WEBVTT 00:00:00.000 --> 00:00:09.050 CHRISTINE BREINER: Welcome back to recitation. 00:00:09.050 --> 00:00:11.860 In this video I want to show how we can use change of 00:00:11.860 --> 00:00:15.030 variables to polar coordinates in particular to evaluate an 00:00:15.030 --> 00:00:17.940 integral that, without the change in variables, we don't 00:00:17.940 --> 00:00:19.770 have the techniques to do. 00:00:19.770 --> 00:00:22.490 So I'm going to show us how to evaluate the integral from 00:00:22.490 --> 00:00:26.210 minus infinity to infinity of e to the minus x squared dx. 00:00:26.210 --> 00:00:28.400 And I'll just point out that if you do anything in 00:00:28.400 --> 00:00:32.520 probability you will see this integral a great deal. 00:00:32.520 --> 00:00:35.290 So this is a distribution and you'll see it a great deal if 00:00:35.290 --> 00:00:37.540 you ever do anything in probability theory. 00:00:37.540 --> 00:00:40.100 But how are we going to use polar 00:00:40.100 --> 00:00:41.380 coordinates to evaluate this? 00:00:41.380 --> 00:00:46.240 Well, the object is going to be to introduce a little bit 00:00:46.240 --> 00:00:51.930 more into this integral, so that when I actually introduce 00:00:51.930 --> 00:00:55.010 that in I'm going to have an r squared term in the exponent 00:00:55.010 --> 00:00:57.440 and I'm going to have --of course by the change of 00:00:57.440 --> 00:00:59.340 variables in the Jacobian-- 00:00:59.340 --> 00:01:00.650 multiplied by r. 00:01:00.650 --> 00:01:02.340 And that's what's going to save us. 00:01:02.340 --> 00:01:05.980 So let me actually just get right to it. 00:01:05.980 --> 00:01:09.310 I'm going to call this integral, this quantity, 00:01:09.310 --> 00:01:12.620 capital I. We'll just call this quantity capital I. And 00:01:12.620 --> 00:01:14.010 then what I'm going to do is introduce 00:01:14.010 --> 00:01:14.830 a couple more things. 00:01:14.830 --> 00:01:18.820 So let me write it down, and then I will justify that it's 00:01:18.820 --> 00:01:21.250 reasonable to even look at this thing. 00:01:21.250 --> 00:01:22.635 And that it will give us something useful. 00:01:29.780 --> 00:01:33.680 OK, so I just took this integral and what I've done is 00:01:33.680 --> 00:01:39.050 I've now taken this quantity and I've multiplied by an e to 00:01:39.050 --> 00:01:41.140 the minus y squared. 00:01:41.140 --> 00:01:44.140 And then I'm now integrating for minus infinity to infinity 00:01:44.140 --> 00:01:45.380 and dy also. 00:01:45.380 --> 00:01:47.920 And what I want to point out, that maybe isn't immediately 00:01:47.920 --> 00:01:50.480 obvious from the way it's written here, but will be 00:01:50.480 --> 00:01:53.060 immediately obvious when I write it a different way, is 00:01:53.060 --> 00:01:55.190 that this quantity is actually just the 00:01:55.190 --> 00:01:57.310 square of this quantity. 00:01:57.310 --> 00:02:00.210 So it's actually just two of these multiplied together. 00:02:00.210 --> 00:02:02.150 And let me point out why that is. 00:02:02.150 --> 00:02:05.510 Because this term right here can be rewritten as e to the 00:02:05.510 --> 00:02:09.910 minus x squared, e to the minus y squared. 00:02:09.910 --> 00:02:13.490 And then I have two integrals, which are for minus infinity 00:02:13.490 --> 00:02:14.300 to infinity both. 00:02:14.300 --> 00:02:16.975 Sorry, that one looks a little off kilter. 00:02:16.975 --> 00:02:20.730 But those two integrals, which the bounds are minus infinity 00:02:20.730 --> 00:02:22.040 to infinity for both. 00:02:22.040 --> 00:02:24.950 And then I have a dx and a dy. 00:02:24.950 --> 00:02:28.070 Now the reason that it is just I squared-- 00:02:28.070 --> 00:02:30.170 this quantity is just I squared-- is because if you 00:02:30.170 --> 00:02:34.770 notice, this first integral, the inside integral, this is 00:02:34.770 --> 00:02:36.850 independent of x. 00:02:36.850 --> 00:02:38.670 So I can move it out. 00:02:38.670 --> 00:02:41.390 So let me actually come up here and write that down. 00:02:41.390 --> 00:02:43.650 So I can move it out and I can actually rewrite the integral. 