1
00:00:00,000 --> 00:00:00,000

2
00:00:00,000 --> 00:00:08,500
CHRISTINE BREINER: Welcome
back to recitation.

3
00:00:08,500 --> 00:00:11,010
In this video I'd like
us to do two things.

4
00:00:11,010 --> 00:00:14,010
The first thing we're going to
do is we're going to graph the

5
00:00:14,010 --> 00:00:18,570
curve, r equals 1 plus cosine
theta over 2 for theta between

6
00:00:18,570 --> 00:00:22,400
0 and 4 pi and we're going to
graph it in the x, y plane.

7
00:00:22,400 --> 00:00:25,040
And then after we've done that,
we're going to take a

8
00:00:25,040 --> 00:00:28,420
look at some components of that
curve and we're going to

9
00:00:28,420 --> 00:00:32,310
calculate the area of some
components that close up.

10
00:00:32,310 --> 00:00:37,190
So what I'd like you to do first
is get a good picture of

11
00:00:37,190 --> 00:00:39,550
this curve in the x, y plane.

12
00:00:39,550 --> 00:00:40,910
I'll give you a little
while to do that.

13
00:00:40,910 --> 00:00:43,320
So why don't you pause the
video, get a good picture of

14
00:00:43,320 --> 00:00:46,230
that curve, then come back when
you're ready and I'll

15
00:00:46,230 --> 00:00:49,440
show you how I graph it, and
then we'll get into these area

16
00:00:49,440 --> 00:00:58,520
problems.

17
00:00:58,520 --> 00:00:58,760
OK.

18
00:00:58,760 --> 00:00:59,720
Welcome back.

19
00:00:59,720 --> 00:01:03,190
So the goal, again, was to
graph a certain curve

20
00:01:03,190 --> 00:01:06,690
described by r and theta,
but in the x, y plane.

21
00:01:06,690 --> 00:01:09,880
And for theta between
0 and pi over 4.

22
00:01:09,880 --> 00:01:15,600
And when I do these problems,
we want to make sure that we

23
00:01:15,600 --> 00:01:18,460
understand how r depends
on theta.

24
00:01:18,460 --> 00:01:21,700
That's sort of the main goal
to graph this curve.

25
00:01:21,700 --> 00:01:25,590
So what I do, actually, is I
look at this not in the x, y

26
00:01:25,590 --> 00:01:28,720
plane, which was what I said
to do in the problem, but I

27
00:01:28,720 --> 00:01:32,070
first look at it in what we
see as the r, theta plane.

28
00:01:32,070 --> 00:01:35,170
And what we mean by that is
we're going to graph this just

29
00:01:35,170 --> 00:01:38,430
like we would if this variable
was x and this variable was y.

30
00:01:38,430 --> 00:01:43,440
So we move out of what we know
about how r relates to x and

31
00:01:43,440 --> 00:01:46,040
y, and how theta relates to x
and y, and we just look at how

32
00:01:46,040 --> 00:01:47,660
r relates to theta.

33
00:01:47,660 --> 00:01:50,430
So let me draw that first and
we'll see if we can sort of

34
00:01:50,430 --> 00:01:54,620
understand what I mean by that,
and then we'll put that,

35
00:01:54,620 --> 00:01:58,030
use that curve to put that
into the x, y plane.

36
00:01:58,030 --> 00:01:58,290
OK.

37
00:01:58,290 --> 00:02:03,500
So, the first thing we do is I'm
going to let this be the

38
00:02:03,500 --> 00:02:07,570
theta-axis and this
be the r-axis.

39
00:02:07,570 --> 00:02:10,350
And let's look at--I want to
write down over here what the

40
00:02:10,350 --> 00:02:12,520
equation is so I don't have
to keep turning around.

41
00:02:12,520 --> 00:02:15,550

42
00:02:15,550 --> 00:02:17,800
So again, this is not, I don't
want to think about this as

43
00:02:17,800 --> 00:02:18,880
the x, y plane.

44
00:02:18,880 --> 00:02:21,820
Because in the x, y plane, theta
has certain values at

45
00:02:21,820 --> 00:02:24,680
each point that are fixed
based on the angle.

46
00:02:24,680 --> 00:02:28,815
But now I'm letting theta vary
in this direction, and r is

47
00:02:28,815 --> 00:02:30,290
varying in this direction.

