ","End":"04:10.775","Text":"1 and because our domain is completely inside this circle,"},{"Start":"04:10.775 ","End":"04:18.665","Text":"we\u0027re okay that f is bigger or equal to g inside all of our domain D. What we have now is"},{"Start":"04:18.665 ","End":"04:23.390","Text":"our volume is equal to the double"},{"Start":"04:23.390 ","End":"04:29.605","Text":"integral over D of f minus g is just f,"},{"Start":"04:29.605 ","End":"04:38.930","Text":"of 1 minus x squared minus y squared and dA."},{"Start":"04:38.930 ","End":"04:42.170","Text":"I should have actually stated in the question,"},{"Start":"04:42.170 ","End":"04:49.890","Text":"do it using the polar transformation of double integrals."},{"Start":"04:49.890 ","End":"04:51.945","Text":"Let\u0027s assume polar was given."},{"Start":"04:51.945 ","End":"04:56.059","Text":"I want to describe this region D in polar terms."},{"Start":"04:56.059 ","End":"05:03.650","Text":"Let me draw a typical point on the boundary of our domain."},{"Start":"05:03.650 ","End":"05:07.480","Text":"You want this point to go all the way around the circle,"},{"Start":"05:07.480 ","End":"05:09.735","Text":"and this will be Theta."},{"Start":"05:09.735 ","End":"05:12.650","Text":"If I start from here and go this way,"},{"Start":"05:12.650 ","End":"05:17.765","Text":"my Theta will be starting off actually here"},{"Start":"05:17.765 ","End":"05:25.230","Text":"at minus 90 degrees and then continuing all the way,"},{"Start":"05:25.230 ","End":"05:30.455","Text":"I\u0027ll end up through various stages in the middle,"},{"Start":"05:30.455 ","End":"05:32.210","Text":"keep getting more and more,"},{"Start":"05:32.210 ","End":"05:35.015","Text":"and as I go around the circle,"},{"Start":"05:35.015 ","End":"05:38.570","Text":"I go from minus 90 to plus 90,"},{"Start":"05:38.570 ","End":"05:42.360","Text":"so Theta goes from here."},{"Start":"05:42.360 ","End":"05:44.300","Text":"Theta equals minus 90,"},{"Start":"05:44.300 ","End":"05:46.475","Text":"which I write as minus Pi over 2,"},{"Start":"05:46.475 ","End":"05:51.980","Text":"and this goes all the way round to Theta equals Pi over 2."},{"Start":"05:51.980 ","End":"05:55.385","Text":"Now, choose one of these Theta as a general one."},{"Start":"05:55.385 ","End":"06:01.055","Text":"R will go always from 0 up to something which is variable."},{"Start":"06:01.055 ","End":"06:03.355","Text":"We\u0027ll call it for the moment r_1."},{"Start":"06:03.355 ","End":"06:06.420","Text":"I want to figure out what r_1 is in terms of Theta."},{"Start":"06:06.420 ","End":"06:08.685","Text":"It will depend on Theta."},{"Start":"06:08.685 ","End":"06:11.490","Text":"It\u0027s not a constant."},{"Start":"06:11.490 ","End":"06:16.400","Text":"Let me just write down what I found out so far about D. I know"},{"Start":"06:16.400 ","End":"06:22.560","Text":"Theta goes from minus Pi over 2 to Pi over 2,"},{"Start":"06:22.560 ","End":"06:27.215","Text":"and I know that r goes from 0 to something."},{"Start":"06:27.215 ","End":"06:30.595","Text":"We call this for the moment r_1."},{"Start":"06:30.595 ","End":"06:33.015","Text":"But I want to find out what r_1 is,"},{"Start":"06:33.015 ","End":"06:36.150","Text":"so I\u0027m going to bring in the equations."},{"Start":"06:36.150 ","End":"06:38.490","Text":"Now we want this circle."},{"Start":"06:38.490 ","End":"06:40.685","Text":"We know its Cartesian equation,"},{"Start":"06:40.685 ","End":"06:45.020","Text":"x squared plus y squared equals x, that\u0027s from here."},{"Start":"06:45.020 ","End":"06:48.730","Text":"I want to now convert this one to polar."