1
00:00:00,000 --> 00:00:09,050
CHRISTINE BREINER: Welcome
back to recitation.

2
00:00:09,050 --> 00:00:11,860
In this video I want to show
how we can use change of

3
00:00:11,860 --> 00:00:15,030
variables to polar coordinates
in particular to evaluate an

4
00:00:15,030 --> 00:00:17,940
integral that, without the
change in variables, we don't

5
00:00:17,940 --> 00:00:19,770
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 --> 00:00:28,400
And I'll just point out that
if you do anything in

9
00:00:28,400 --> 00:00:32,520
probability you will see this
integral a great deal.

10
00:00:32,520 --> 00:00:35,290
So this is a distribution and
you'll see it a great deal if

11
00:00:35,290 --> 00:00:37,540
you ever do anything in
probability theory.

12
00:00:37,540 --> 00:00:40,100
But how are we going
to use polar

13
00:00:40,100 --> 00:00:41,380
coordinates to evaluate this?

14
00:00:41,380 --> 00:00:46,240
Well, the object is going to be
to introduce a little bit

15
00:00:46,240 --> 00:00:51,930
more into this integral, so that
when I actually introduce

16
00:00:51,930 --> 00:00:55,010
that in I'm going to have an r
squared term in the exponent

17
00:00:55,010 --> 00:00:57,440
and I'm going to have --of
course by the change of

18
00:00:57,440 --> 00:00:59,340
variables in the Jacobian--

19
00:00:59,340 --> 00:01:00,650
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 --> 00:01:12,620
capital I. We'll just call this
quantity capital I. And

24
00:01:12,620 --> 00:01:14,010
then what I'm going
to do is introduce

25
00:01:14,010 --> 00:01:14,830
a couple more things.

26
00:01:14,830 --> 00:01:18,820
So let me write it down, and
then I will justify that it's

27
00:01:18,820 --> 00:01:21,250
reasonable to even look
at this thing.

28
00:01:21,250 --> 00:01:22,635
And that it will give
us something useful.

29
00:01:22,635 --> 00:01:29,780

30
00:01:29,780 --> 00:01:33,680
OK, so I just took this integral
and what I've done is

31
00:01:33,680 --> 00:01:39,050
I've now taken this quantity and
I've multiplied by an e to

32
00:01:39,050 --> 00:01:41,140
the minus y squared.

33
00:01:41,140 --> 00:01:44,140
And then I'm now integrating for
minus infinity to infinity

34
00:01:44,140 --> 00:01:45,380
and dy also.

35
00:01:45,380 --> 00:01:47,920
And what I want to point out,
that maybe isn't immediately

36
00:01:47,920 --> 00:01:50,480
obvious from the way it's
written here, but will be

37
00:01:50,480 --> 00:01:53,060
immediately obvious when I write
it a different way, is

38
00:01:53,060 --> 00:01:55,190
that this quantity is
actually just the

39
00:01:55,190 --> 00:01:57,310
square of this quantity.

40
00:01:57,310 --> 00:02:00,210
So it's actually just two of
these multiplied together.

41
00:02:00,210 --> 00:02:02,150
And let me point out
why that is.

42
00:02:02,150 --> 00:02:05,510
Because this term right here can
be rewritten as e to the

43
00:02:05,510 --> 00:02:09,910
minus x squared, e to
the minus y squared.

44
00:02:09,910 --> 00:02:13,490
And then I have two integrals,
which are for minus infinity

45
00:02:13,490 --> 00:02:14,300
to infinity both.

46
00:02:14,300 --> 00:02:16,975
Sorry, that one looks
a little off kilter.

47
00:02:16,975 --> 00:02:20,730
But those two integrals, which
the bounds are minus infinity

48
00:02:20,730 --> 00:02:22,040
to infinity for both.

49
00:02:22,040 --> 00:02:24,950
And then I have a dx and a dy.

50
00:02:24,950 --> 00:02:28,070
Now the reason that it
is just I squared--

51
00:02:28,070 --> 00:02:30,170
this quantity is just I
squared-- is because if you

52
00:02:30,170 --> 00:02:34,770
notice, this first integral, the
inside integral, this is

53
00:02:34,770 --> 00:02:36,850
independent of x.

54
00:02:36,850 --> 00:02:38,670
So I can move it out.

55
00:02:38,670 --> 00:02:41,390
So let me actually come up
here and write that down.

56
00:02:41,390 --> 00:02:43,650
So I can move it out and I can
actually rewrite the integral.