00:02:43.650 --> 00:02:46.240 And I'll put the bounds here, because it's high up enough 00:02:46.240 --> 00:02:47.970 that I can actually write it. 00:02:47.970 --> 00:02:49.530 e to the minus y squared. 00:02:49.530 --> 00:02:53.560 And then integral for minus infinity to infinity. 00:02:53.560 --> 00:02:57.950 e to the minus x squared dx dy. 00:02:57.950 --> 00:02:58.640 Right? 00:02:58.640 --> 00:03:00.860 So let's point out again what I did. 00:03:00.860 --> 00:03:05.975 This quantity I circled was the constant when I considered 00:03:05.975 --> 00:03:06.920 x the variable. 00:03:06.920 --> 00:03:07.180 Right? 00:03:07.180 --> 00:03:09.880 In terms of x, this is the constant, so I can move it in 00:03:09.880 --> 00:03:12.840 front of that integral, which is the dx integral. 00:03:12.840 --> 00:03:14.280 And so now what do I see? 00:03:14.280 --> 00:03:18.430 Well, this quantity is what I've called I. And so now 00:03:18.430 --> 00:03:19.480 that's a constant. 00:03:19.480 --> 00:03:21.660 So I can move that out to the front. 00:03:21.660 --> 00:03:22.750 That's a constant. 00:03:22.750 --> 00:03:25.370 That's I, as designated. 00:03:25.370 --> 00:03:27.870 And so now I have the integral for minus infinity to infinity 00:03:27.870 --> 00:03:31.560 of either the minus y squared dy. 00:03:31.560 --> 00:03:35.010 And then that again is another I. Because if you notice, I 00:03:35.010 --> 00:03:37.990 mean, this was either the minus x squared dx-- 00:03:37.990 --> 00:03:40.510 and so if I keep this as either the minus y squared dy, 00:03:40.510 --> 00:03:42.870 it's going to be exactly the same value. 00:03:42.870 --> 00:03:46.560 So this is actually equal to I squared. 00:03:46.560 --> 00:03:49.260 So the point I want to make, if you come back over here to 00:03:49.260 --> 00:03:57.110 this integral, is that this quantity I'm going to be able 00:03:57.110 --> 00:03:58.430 to integrate very easily. 00:03:58.430 --> 00:04:01.090 And when I do that, if I take the square root, I'm going to 00:04:01.090 --> 00:04:03.270 get this value. 00:04:03.270 --> 00:04:05.010 Because I just showed this thing in the box 00:04:05.010 --> 00:04:07.410 was equal to I squared. 00:04:07.410 --> 00:04:10.140 So let me write it down. 00:04:10.140 --> 00:04:13.250 On this last part of the board I will write down that thing 00:04:13.250 --> 00:04:14.940 again and then we'll actually evaluate it and 00:04:14.940 --> 00:04:17.592 we'll see what we get. 00:04:17.592 --> 00:04:18.860 So let me rewrite. 00:04:18.860 --> 00:04:20.110 This is the thing we want to evaluate. 00:04:30.550 --> 00:04:32.730 And I'm going to remind myself that 00:04:32.730 --> 00:04:34.610 that's equal to I squared. 00:04:34.610 --> 00:04:36.740 I squared is equal to this quantity. 00:04:36.740 --> 00:04:38.560 And I mentioned at the beginning that we're going to 00:04:38.560 --> 00:04:39.200 use a trick. 00:04:39.200 --> 00:04:41.020 We're going to change variables into polar 00:04:41.020 --> 00:04:42.240 coordinates. 00:04:42.240 --> 00:04:45.700 And so notice that the region we're integrating over is the 00:04:45.700 --> 00:04:47.700 entire xy plane. 00:04:47.700 --> 00:04:50.220 And in polar coordinates, what's that going to be? 00:04:50.220 --> 00:04:54.240 Theta is going to run from 0 all the way around to 2 pi, 00:04:54.240 --> 00:04:57.160 and r is going to run from 0 to infinity. 00:04:57.160 --> 00:04:59.660 And so those will be our bounds for r and theta. 