},{"Start":"06:48.730 ","End":"06:52.880","Text":"Looking at this x squared plus y squared is r squared,"},{"Start":"06:52.880 ","End":"07:00.335","Text":"so I have that r squared equals x is,"},{"Start":"07:00.335 ","End":"07:03.350","Text":"here it is, r cosine Theta."},{"Start":"07:03.350 ","End":"07:08.330","Text":"But we\u0027re talking about our particular point, so it\u0027s r_1."},{"Start":"07:08.330 ","End":"07:12.709","Text":"Now I can divide both sides by r_1,"},{"Start":"07:12.709 ","End":"07:16.894","Text":"so r_1 is cosine Theta."},{"Start":"07:16.894 ","End":"07:21.380","Text":"We\u0027ve talked previously about why it\u0027s okay to divide by r_1,"},{"Start":"07:21.380 ","End":"07:24.140","Text":"that even if r_1 is 0, it still works out."},{"Start":"07:24.140 ","End":"07:31.410","Text":"Now I can plug that here, cosine Theta,"},{"Start":"07:31.410 ","End":"07:40.085","Text":"and then I can express my integral in terms of this region D in polar form."},{"Start":"07:40.085 ","End":"07:44.060","Text":"What I get is that the volume is equal to"},{"Start":"07:44.060 ","End":"07:49.805","Text":"the integral Theta from minus Pi over 2 to Pi over 2,"},{"Start":"07:49.805 ","End":"07:51.575","Text":"let me write that, that\u0027s Theta."},{"Start":"07:51.575 ","End":"07:58.320","Text":"R goes from 0 to cosine Theta."},{"Start":"07:58.320 ","End":"07:59.895","Text":"Then I need this,"},{"Start":"07:59.895 ","End":"08:02.100","Text":"x squared plus y squared is r squared,"},{"Start":"08:02.100 ","End":"08:04.905","Text":"so this is 1 minus r squared."},{"Start":"08:04.905 ","End":"08:06.315","Text":"That\u0027s from this formula,"},{"Start":"08:06.315 ","End":"08:10.510","Text":"and dA here it is, rdrd Theta."},{"Start":"08:11.570 ","End":"08:15.585","Text":"As usual, we begin from the inside,"},{"Start":"08:15.585 ","End":"08:21.310","Text":"so we\u0027ll be wanting to do this one first."},{"Start":"08:21.800 ","End":"08:25.170","Text":"I\u0027d like to do this inner integral at the side,"},{"Start":"08:25.170 ","End":"08:27.965","Text":"I\u0027ll call it asterisk and I\u0027ll continue over here."},{"Start":"08:27.965 ","End":"08:34.970","Text":"What I have is the integral from 0 to cosine Theta."},{"Start":"08:34.970 ","End":"08:36.710","Text":"If I multiply it out,"},{"Start":"08:36.710 ","End":"08:43.850","Text":"I have r minus r cubed dr,"},{"Start":"08:43.850 ","End":"08:46.835","Text":"and this is equal to,"},{"Start":"08:46.835 ","End":"08:52.410","Text":"let see, integral of r is a 1/2r squared,"},{"Start":"08:52.410 ","End":"08:57.875","Text":"the integral of r cubed is 1/4r^4,"},{"Start":"08:57.875 ","End":"09:05.250","Text":"and I have to take this from 0 to cosine Theta."},{"Start":"09:05.250 ","End":"09:08.840","Text":"Notice that if I plug in r equals 0,"},{"Start":"09:08.840 ","End":"09:12.260","Text":"I get 0, so I just need the cosine Theta."},{"Start":"09:12.260 ","End":"09:17.165","Text":"I have 1/2 cosine squared"},{"Start":"09:17.165 ","End":"09:24.390","Text":"Theta minus 1/4 cosine^4 Theta."},{"Start":"09:24.390 ","End":"09:27.160","Text":"That\u0027s the answer to this inner one."},{"Start":"09:27.160 ","End":"09:30.625","Text":"Now I\u0027m returning here, more space now."},{"Start":"09:30.625 ","End":"09:39.265","Text":"I get that V is equal to the integral from minus Pi over 2 to Pi over 2."},{"Start":"09:39.265 ","End":"09:40.990","Text":"Instead of just copying,"},{"Start":"09:40.990 ","End":"09:45.490","Text":"maybe I\u0027ll put a common denominator over 4 and bring the 1/4 out."