1 00:00:00,000 --> 00:00:09,050 CHRISTINE BREINER: Welcome back to recitation. 2 00:00:09,050 --> 00:00:12,390 In this video I want to show how we can use change of variables, 3 00:00:12,390 --> 00:00:14,230 to polar coordinates in particular, 4 00:00:14,230 --> 00:00:16,620 to evaluate an integral that, without the change 5 00:00:16,620 --> 00:00:19,770 in variables, we don't have the techniques to do. 6 00:00:19,770 --> 00:00:22,490 So I'm going to show us how to evaluate the integral from 7 00:00:22,490 --> 00:00:26,210 minus infinity to infinity of e to the minus x squared dx. 8 00:00:26,210 --> 00:00:29,110 And I'll just point out that if you do anything in probability 9 00:00:29,110 --> 00:00:32,520 you will see this integral a great deal. 10 00:00:32,520 --> 00:00:34,179 So this is a distribution and you'll 11 00:00:34,179 --> 00:00:35,970 see it a great deal if you ever do anything 12 00:00:35,970 --> 00:00:37,540 in probability theory. 13 00:00:37,540 --> 00:00:40,450 But how are we going to use polar coordinates 14 00:00:40,450 --> 00:00:41,380 to evaluate this? 15 00:00:41,380 --> 00:00:47,960 Well, the object is going to be to introduce a little bit more 16 00:00:47,960 --> 00:00:51,484 into this integral, so that when I actually 17 00:00:51,484 --> 00:00:53,150 introduce that in I'm going to have an r 18 00:00:53,150 --> 00:00:56,890 squared term in the exponent and I'm going to have-- of course 19 00:00:56,890 --> 00:01:00,650 by the change of variables in the Jacobian-- multiplied by r. 20 00:01:00,650 --> 00:01:02,340 And that's what's going to save us. 21 00:01:02,340 --> 00:01:05,980 So let me actually just get right to it. 22 00:01:05,980 --> 00:01:09,310 I'm going to call this integral, this quantity, 23 00:01:09,310 --> 00:01:12,164 capital I. We'll just call this quantity capital I. 24 00:01:12,164 --> 00:01:14,830 And then what I'm going to do is introduce a couple more things. 25 00:01:14,830 --> 00:01:17,020 So let me write it down, and then I 26 00:01:17,020 --> 00:01:20,050 will justify that it's reasonable to even look 27 00:01:20,050 --> 00:01:20,885 at this thing. 28 00:01:20,885 --> 00:01:22,635 And that it will give us something useful. 29 00:01:29,780 --> 00:01:33,570 OK, so I just took this integral and what I've done 30 00:01:33,570 --> 00:01:38,970 is I've now taken this quantity and I've multiplied by an e 31 00:01:38,970 --> 00:01:41,140 to the minus y squared. 32 00:01:41,140 --> 00:01:43,590 And then I'm now integrating from minus infinity 33 00:01:43,590 --> 00:01:45,380 to infinity in dy also. 34 00:01:45,380 --> 00:01:47,920 And what I want to point out, that maybe isn't immediately 35 00:01:47,920 --> 00:01:49,900 obvious from the way it's written here, 36 00:01:49,900 --> 00:01:51,450 but will be immediately obvious when 37 00:01:51,450 --> 00:01:54,830 I write it a different way, is that this quantity is actually 38 00:01:54,830 --> 00:01:57,310 just the square of this quantity. 39 00:01:57,310 --> 00:02:00,210 So it's actually just two of these multiplied together. 40 00:02:00,210 --> 00:02:02,150 And let me point out why that is. 41 00:02:02,150 --> 00:02:05,840 Because this term right here can be rewritten as e to the minus 42 00:02:05,840 --> 00:02:09,910 x squared, e to the minus y squared. 43 00:02:09,910 --> 00:02:13,040 And then I have two integrals, which are from minus 44 00:02:13,040 --> 00:02:14,300 infinity to infinity both. 45 00:02:14,300 --> 00:02:16,975 Sorry, that one looks a little off kilter. 46 00:02:16,975 --> 00:02:20,270 But those two integrals, which the bounds are minus 47 00:02:20,270 --> 00:02:22,040 infinity to infinity for both. 48 00:02:22,040 --> 00:02:24,950 And then I have a dx and a dy. 49 00:02:24,950 --> 00:02:28,930 Now the reason that it is just I squared-- this quantity is just 50 00:02:28,930 --> 00:02:31,510 I squared-- is because if you notice, 51 00:02:31,510 --> 00:02:34,580 this first integral, the inside integral, this 52 00:02:34,580 --> 00:02:36,850 is independent of x. 53 00:02:36,850 --> 00:02:38,670 So I can move it out. 54 00:02:38,670 --> 00:02:41,109 So let me actually come up here and write that down. 55 00:02:41,109 --> 00:02:43,650 So I can move it out and I can actually rewrite the integral. 