57
00:02:43,650 --> 00:02:46,240
And I'll put the bounds here,
because it's high up enough

58
00:02:46,240 --> 00:02:47,970
that I can actually write it.

59
00:02:47,970 --> 00:02:49,530
e to the minus y squared.

60
00:02:49,530 --> 00:02:53,560
And then integral for minus
infinity to infinity.

61
00:02:53,560 --> 00:02:57,950
e to the minus x
squared dx dy.

62
00:02:57,950 --> 00:02:58,640
Right?

63
00:02:58,640 --> 00:03:00,860
So let's point out
again what I did.

64
00:03:00,860 --> 00:03:05,975
This quantity I circled was the
constant when I considered

65
00:03:05,975 --> 00:03:06,920
x the variable.

66
00:03:06,920 --> 00:03:07,180
Right?

67
00:03:07,180 --> 00:03:09,880
In terms of x, this is the
constant, so I can move it in

68
00:03:09,880 --> 00:03:12,840
front of that integral, which
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 --> 00:03:18,430
Well, this quantity is what
I've called I. And so now

71
00:03:18,430 --> 00:03:19,480
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 --> 00:03:25,370
That's I, as designated.

75
00:03:25,370 --> 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 either the minus
y squared dy.

77
00:03:31,560 --> 00:03:35,010
And then that again is another
I. Because if you notice, I

78
00:03:35,010 --> 00:03:37,990
mean, this was either the
minus x squared dx--

79
00:03:37,990 --> 00:03:40,510
and so if I keep this as either
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 --> 00:03:49,260
So the point I want to make, if
you come back over here to

83
00:03:49,260 --> 00:03:57,110
this integral, is that this
quantity I'm going to be able

84
00:03:57,110 --> 00:03:58,430
to integrate very easily.

85
00:03:58,430 --> 00:04:01,090
And when I do that, if I take
the square root, I'm going to

86
00:04:01,090 --> 00:04:03,270
get this value.

87
00:04:03,270 --> 00:04:05,010
Because I just showed this
thing in the box

88
00:04:05,010 --> 00:04:07,410
was equal to I squared.

89
00:04:07,410 --> 00:04:10,140
So let me write it down.

90
00:04:10,140 --> 00:04:13,250
On this last part of the board
I will write down that thing

91
00:04:13,250 --> 00:04:14,940
again and then we'll actually
evaluate it and

92
00:04:14,940 --> 00:04:17,592
we'll see what we get.

93
00:04:17,592 --> 00:04:18,860
So let me rewrite.

94
00:04:18,860 --> 00:04:20,110
This is the thing we
want to evaluate.

95
00:04:20,110 --> 00:04:30,550

96
00:04:30,550 --> 00:04:32,730
And I'm going to remind
myself that

97
00:04:32,730 --> 00:04:34,610
that's equal to I squared.

98
00:04:34,610 --> 00:04:36,740
I squared is equal
to this quantity.

99
00:04:36,740 --> 00:04:38,560
And I mentioned at the beginning
that we're going to

100
00:04:38,560 --> 00:04:39,200
use a trick.

101
00:04:39,200 --> 00:04:41,020
We're going to change
variables into polar

102
00:04:41,020 --> 00:04:42,240
coordinates.

103
00:04:42,240 --> 00:04:45,700
And so notice that the region
we're integrating over is the

104
00:04:45,700 --> 00:04:47,700
entire xy plane.

105
00:04:47,700 --> 00:04:50,220
And in polar coordinates,
what's that going to be?

106
00:04:50,220 --> 00:04:54,240
Theta is going to run from 0
all the way around to 2 pi,

107
00:04:54,240 --> 00:04:57,160
and r is going to run
from 0 to infinity.

108
00:04:57,160 --> 00:04:59,660
And so those will be our
bounds for r and theta.

109
00:04:59,660 --> 00:05:03,550
Because if you want to get the
entire xy plane, in terms of r

110
00:05:03,550 --> 00:05:06,750
and theta, that is what
you have to do.

111
00:05:06,750 --> 00:05:09,900
And I want to mention also, what
is this quantity going to

112
00:05:09,900 --> 00:05:12,220
become in terms of
r and theta?

113
00:05:12,220 --> 00:05:14,360
Well, notice that this
is e to the minus x

114
00:05:14,360 --> 00:05:15,510
squared minus y squared.

115
00:05:15,510 --> 00:05:17,990
It's just really e to
the minus quantity x

116
00:05:17,990 --> 00:05:19,030
squared plus y squared.