},{"Start":"09:45.490 ","End":"09:48.285","Text":"I can put 1/4 here,"},{"Start":"09:48.285 ","End":"09:53.310","Text":"and then I have 2 cosine squared"},{"Start":"09:53.310 ","End":"10:00.700","Text":"Theta minus cosine^4 Theta d Theta."},{"Start":"10:00.860 ","End":"10:06.690","Text":"Now, I\u0027d like to bring the solutions"},{"Start":"10:06.690 ","End":"10:10.110","Text":"for the integrals of cosine squared Theta"},{"Start":"10:10.110 ","End":"10:13.620","Text":"and cosine to the 4th Theta from the formula sheet."},{"Start":"10:13.620 ","End":"10:16.710","Text":"I\u0027ll do that but at the end,"},{"Start":"10:16.710 ","End":"10:19.320","Text":"I\u0027ll also show you how to derive them in case you\u0027re not"},{"Start":"10:19.320 ","End":"10:24.070","Text":"allowed to bring formula sheets into the exam."},{"Start":"10:24.260 ","End":"10:28.110","Text":"I just pulled these 2 out from the formula sheet,"},{"Start":"10:28.110 ","End":"10:32.235","Text":"the indefinite integral of cosine squared and cosine to the 4th."},{"Start":"10:32.235 ","End":"10:34.770","Text":"Like I said, at the end,"},{"Start":"10:34.770 ","End":"10:41.070","Text":"I owe you to show you how I get this in case you are not allowed to use a formula sheet."},{"Start":"10:41.070 ","End":"10:44.065","Text":"I\u0027m going to continue over here."},{"Start":"10:44.065 ","End":"10:49.680","Text":"V equals 2 of these minus 1 of this,"},{"Start":"10:49.680 ","End":"10:51.570","Text":"I don\u0027t need the constant."},{"Start":"10:51.570 ","End":"10:54.060","Text":"Let\u0027s combine like terms."},{"Start":"10:54.060 ","End":"10:55.290","Text":"First of all, notice that here,"},{"Start":"10:55.290 ","End":"10:58.260","Text":"I have Theta and here, I have Theta."},{"Start":"10:58.260 ","End":"11:00.525","Text":"Here, I have sine 2 Theta,"},{"Start":"11:00.525 ","End":"11:02.565","Text":"here, I have sine 2 Theta."},{"Start":"11:02.565 ","End":"11:05.730","Text":"I want to take 2 of these minus 1 of those."},{"Start":"11:05.730 ","End":"11:13.500","Text":"For theta, 2 of these minus that is twice a 1/2 is 1 minus 3/8 is 5/8."},{"Start":"11:13.500 ","End":"11:16.830","Text":"I must remember the 1/4."},{"Start":"11:16.830 ","End":"11:22.200","Text":"So it\u0027s 1/4, 5/8 Theta."},{"Start":"11:22.200 ","End":"11:25.800","Text":"Then let\u0027s see, the sine 2 Theta,"},{"Start":"11:25.800 ","End":"11:34.665","Text":"2 of these minus 1 of those is 1/4 sine of 2 Theta."},{"Start":"11:34.665 ","End":"11:38.955","Text":"This one is on its own but it\u0027s minus."},{"Start":"11:38.955 ","End":"11:46.660","Text":"Minus 1 over 32 sine of 4 Theta."},{"Start":"11:47.480 ","End":"11:53.745","Text":"This, I have to take between the limits as they were,"},{"Start":"11:53.745 ","End":"12:00.310","Text":"minus Pi over 2 to Pi over 2."},{"Start":"12:01.370 ","End":"12:04.560","Text":"I get 1/4. Now, let\u0027s see."},{"Start":"12:04.560 ","End":"12:06.915","Text":"First of all, let\u0027s do the Pi over 2."},{"Start":"12:06.915 ","End":"12:10.200","Text":"If Theta is Pi over 2, here,"},{"Start":"12:10.200 ","End":"12:16.070","Text":"I have 5Pi over 16,"},{"Start":"12:16.070 ","End":"12:18.200","Text":"Thetas Pi over 2,"},{"Start":"12:18.200 ","End":"12:20.785","Text":"2 Thetas Pi, that\u0027s 0,"},{"Start":"12:20.785 ","End":"12:25.365","Text":"and 4 Theta is 2 Pi,"},{"Start":"12:25.365 ","End":"12:28.540","Text":"also the sign of that is 0."},{"Start":"12:29.360 ","End":"12:32.295","Text":"That\u0027s for the Pi over 2."},{"Start":"12:32.295 ","End":"12:34.980","Text":"For the minus Pi over 2,"},{"Start":"12:34.980 ","End":"12:37.470","Text":"when I plug in minus Pi over 2,"},{"Start":"12:37.470 ","End":"12:42.120","Text":"I just get everything like here but with opposite signs."