00:04:59.660 --> 00:05:03.550 Because if you want to get the entire xy plane, in terms of r 00:05:03.550 --> 00:05:06.750 and theta, that is what you have to do. 00:05:06.750 --> 00:05:09.900 And I want to mention also, what is this quantity going to 00:05:09.900 --> 00:05:12.220 become in terms of r and theta? 00:05:12.220 --> 00:05:14.360 Well, notice that this is e to the minus x 00:05:14.360 --> 00:05:15.510 squared minus y squared. 00:05:15.510 --> 00:05:17.990 It's just really e to the minus quantity x 00:05:17.990 --> 00:05:19.030 squared plus y squared. 00:05:19.030 --> 00:05:21.330 It's really e to the minus r squared. 00:05:21.330 --> 00:05:23.790 So what we get when we do a change of 00:05:23.790 --> 00:05:25.760 variables, is we do-- 00:05:25.760 --> 00:05:30.500 I'll put the theta on the inside actually, because 00:05:30.500 --> 00:05:34.420 that'll be easy to do first. So the r is going 00:05:34.420 --> 00:05:35.910 to be on the outside. 00:05:35.910 --> 00:05:38.100 The theta is going to be on the inside. 00:05:38.100 --> 00:05:40.870 And then e to the minus r squared, that's a direct 00:05:40.870 --> 00:05:41.420 substitution. 00:05:41.420 --> 00:05:44.220 x squared plus y squared equals r squared. 00:05:44.220 --> 00:05:47.330 And then I get r d theta d r. 00:05:47.330 --> 00:05:50.450 And this comes from the Jacobian that you computed-- 00:05:50.450 --> 00:05:53.160 I believe in lecture even-- 00:05:53.160 --> 00:05:55.810 to show how you change from xy 00:05:55.810 --> 00:05:57.660 variables to r theta variables. 00:05:57.660 --> 00:05:59.950 And I put the d theta first here because I wanted to 00:05:59.950 --> 00:06:02.770 integrate in theta first. And the reason I want to integrate 00:06:02.770 --> 00:06:04.400 in theta first is notice that-- 00:06:04.400 --> 00:06:06.610 because nothing here depends on theta-- 00:06:06.610 --> 00:06:09.400 all I pick up is a 2 pi. 00:06:09.400 --> 00:06:09.730 I just pick up --when I integrate-- 00:06:09.730 --> 00:06:10.060 I get a theta. 00:06:10.060 --> 00:06:13.570 I evaluate it at 2 pi, and then I evaluate it at 0 and I 00:06:13.570 --> 00:06:14.740 take the difference. 00:06:14.740 --> 00:06:17.620 And so if I do that line all I get is-- well, I should 00:06:17.620 --> 00:06:18.890 probably write it in front-- 00:06:18.890 --> 00:06:22.110 all I get is 2 pi from the theta, and then the integral 00:06:22.110 --> 00:06:24.410 from 0 to infinity. 00:06:24.410 --> 00:06:28.590 e to the minus r squared times r dr. 00:06:28.590 --> 00:06:32.560 Now this is a much easier quantity to evaluate than e to 00:06:32.560 --> 00:06:34.790 the minus x squared dx. 00:06:34.790 --> 00:06:36.660 Because now we have an r here. 00:06:36.660 --> 00:06:39.810 So now it's a natural substitution type of problem. 00:06:39.810 --> 00:06:41.110 And we can do it right away. 00:06:41.110 --> 00:06:43.380 I'm going to write it down and then we'll check and make sure 00:06:43.380 --> 00:06:44.760 I didn't make a mistake. 00:06:44.760 --> 00:06:51.460 We should get something like e to the minus r squared over 2 00:06:51.460 --> 00:06:53.740 with a negative sign in front. 00:06:53.740 --> 00:06:55.570 So let me make sure when I take this derivative-- 00:06:55.570 --> 00:06:57.570 you could just do a substitution to check, but you 00:06:57.570 --> 00:06:59.480 should get exactly this kind of thing. 00:06:59.480 --> 00:07:02.140 When I take the derivative of e to the minus r squared, I 00:07:02.140 --> 00:07:06.560 get a negative 2r and then-- 00:07:06.560 --> 00:07:07.750 did I do something wrong here? 00:07:07.750 --> 00:07:08.790 Oh yeah. 00:07:08.790 --> 00:07:09.310 I'm good. 00:07:09.310 --> 00:07:11.780 I get a negative 2r. 00:07:11.780 --> 00:07:14.770 And so the negatives kill off, the 2s divide out, and I get 00:07:14.770 --> 00:07:17.880 my e to the minus r squared times r. 00:07:17.880 --> 00:07:19.040 So that's good. 00:07:19.040 --> 00:07:22.290 And now I have to evaluate it at the bounds. 