56 00:02:43,650 --> 00:02:45,820 And I'll put the bounds here, because it's high 57 00:02:45,820 --> 00:02:47,970 up enough that I can actually write it. 58 00:02:47,970 --> 00:02:49,530 e to the minus y squared. 59 00:02:49,530 --> 00:02:53,560 And then integral from minus infinity to infinity 60 00:02:53,560 --> 00:02:57,950 e to the minus x squared dx dy. 61 00:02:57,950 --> 00:02:58,640 Right? 62 00:02:58,640 --> 00:03:00,860 So let's point out again what I did. 63 00:03:00,860 --> 00:03:04,900 This quantity I circled was the constant when 64 00:03:04,900 --> 00:03:06,680 I considered x the variable. 65 00:03:06,680 --> 00:03:07,180 Right? 66 00:03:07,180 --> 00:03:08,680 In terms of x, this is the constant, 67 00:03:08,680 --> 00:03:11,210 so I can move it in front of that integral, which 68 00:03:11,210 --> 00:03:12,840 is the dx integral. 69 00:03:12,840 --> 00:03:14,280 And so now what do I see? 70 00:03:14,280 --> 00:03:18,010 Well, this quantity is what I've called I. 71 00:03:18,010 --> 00:03:19,480 And so now that's a constant. 72 00:03:19,480 --> 00:03:21,660 So I can move that out to the front. 73 00:03:21,660 --> 00:03:22,750 That's a constant. 74 00:03:22,750 --> 00:03:25,329 That's I, as designated. 75 00:03:25,329 --> 00:03:27,870 And so now I have the integral for minus infinity to infinity 76 00:03:27,870 --> 00:03:31,560 of e to the minus y squared dy. 77 00:03:31,560 --> 00:03:34,940 And then that again is another I. Because if you notice, 78 00:03:34,940 --> 00:03:37,990 I mean, this was e to the minus x squared dx, 79 00:03:37,990 --> 00:03:40,510 and so if I keep this as e to the minus y squared dy, 80 00:03:40,510 --> 00:03:42,870 it's going to be exactly the same value. 81 00:03:42,870 --> 00:03:46,560 So this is actually equal to I squared. 82 00:03:46,560 --> 00:03:49,120 So the point I want to make, if you come back over here 83 00:03:49,120 --> 00:03:56,670 to this integral, is that this quantity I'm 84 00:03:56,670 --> 00:03:58,430 going to be able to integrate very easily. 85 00:03:58,430 --> 00:04:00,740 And when I do that, if I take the square root, 86 00:04:00,740 --> 00:04:03,194 I'm going to get this value. 87 00:04:03,194 --> 00:04:05,610 Because I just showed this thing in the box was equal to I 88 00:04:05,610 --> 00:04:07,410 squared. 89 00:04:07,410 --> 00:04:10,140 So let me write it down. 90 00:04:10,140 --> 00:04:13,142 On this last part of the board I will write down that thing 91 00:04:13,142 --> 00:04:14,850 again and then we'll actually evaluate it 92 00:04:14,850 --> 00:04:17,592 and we'll see what we get. 93 00:04:17,592 --> 00:04:18,511 So let me rewrite. 94 00:04:18,511 --> 00:04:20,094 This is the thing we want to evaluate. 95 00:04:30,550 --> 00:04:34,610 And I'm going to remind myself that that's equal to I squared. 96 00:04:34,610 --> 00:04:36,549 I squared is equal to this quantity. 97 00:04:36,549 --> 00:04:38,340 And I mentioned at the beginning that we're 98 00:04:38,340 --> 00:04:39,215 going to use a trick. 99 00:04:39,215 --> 00:04:42,240 We're going to change variables into polar coordinates. 100 00:04:42,240 --> 00:04:45,410 And so notice that the region we're integrating over 101 00:04:45,410 --> 00:04:47,700 is the entire xy-plane. 102 00:04:47,700 --> 00:04:50,220 And in polar coordinates, what's that going to be? 103 00:04:50,220 --> 00:04:54,240 Theta is going to run from 0 all the way around to 2 pi, 104 00:04:54,240 --> 00:04:57,160 and r is going to run from 0 to infinity. 105 00:04:57,160 --> 00:04:59,660 And so those will be our bounds for r and theta. 