},{"Start":"12:42.120 ","End":"12:46.360","Text":"If I summarize, I have"},{"Start":"12:46.970 ","End":"12:54.480","Text":"5Pi over 16 and another 5Pi over 16 is 5Pi over 8."},{"Start":"12:54.480 ","End":"13:01.180","Text":"It\u0027s 5Pi over 8 times 4 over 32."},{"Start":"13:02.090 ","End":"13:05.684","Text":"It looks like that\u0027s the answer."},{"Start":"13:05.684 ","End":"13:09.180","Text":"I just want to make a room. Well, 2 things."},{"Start":"13:09.180 ","End":"13:11.745","Text":"First of all, a remark,"},{"Start":"13:11.745 ","End":"13:14.040","Text":"how we might have taken a shortcut."},{"Start":"13:14.040 ","End":"13:19.305","Text":"Secondly, I\u0027ll also be doing the IoU for these 2 integrals."},{"Start":"13:19.305 ","End":"13:23.340","Text":"I could have saved a bit of work here by noticing"},{"Start":"13:23.340 ","End":"13:26.999","Text":"that the thing to be integrated is an even function,"},{"Start":"13:26.999 ","End":"13:30.000","Text":"this 2 cosine squared minus cosine to the 4th."},{"Start":"13:30.000 ","End":"13:32.730","Text":"If I put Theta instead of minus Theta,"},{"Start":"13:32.730 ","End":"13:35.745","Text":"it\u0027s the same thing and these are symmetrical limits."},{"Start":"13:35.745 ","End":"13:42.630","Text":"In general, the integral from minus something to plus something of"},{"Start":"13:42.630 ","End":"13:48.765","Text":"an even function of x dx is just"},{"Start":"13:48.765 ","End":"13:55.140","Text":"twice the integral from 0 to a of the same function."},{"Start":"13:55.140 ","End":"13:58.830","Text":"I\u0027ll call it even dx."},{"Start":"13:58.830 ","End":"14:03.885","Text":"Having a limit of 0 is easier than having a limit of minus a, usually."},{"Start":"14:03.885 ","End":"14:06.765","Text":"I could have used that to save a bit of time."},{"Start":"14:06.765 ","End":"14:14.325","Text":"The last thing I have to do if you want to stay is to show you how I got this and this,"},{"Start":"14:14.325 ","End":"14:17.910","Text":"how I would get it if I didn\u0027t have a formula sheet."},{"Start":"14:17.910 ","End":"14:21.495","Text":"Now, let\u0027s do these 2 integrals."},{"Start":"14:21.495 ","End":"14:27.370","Text":"I\u0027ll start with the first one with the cosine squared."},{"Start":"14:28.490 ","End":"14:36.675","Text":"The integral of cosine squared Theta d Theta."},{"Start":"14:36.675 ","End":"14:40.260","Text":"For this, I\u0027ll use a trigonometric identity."},{"Start":"14:40.260 ","End":"14:48.280","Text":"That cosine squared Theta is 1/2 of 1 plus cosine 2 Theta."},{"Start":"14:48.280 ","End":"14:51.635","Text":"I\u0027m assuming, you know trigonometrical identities,"},{"Start":"14:51.635 ","End":"14:53.810","Text":"I can\u0027t do all the mathematics from scratch."},{"Start":"14:53.810 ","End":"14:57.890","Text":"Let\u0027s assume that you have at least some trigonometric identities."},{"Start":"14:57.890 ","End":"15:01.160","Text":"I can take the half outside."},{"Start":"15:01.160 ","End":"15:04.925","Text":"I\u0027ll just imagine that it\u0027s written here."},{"Start":"15:04.925 ","End":"15:09.640","Text":"I get 1/2."},{"Start":"15:09.640 ","End":"15:18.044","Text":"The integral of 1 d Theta is just the Theta and the integral of cosine 2 Theta."},{"Start":"15:18.044 ","End":"15:22.260","Text":"The integral of cosine Theta would be sine Theta."