48
00:02:30,290 --> 00:02:34,620
And my theta, I said, was
between 0 and pi over 4--

49
00:02:34,620 --> 00:02:37,000
I'm sorry, not pi over 4.

50
00:02:37,000 --> 00:02:38,350
4 pi.

51
00:02:38,350 --> 00:02:40,200
0 and 4 pi.

52
00:02:40,200 --> 00:02:42,730
pi over 4 would be a very small
component of this that

53
00:02:42,730 --> 00:02:43,920
I'm interested in.

54
00:02:43,920 --> 00:02:47,290
And let's think about what
happens, what kind of

55
00:02:47,290 --> 00:02:50,780
transformations have been done
to the normal cosine function.

56
00:02:50,780 --> 00:02:55,860
So if I took the normal cosine
function and I take it instead

57
00:02:55,860 --> 00:02:59,260
of cosine theta, I look it
cosine theta over 2, what

58
00:02:59,260 --> 00:03:01,250
that's doing is that's
stretching it

59
00:03:01,250 --> 00:03:02,900
horizontally out.

60
00:03:02,900 --> 00:03:05,820
So think about the period
of the cosine function

61
00:03:05,820 --> 00:03:07,910
usually is 2 pi.

62
00:03:07,910 --> 00:03:10,630
But notice what happens when I
put in 2 pi for theta, I'm

63
00:03:10,630 --> 00:03:12,700
getting cosine of pi.

64
00:03:12,700 --> 00:03:15,200
If I want to get cosine of 2 pi,
I have to let theta go to

65
00:03:15,200 --> 00:03:19,080
4 pi, which is why I'm letting
theta be between 0 and 4 pi.

66
00:03:19,080 --> 00:03:23,870
So dividing the input value
by 2 doubles your period.

67
00:03:23,870 --> 00:03:26,280
So the period is now 4 pi.

68
00:03:26,280 --> 00:03:28,190
So to get all the way through,
I'm going to have

69
00:03:28,190 --> 00:03:29,900
to go up to 4 pi.

70
00:03:29,900 --> 00:03:34,440
So let me just make a 2
pi here, and this is

71
00:03:34,440 --> 00:03:36,910
about a 4 pi here.

72
00:03:36,910 --> 00:03:40,620
So here's around 3 pi,
and here's around pi.

73
00:03:40,620 --> 00:03:44,010
So that now, instead of the
usual cosine function, it's

74
00:03:44,010 --> 00:03:46,380
going to take twice as long to
get all the way through.

75
00:03:46,380 --> 00:03:47,660
That's one thing we know.

76
00:03:47,660 --> 00:03:49,730
What else have we done to the
usual cosine function?

77
00:03:49,730 --> 00:03:51,420
We've moved it up by 1.

78
00:03:51,420 --> 00:03:56,550
And so instead of starting out
when your input is 0, starting

79
00:03:56,550 --> 00:03:59,910
out at height 1, when you're
input is 0, you start out at

80
00:03:59,910 --> 00:04:00,920
height 1 plus 1.

81
00:04:00,920 --> 00:04:03,180
You start out at height 2.

82
00:04:03,180 --> 00:04:09,250
So in fact, this function, let
me point out, it's going to

83
00:04:09,250 --> 00:04:12,030
start at 2, which
means it also is

84
00:04:12,030 --> 00:04:13,520
going to end over here.

85
00:04:13,520 --> 00:04:16,680
Because it's periodic it's
going to end at 2.

86
00:04:16,680 --> 00:04:18,870
And let's think about
what else we know.

87
00:04:18,870 --> 00:04:21,070
We know that the usual
cosine function goes

88
00:04:21,070 --> 00:04:22,800
down to negative 1.

89
00:04:22,800 --> 00:04:25,257
But I've added 1 to it, so now
it only goes down to 0.

90
00:04:25,257 --> 00:04:26,880
OK?

91
00:04:26,880 --> 00:04:27,790
Hopefully that makes sense.

92
00:04:27,790 --> 00:04:31,090
Maybe I should even--

93
00:04:31,090 --> 00:04:32,010
hmm.

94
00:04:32,010 --> 00:04:34,170
I don't want to draw the actual
cosine function again

95
00:04:34,170 --> 00:04:35,100
right on here.

96
00:04:35,100 --> 00:04:38,080
But let me draw the regular
cosine function here.