00:07:22.290 --> 00:07:28.000 So again, this was much easier to evaluate if I do a 00:07:28.000 --> 00:07:30.900 substitution and I let r-- 00:07:30.900 --> 00:07:33.070 I think I want to let r squared equal u or 00:07:33.070 --> 00:07:34.080 something like this. 00:07:34.080 --> 00:07:34.670 I don't even know. 00:07:34.670 --> 00:07:36.590 I just did the substitution without thinking about it. 00:07:36.590 --> 00:07:38.610 So you can figure out what you need to substitute. 00:07:38.610 --> 00:07:40.050 But this is what you get. 00:07:40.050 --> 00:07:42.430 And so let me actually just evaluate at the bounds and see 00:07:42.430 --> 00:07:43.380 what happens. 00:07:43.380 --> 00:07:47.060 So as r goes to infinity, I get e to the minus r squared. 00:07:47.060 --> 00:07:51.200 As r goes to infinity, e to the minus r squared goes to 0. 00:07:51.200 --> 00:07:54.000 And so the first term is 0 when I evaluate. 00:07:54.000 --> 00:07:59.480 And then so the second term I get is, I do 0 minus 2 pi 00:07:59.480 --> 00:08:02.380 times-- if I evaluate at e to the 0 I get 1-- 00:08:02.380 --> 00:08:07.350 so I get negative 1 over 2. 00:08:07.350 --> 00:08:10.370 The negative 1 comes from e to the 0. 00:08:10.370 --> 00:08:12.490 And then I divide by 2 so I get my 2 there. 00:08:12.490 --> 00:08:15.200 So a negative and a negative gives me a positive. 00:08:15.200 --> 00:08:16.380 2 divided by 2. 00:08:16.380 --> 00:08:18.870 And so the whole thing equals pi. 00:08:18.870 --> 00:08:19.160 Whew! 00:08:19.160 --> 00:08:20.350 Just enough room. 00:08:20.350 --> 00:08:23.010 And so I just want to remind us where we came from. 00:08:23.010 --> 00:08:25.790 We came from pi-- 00:08:25.790 --> 00:08:29.380 we wanted to show was equal-- well, the pi, 00:08:29.380 --> 00:08:30.200 where did it come from? 00:08:30.200 --> 00:08:34.380 It's equal if we go all the way back up to I squared. 00:08:34.380 --> 00:08:37.310 And so we wanted to show what I was equal to. 00:08:37.310 --> 00:08:40.620 I then is equal to just the square root of pi. 00:08:40.620 --> 00:08:42.850 So I can come back over to where I started 00:08:42.850 --> 00:08:44.230 which is over here. 00:08:44.230 --> 00:08:49.270 And I can say this is equal to the square root of pi. 00:08:49.270 --> 00:08:52.080 And again, I just want to mention what it came down to. 00:08:52.080 --> 00:08:55.870 It came down to introducing a little bit more. 00:08:55.870 --> 00:08:59.040 Essentially we took something that was a single variable 00:08:59.040 --> 00:09:01.720 problem, we made it a multivariable problem. 00:09:01.720 --> 00:09:04.630 But what that allowed us to do is to do a change of 00:09:04.630 --> 00:09:07.730 coordinates from xy into the polar coordinate system. 00:09:07.730 --> 00:09:10.190 And the trick here-- as you come through-- the trick where 00:09:10.190 --> 00:09:12.760 it actually happens is right here. 00:09:12.760 --> 00:09:15.120 Is because this is easy to integrate now 00:09:15.120 --> 00:09:16.870 that I have an r here. 00:09:16.870 --> 00:09:19.690 If I didn't have-- my problem initially was I didn't have an 00:09:19.690 --> 00:09:22.330 x when I was trying to find an antiderivative. 00:09:22.330 --> 00:09:24.140 But when I do the change of variables, I get an e to the 00:09:24.140 --> 00:09:25.250 minus r squared. 00:09:25.250 --> 00:09:27.900 And then with an r here, that's much easier to find an 00:09:27.900 --> 00:09:29.470 antiderivative. 00:09:29.470 --> 00:09:30.380 OK. 00:09:30.380 --> 00:09:33.250 So I think that is where I'll stop.