106 00:04:59,660 --> 00:05:02,890 Because if you want to get the entire xy-plane, 107 00:05:02,890 --> 00:05:06,750 in terms of r and theta, that is what you have to do. 108 00:05:06,750 --> 00:05:09,020 And I want to mention also, what is 109 00:05:09,020 --> 00:05:12,220 this quantity going to become in terms of r and theta? 110 00:05:12,220 --> 00:05:14,360 Well, notice that this is e to the minus x 111 00:05:14,360 --> 00:05:15,510 squared minus y squared. 112 00:05:15,510 --> 00:05:17,570 It's just really e to the minus quantity 113 00:05:17,570 --> 00:05:19,030 x squared plus y squared. 114 00:05:19,030 --> 00:05:21,330 It's really e to the minus r squared. 115 00:05:21,330 --> 00:05:24,350 So what we get when we do a change of variables, 116 00:05:24,350 --> 00:05:29,820 is we do-- I'll put the theta on the inside 117 00:05:29,820 --> 00:05:33,300 actually, because that'll be easy to do first. 118 00:05:33,300 --> 00:05:35,910 So the r is going to be on the outside. 119 00:05:35,910 --> 00:05:38,100 The theta is going to be on the inside. 120 00:05:38,100 --> 00:05:40,212 And then e to the minus r squared, 121 00:05:40,212 --> 00:05:41,420 that's a direct substitution. 122 00:05:41,420 --> 00:05:44,220 x squared plus y squared equals r squared. 123 00:05:44,220 --> 00:05:47,330 And then I get r d theta dr. 124 00:05:47,330 --> 00:05:50,450 And this comes from the Jacobian that you computed-- 125 00:05:50,450 --> 00:05:53,980 I believe in lecture even-- to show 126 00:05:53,980 --> 00:05:56,370 how you change from x, y variables 127 00:05:56,370 --> 00:05:57,660 to r, theta variables. 128 00:05:57,660 --> 00:05:59,880 And I put the d theta first here because I wanted 129 00:05:59,880 --> 00:06:01,259 to integrate in theta first. 130 00:06:01,259 --> 00:06:03,050 And the reason I want to integrate in theta 131 00:06:03,050 --> 00:06:06,610 first is notice that-- because nothing here depends on theta-- 132 00:06:06,610 --> 00:06:08,019 all I pick up is a 2 pi. 133 00:06:08,019 --> 00:06:10,060 I just pick up-- when I integrate, I get a theta. 134 00:06:10,060 --> 00:06:13,270 I evaluate it at 2 pi, and then I evaluate it at 0 135 00:06:13,270 --> 00:06:14,740 and I take the difference. 136 00:06:14,740 --> 00:06:17,349 And so if I do that line all I get is-- well, 137 00:06:17,349 --> 00:06:18,890 I should probably write it in front-- 138 00:06:18,890 --> 00:06:21,650 all I get is 2 pi from the theta, and then 139 00:06:21,650 --> 00:06:24,580 the integral from 0 to infinity, e 140 00:06:24,580 --> 00:06:28,590 to the minus r squared times r dr. 141 00:06:28,590 --> 00:06:30,750 Now this is a much easier quantity 142 00:06:30,750 --> 00:06:34,790 to evaluate than e to the minus x squared dx. 143 00:06:34,790 --> 00:06:36,660 Because now we have an r here. 144 00:06:36,660 --> 00:06:39,810 So now it's a natural substitution type of problem. 145 00:06:39,810 --> 00:06:41,110 And we can do it right away. 146 00:06:41,110 --> 00:06:42,630 I'm going to write it down and then 147 00:06:42,630 --> 00:06:44,760 we'll check and make sure I didn't make a mistake. 148 00:06:44,760 --> 00:06:51,460 We should get something like e to the minus r squared over 2 149 00:06:51,460 --> 00:06:53,352 with a negative sign in front. 150 00:06:53,352 --> 00:06:55,810 So let me make sure when I take this derivative-- you could 151 00:06:55,810 --> 00:06:57,790 just do a substitution to check, but you should 152 00:06:57,790 --> 00:06:59,480 get exactly this kind of thing. 153 00:06:59,480 --> 00:07:01,810 When I take the derivative of e to the minus r squared, 154 00:07:01,810 --> 00:07:07,750 I get a negative 2r and then-- did I do something wrong here? 155 00:07:07,750 --> 00:07:08,790 Oh yeah. 156 00:07:08,790 --> 00:07:09,310 I'm good. 157 00:07:09,310 --> 00:07:11,780 I get a negative 2r. 158 00:07:11,780 --> 00:07:14,400 And so the negatives kill off, the 2s divide out, 159 00:07:14,400 --> 00:07:17,880 and I get my e to the minus r squared times r. 160 00:07:17,880 --> 00:07:19,040 So that's good. 161 00:07:19,040 --> 00:07:22,290 And now I have to evaluate it at the bounds. 