117
00:05:19,030 --> 00:05:21,330
It's really e to the
minus r squared.

118
00:05:21,330 --> 00:05:23,790
So what we get when
we do a change of

119
00:05:23,790 --> 00:05:25,760
variables, is we do--

120
00:05:25,760 --> 00:05:30,500
I'll put the theta on the
inside actually, because

121
00:05:30,500 --> 00:05:34,420
that'll be easy to do first.
So the r is going

122
00:05:34,420 --> 00:05:35,910
to be on the outside.

123
00:05:35,910 --> 00:05:38,100
The theta is going to
be on the inside.

124
00:05:38,100 --> 00:05:40,870
And then e to the minus r
squared, that's a direct

125
00:05:40,870 --> 00:05:41,420
substitution.

126
00:05:41,420 --> 00:05:44,220
x squared plus y squared
equals r squared.

127
00:05:44,220 --> 00:05:47,330
And then I get r d theta d r.

128
00:05:47,330 --> 00:05:50,450
And this comes from the Jacobian
that you computed--

129
00:05:50,450 --> 00:05:53,160
I believe in lecture even--

130
00:05:53,160 --> 00:05:55,810
to show how you change from xy

131
00:05:55,810 --> 00:05:57,660
variables to r theta variables.

132
00:05:57,660 --> 00:05:59,950
And I put the d theta first
here because I wanted to

133
00:05:59,950 --> 00:06:02,770
integrate in theta first. And
the reason I want to integrate

134
00:06:02,770 --> 00:06:04,400
in theta first is
notice that--

135
00:06:04,400 --> 00:06:06,610
because nothing here
depends on theta--

136
00:06:06,610 --> 00:06:09,400
all I pick up is a 2 pi.

137
00:06:09,400 --> 00:06:09,730
I just pick up --when
I integrate--

138
00:06:09,730 --> 00:06:10,060
I get a theta.

139
00:06:10,060 --> 00:06:13,570
I evaluate it at 2 pi, and then
I evaluate it at 0 and I

140
00:06:13,570 --> 00:06:14,740
take the difference.

141
00:06:14,740 --> 00:06:17,620
And so if I do that line all
I get is-- well, I should

142
00:06:17,620 --> 00:06:18,890
probably write it in front--

143
00:06:18,890 --> 00:06:22,110
all I get is 2 pi from the
theta, and then the integral

144
00:06:22,110 --> 00:06:24,410
from 0 to infinity.

145
00:06:24,410 --> 00:06:28,590
e to the minus r squared
times r dr.

146
00:06:28,590 --> 00:06:32,560
Now this is a much easier
quantity to evaluate than e to

147
00:06:32,560 --> 00:06:34,790
the minus x squared dx.

148
00:06:34,790 --> 00:06:36,660
Because now we have an r here.

149
00:06:36,660 --> 00:06:39,810
So now it's a natural
substitution type of problem.

150
00:06:39,810 --> 00:06:41,110
And we can do it right away.

151
00:06:41,110 --> 00:06:43,380
I'm going to write it down and
then we'll check and make sure

152
00:06:43,380 --> 00:06:44,760
I didn't make a mistake.

153
00:06:44,760 --> 00:06:51,460
We should get something like e
to the minus r squared over 2

154
00:06:51,460 --> 00:06:53,740
with a negative sign in front.

155
00:06:53,740 --> 00:06:55,570
So let me make sure when I
take this derivative--

156
00:06:55,570 --> 00:06:57,570
you could just do a substitution
to check, but you

157
00:06:57,570 --> 00:06:59,480
should get exactly this
kind of thing.

158
00:06:59,480 --> 00:07:02,140
When I take the derivative of
e to the minus r squared, I

159
00:07:02,140 --> 00:07:06,560
get a negative 2r and then--

160
00:07:06,560 --> 00:07:07,750
did I do something wrong here?

161
00:07:07,750 --> 00:07:08,790
Oh yeah.

162
00:07:08,790 --> 00:07:09,310
I'm good.

163
00:07:09,310 --> 00:07:11,780
I get a negative 2r.

164
00:07:11,780 --> 00:07:14,770
And so the negatives kill off,
the 2s divide out, and I get

165
00:07:14,770 --> 00:07:17,880
my e to the minus r
squared times r.

166
00:07:17,880 --> 00:07:19,040
So that's good.

167
00:07:19,040 --> 00:07:22,290
And now I have to evaluate
it at the bounds.