},{"Start":"15:22.260 ","End":"15:25.005","Text":"We start off with sine of 2 Theta,"},{"Start":"15:25.005 ","End":"15:27.060","Text":"but because of the inner derivative,"},{"Start":"15:27.060 ","End":"15:29.985","Text":"this is a linear function of Theta,"},{"Start":"15:29.985 ","End":"15:34.140","Text":"so I need the coefficient or the derivative is 2,"},{"Start":"15:34.140 ","End":"15:36.030","Text":"I need to divide by the 2."},{"Start":"15:36.030 ","End":"15:42.310","Text":"I have an extra half here and plus the constant."},{"Start":"15:42.950 ","End":"15:47.100","Text":"Finally, I just collected together and say, okay,"},{"Start":"15:47.100 ","End":"15:55.155","Text":"this is 1/2 Theta plus 1/4 sine 2 Theta plus constant."},{"Start":"15:55.155 ","End":"15:57.135","Text":"This is what we have,"},{"Start":"15:57.135 ","End":"16:01.065","Text":"what we produced from the formula sheet here, this is fine."},{"Start":"16:01.065 ","End":"16:03.240","Text":"Now, let\u0027s get to the other one."},{"Start":"16:03.240 ","End":"16:09.690","Text":"The integral of cosine to the 4th Theta d Theta."},{"Start":"16:09.690 ","End":"16:11.940","Text":"There\u0027s something I didn\u0027t mention,"},{"Start":"16:11.940 ","End":"16:13.095","Text":"I guess it\u0027s obvious."},{"Start":"16:13.095 ","End":"16:14.370","Text":"In the formula sheets,"},{"Start":"16:14.370 ","End":"16:15.810","Text":"they don\u0027t use the letter Theta,"},{"Start":"16:15.810 ","End":"16:21.090","Text":"they would use the letter x. I would replace Theta by x,"},{"Start":"16:21.090 ","End":"16:24.360","Text":"you\u0027d look up the integral of cosine squared x dx."},{"Start":"16:24.360 ","End":"16:29.715","Text":"When I copied it, I just switched all the x\u0027s to Thetas."},{"Start":"16:29.715 ","End":"16:32.340","Text":"I\u0027ll continue here with Theta though really,"},{"Start":"16:32.340 ","End":"16:36.540","Text":"I could have also done this with x, doesn\u0027t really matter."},{"Start":"16:36.540 ","End":"16:39.720","Text":"For this one, what we\u0027re going to do is again,"},{"Start":"16:39.720 ","End":"16:45.090","Text":"use trigonometric formulas, but they already have the formula for cosine squared here."},{"Start":"16:45.090 ","End":"16:49.890","Text":"What I\u0027ll do is use this formula and square it."},{"Start":"16:49.890 ","End":"16:54.930","Text":"What I have is 1/2."},{"Start":"16:54.930 ","End":"16:58.680","Text":"I\u0027ll just take this thing that\u0027s here and square it."},{"Start":"16:58.680 ","End":"17:05.685","Text":"Already, this time, put the quarter in front of the integral and this thing squared is 1."},{"Start":"17:05.685 ","End":"17:07.980","Text":"I\u0027ll write it, first of all, like this,"},{"Start":"17:07.980 ","End":"17:11.730","Text":"cosine 2 Theta squared d Theta."},{"Start":"17:11.730 ","End":"17:16.980","Text":"Now, we\u0027ll actually do the squaring which is 1 plus twice this times this is"},{"Start":"17:16.980 ","End":"17:24.374","Text":"2 cosine 2 Theta plus this thing squared is cosine squared 2 Theta."},{"Start":"17:24.374 ","End":"17:26.655","Text":"All this d Theta."},{"Start":"17:26.655 ","End":"17:28.575","Text":"Yeah, I guess should\u0027ve been working with x\u0027s,"},{"Start":"17:28.575 ","End":"17:31.030","Text":"but I like the letter Theta."},{"Start":"17:31.640 ","End":"17:36.840","Text":"Then we get 1/4 times."},{"Start":"17:36.840 ","End":"17:41.520","Text":"The integral of 1 is just Theta."},{"Start":"17:41.520 ","End":"17:48.900","Text":"The integral of cosine 2 Theta is sine 2 Theta over 2."},{"Start":"17:48.900 ","End":"17:53.655","Text":"This 2 disappears and I just get sine 2 Theta."