162 00:07:22,290 --> 00:07:24,680 So again, this was much easier to evaluate 163 00:07:24,680 --> 00:07:30,900 if I do a substitution and I let r-- 164 00:07:30,900 --> 00:07:33,920 I think I want to let r squared equal u or something like this. 165 00:07:33,920 --> 00:07:34,670 I don't even know. 166 00:07:34,670 --> 00:07:36,590 I just did the substitution without thinking about it. 167 00:07:36,590 --> 00:07:38,673 So you can figure out what you need to substitute. 168 00:07:38,673 --> 00:07:40,050 But this is what you get. 169 00:07:40,050 --> 00:07:42,180 And so let me actually just evaluate at the bounds 170 00:07:42,180 --> 00:07:43,380 and see what happens. 171 00:07:43,380 --> 00:07:47,060 So as r goes to infinity, I get e to the minus r squared. 172 00:07:47,060 --> 00:07:51,200 As r goes to infinity, e to the minus r squared goes to 0. 173 00:07:51,200 --> 00:07:54,000 And so the first term is 0 when I evaluate. 174 00:07:54,000 --> 00:07:55,510 And then so the second term I get 175 00:07:55,510 --> 00:08:01,530 is, I do 0 minus 2 pi times-- if I evaluate at e to the 0 176 00:08:01,530 --> 00:08:07,350 I get 1-- so I get negative 1 over 2. 177 00:08:07,350 --> 00:08:10,370 The negative 1 comes from e to the 0. 178 00:08:10,370 --> 00:08:12,490 And then I divide by 2 so I get my 2 there. 179 00:08:12,490 --> 00:08:15,200 So a negative and a negative gives me a positive. 180 00:08:15,200 --> 00:08:16,380 2 divided by 2. 181 00:08:16,380 --> 00:08:18,660 And so the whole thing equals pi. 182 00:08:18,660 --> 00:08:19,160 Whew! 183 00:08:19,160 --> 00:08:20,350 Just enough room. 184 00:08:20,350 --> 00:08:23,010 And so I just want to remind us where we came from. 185 00:08:23,010 --> 00:08:29,242 We came from pi-- we wanted to show was equal-- well, the pi, 186 00:08:29,242 --> 00:08:30,200 where did it come from? 187 00:08:30,200 --> 00:08:34,380 It's equal, if we go all the way back up, to I squared. 188 00:08:34,380 --> 00:08:37,310 And so we wanted to show what I was equal to. 189 00:08:37,310 --> 00:08:40,620 I then is equal to just the square root of pi. 190 00:08:40,620 --> 00:08:44,230 So I can come back over to where I started, which is over here. 191 00:08:44,230 --> 00:08:49,270 And I can say this is equal to the square root of pi. 192 00:08:49,270 --> 00:08:52,080 And again, I just want to mention what it came down to. 193 00:08:52,080 --> 00:08:55,870 It came down to introducing a little bit more. 194 00:08:55,870 --> 00:08:57,580 Essentially we took something that 195 00:08:57,580 --> 00:08:59,960 was a single variable problem, we made it 196 00:08:59,960 --> 00:09:01,720 a multivariable problem. 197 00:09:01,720 --> 00:09:03,930 But what that allowed us to do is 198 00:09:03,930 --> 00:09:05,770 to do a change of coordinates from x, y 199 00:09:05,770 --> 00:09:07,730 into the polar coordinate system. 200 00:09:07,730 --> 00:09:09,980 And the trick here-- as you come through-- the trick 201 00:09:09,980 --> 00:09:12,760 where it actually happens is right here. 202 00:09:12,760 --> 00:09:15,750 Is because this is easy to integrate now that I have an r 203 00:09:15,750 --> 00:09:16,870 here. 204 00:09:16,870 --> 00:09:18,940 If I didn't have-- my problem initially 205 00:09:18,940 --> 00:09:21,150 was I didn't have an x when I was trying 206 00:09:21,150 --> 00:09:22,234 to find an antiderivative. 207 00:09:22,234 --> 00:09:23,816 But when I do the change of variables, 208 00:09:23,816 --> 00:09:25,250 I get an e to the minus r squared. 209 00:09:25,250 --> 00:09:27,280 And then with an r here, that's much easier 210 00:09:27,280 --> 00:09:29,470 to find an antiderivative. 211 00:09:29,470 --> 00:09:30,380 OK. 212 00:09:30,380 --> 00:09:33,250 So I think that is where I'll stop.