168
00:07:22,290 --> 00:07:28,000
So again, this was much easier
to evaluate if I do a

169
00:07:28,000 --> 00:07:30,900
substitution and I let r--

170
00:07:30,900 --> 00:07:33,070
I think I want to let
r squared equal u or

171
00:07:33,070 --> 00:07:34,080
something like this.

172
00:07:34,080 --> 00:07:34,670
I don't even know.

173
00:07:34,670 --> 00:07:36,590
I just did the substitution
without thinking about it.

174
00:07:36,590 --> 00:07:38,610
So you can figure out what
you need to substitute.

175
00:07:38,610 --> 00:07:40,050
But this is what you get.

176
00:07:40,050 --> 00:07:42,430
And so let me actually just
evaluate at the bounds and see

177
00:07:42,430 --> 00:07:43,380
what happens.

178
00:07:43,380 --> 00:07:47,060
So as r goes to infinity, I get
e to the minus r squared.

179
00:07:47,060 --> 00:07:51,200
As r goes to infinity, e to the
minus r squared goes to 0.

180
00:07:51,200 --> 00:07:54,000
And so the first term is
0 when I evaluate.

181
00:07:54,000 --> 00:07:59,480
And then so the second term I
get is, I do 0 minus 2 pi

182
00:07:59,480 --> 00:08:02,380
times-- if I evaluate at
e to the 0 I get 1--

183
00:08:02,380 --> 00:08:07,350
so I get negative 1 over 2.

184
00:08:07,350 --> 00:08:10,370
The negative 1 comes
from e to the 0.

185
00:08:10,370 --> 00:08:12,490
And then I divide by 2
so I get my 2 there.

186
00:08:12,490 --> 00:08:15,200
So a negative and a negative
gives me a positive.

187
00:08:15,200 --> 00:08:16,380
2 divided by 2.

188
00:08:16,380 --> 00:08:18,870
And so the whole thing
equals pi.

189
00:08:18,870 --> 00:08:19,160
Whew!

190
00:08:19,160 --> 00:08:20,350
Just enough room.

191
00:08:20,350 --> 00:08:23,010
And so I just want to remind
us where we came from.

192
00:08:23,010 --> 00:08:25,790
We came from pi--

193
00:08:25,790 --> 00:08:29,380
we wanted to show was equal--
well, the pi,

194
00:08:29,380 --> 00:08:30,200
where did it come from?

195
00:08:30,200 --> 00:08:34,380
It's equal if we go all the
way back up to I squared.

196
00:08:34,380 --> 00:08:37,310
And so we wanted to show
what I was equal to.

197
00:08:37,310 --> 00:08:40,620
I then is equal to just
the square root of pi.

198
00:08:40,620 --> 00:08:42,850
So I can come back over
to where I started

199
00:08:42,850 --> 00:08:44,230
which is over here.

200
00:08:44,230 --> 00:08:49,270
And I can say this is equal
to the square root of pi.

201
00:08:49,270 --> 00:08:52,080
And again, I just want to
mention what it came down to.

202
00:08:52,080 --> 00:08:55,870
It came down to introducing
a little bit more.

203
00:08:55,870 --> 00:08:59,040
Essentially we took something
that was a single variable

204
00:08:59,040 --> 00:09:01,720
problem, we made it a
multivariable problem.

205
00:09:01,720 --> 00:09:04,630
But what that allowed us to
do is to do a change of

206
00:09:04,630 --> 00:09:07,730
coordinates from xy into the
polar coordinate system.

207
00:09:07,730 --> 00:09:10,190
And the trick here-- as you come
through-- the trick where

208
00:09:10,190 --> 00:09:12,760
it actually happens
is right here.

209
00:09:12,760 --> 00:09:15,120
Is because this is easy
to integrate now

210
00:09:15,120 --> 00:09:16,870
that I have an r here.

211
00:09:16,870 --> 00:09:19,690
If I didn't have-- my problem
initially was I didn't have an

212
00:09:19,690 --> 00:09:22,330
x when I was trying to find
an antiderivative.

213
00:09:22,330 --> 00:09:24,140
But when I do the change of
variables, I get an e to the

214
00:09:24,140 --> 00:09:25,250
minus r squared.

215
00:09:25,250 --> 00:09:27,900
And then with an r here, that's
much easier to find an

216
00:09:27,900 --> 00:09:29,470
antiderivative.

217
00:09:29,470 --> 00:09:30,380
OK.

218
00:09:30,380 --> 00:09:33,250
So I think that is
where I'll stop.

219
00:09:33,250 --> 00:09:33,774