},{"Start":"17:53.655 ","End":"17:55.320","Text":"You can check, differentiate this,"},{"Start":"17:55.320 ","End":"17:58.200","Text":"and you get cosine 2 Theta but times the 2."},{"Start":"17:58.200 ","End":"18:05.550","Text":"Now, this the last part here which I have to complete."},{"Start":"18:05.550 ","End":"18:10.305","Text":"I want to do this one here at the side."},{"Start":"18:10.305 ","End":"18:13.245","Text":"This integral, as I said,"},{"Start":"18:13.245 ","End":"18:15.705","Text":"I put asterisk, I\u0027ll do it at the side."},{"Start":"18:15.705 ","End":"18:18.630","Text":"Just the cosine squared 2 Theta part."},{"Start":"18:18.630 ","End":"18:25.725","Text":"The integral of cosine squared 2 Theta d Theta will equal."},{"Start":"18:25.725 ","End":"18:32.130","Text":"Now, if I use the formula for cosine squared Theta and just replace Theta by 2 Theta,"},{"Start":"18:32.130 ","End":"18:36.800","Text":"that will be okay as long as I remember to divide by 2 at the end,"},{"Start":"18:36.800 ","End":"18:40.745","Text":"so I\u0027ll just let you remember it by writing 1/2."},{"Start":"18:40.745 ","End":"18:43.825","Text":"Then I\u0027ll use this formula here,"},{"Start":"18:43.825 ","End":"18:46.035","Text":"or if you like from here,"},{"Start":"18:46.035 ","End":"18:48.765","Text":"with 2 Theta instead of Theta."},{"Start":"18:48.765 ","End":"18:53.575","Text":"I have 1/2 of 1/2."},{"Start":"18:53.575 ","End":"18:59.150","Text":"I\u0027m copying and substituting 2 Theta instead of Theta because I have 2 Theta here."},{"Start":"18:59.150 ","End":"19:02.885","Text":"It\u0027s 1/2 times 2 Theta"},{"Start":"19:02.885 ","End":"19:11.100","Text":"plus 1/4 sine of 4 Theta,"},{"Start":"19:11.450 ","End":"19:15.490","Text":"because Theta is replaced by 2 Theta."},{"Start":"19:15.950 ","End":"19:24.230","Text":"We get 1/2 Theta because 1/2 times 1/2 times 2 is 1/2."},{"Start":"19:24.230 ","End":"19:26.420","Text":"Then 1/2 times 1/4 is 1/8."},{"Start":"19:26.420 ","End":"19:32.890","Text":"Plus 1/8 sine 4 Theta."},{"Start":"19:32.890 ","End":"19:36.480","Text":"Now, this bit, this is the asterisk,"},{"Start":"19:36.480 ","End":"19:38.650","Text":"I\u0027ll put it here."},{"Start":"19:38.930 ","End":"19:48.970","Text":"Plus 1/2 Theta plus 1/8 sine 4 Theta."},{"Start":"19:52.400 ","End":"19:56.430","Text":"Let\u0027s open up the brackets."},{"Start":"19:56.430 ","End":"20:04.425","Text":"Theta plus 1/2 Theta is 1 1/2 Theta."},{"Start":"20:04.425 ","End":"20:08.835","Text":"If I divide that by 4 over here, this equals,"},{"Start":"20:08.835 ","End":"20:14.445","Text":"like I said, 1 1/2 divided by 4 is 3/8 Theta."},{"Start":"20:14.445 ","End":"20:23.505","Text":"Now, let\u0027s see, plus sine 2 Theta with 1/4 sine 2 Theta."},{"Start":"20:23.505 ","End":"20:33.045","Text":"The 1/8 with the 1/4 gives me 1 over 32 sine 4 Theta."},{"Start":"20:33.045 ","End":"20:36.210","Text":"If you want to, plus C. Now,"},{"Start":"20:36.210 ","End":"20:39.525","Text":"let\u0027s compare this to this."},{"Start":"20:39.525 ","End":"20:42.315","Text":"Slightly different order, that\u0027s all."},{"Start":"20:42.315 ","End":"20:44.700","Text":"3/8 Theta is here and here,"},{"Start":"20:44.700 ","End":"20:46.800","Text":"1/4 sine 2 Theta here,"},{"Start":"20:46.800 ","End":"20:48.345","Text":"1 over 32 sine."},{"Start":"20:48.345 ","End":"20:50.325","Text":"That\u0027s okay then also,"},{"Start":"20:50.325 ","End":"20:55.130","Text":"that\u0027s also been verified and that\u0027s settled my IoU,"},{"Start":"20:55.130 ","End":"20:57.390","Text":"so we are done."}],"ID":8713}],"Thumbnail":null,"ID":4969}]

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