97
00:04:38,080 --> 00:04:42,710
So we have it, the regular
cosine function--

98
00:04:42,710 --> 00:04:45,620
because I keep talking
about it--

99
00:04:45,620 --> 00:04:46,720
does something like this.

100
00:04:46,720 --> 00:04:50,380
It goes at 1 here, and
it's at 1 again here.

101
00:04:50,380 --> 00:04:53,530
And so it's at minus 1 at pi.

102
00:04:53,530 --> 00:04:56,290
And so, very roughly, it looks
something like this.

103
00:04:56,290 --> 00:04:57,680
Right?

104
00:04:57,680 --> 00:04:59,970
So I keep referencing the cosine
function, so this is

105
00:04:59,970 --> 00:05:01,730
the part I'm referencing.

106
00:05:01,730 --> 00:05:05,620
So we have to stretch it by 2,
and then shift it up by 1.

107
00:05:05,620 --> 00:05:09,680
And so we see what was at pi,
negative 1, I'm now going to

108
00:05:09,680 --> 00:05:13,110
be at 2 pi, 0.

109
00:05:13,110 --> 00:05:14,500
And then where do
these points go?

110
00:05:14,500 --> 00:05:16,540
This was pi over 2.

111
00:05:16,540 --> 00:05:19,160
The x value is going
to be doubled.

112
00:05:19,160 --> 00:05:20,950
I'm going to be at pi,
and the y value is

113
00:05:20,950 --> 00:05:22,420
going to go up by 1.

114
00:05:22,420 --> 00:05:26,050
So I'm at pi, 1 and 3, pi 1.

115
00:05:26,050 --> 00:05:30,410
And so the curve will look
something like this.

116
00:05:30,410 --> 00:05:35,500
I'm not an expert here, but
hopefully that looks something

117
00:05:35,500 --> 00:05:36,780
like the cosine time function.

118
00:05:36,780 --> 00:05:43,230
But with a stretch
and a shift.

119
00:05:43,230 --> 00:05:48,700
So that is the curve, r equals
1 plus cosine theta over 2 in

120
00:05:48,700 --> 00:05:51,510
what we consider the
r, theta plane.

121
00:05:51,510 --> 00:05:54,560
So theta is varying in this
direction and r is varying in

122
00:05:54,560 --> 00:05:55,840
this direction.

123
00:05:55,840 --> 00:05:58,790
But how do I transfer that
to the x, y plane?

124
00:05:58,790 --> 00:06:00,630
That's the real point
that I want to

125
00:06:00,630 --> 00:06:02,830
make about this problem.

126
00:06:02,830 --> 00:06:06,980
So let's look at what's
happening in the x, y plane.

127
00:06:06,980 --> 00:06:10,160
So this will be x and
this will be y.

128
00:06:10,160 --> 00:06:11,560
Let's pick some points
and try to

129
00:06:11,560 --> 00:06:13,120
figure out what's happening.

130
00:06:13,120 --> 00:06:15,150
So where is this point?

131
00:06:15,150 --> 00:06:17,360
This is the point
zero comma two.

132
00:06:17,360 --> 00:06:20,820
When theta is 0, r equals 2.

133
00:06:20,820 --> 00:06:23,350
Where is theta equal to
0 in this picture?

134
00:06:23,350 --> 00:06:24,525
It's in the x direction.

135
00:06:24,525 --> 00:06:25,310
Right?

136
00:06:25,310 --> 00:06:31,270
That's theta equals 0, and
also 2 pi and 4 pi.

137
00:06:31,270 --> 00:06:36,360
Any multiple of 2 pi, theta is
pointing in this direction.

138
00:06:36,360 --> 00:06:38,470
And r equals 2 there.

139
00:06:38,470 --> 00:06:41,830
So in a strange twist,
this is the point, 2,

140
00:06:41,830 --> 00:06:43,540
0 on the x, y plane.

141
00:06:43,540 --> 00:06:46,210
But that's no going to, that's
just a coincidence, OK?

142
00:06:46,210 --> 00:06:48,560
Don't think, oh, I'm just
going to flip the values

143
00:06:48,560 --> 00:06:49,020
everywhere.

144
00:06:49,020 --> 00:06:49,795
That's not going to happen.

145
00:06:49,795 --> 00:06:51,540
OK?

146
00:06:51,540 --> 00:06:53,520
Actually, and also, before I
make a mistake, I'm going to

147
00:06:53,520 --> 00:06:55,190
make this a little bit bigger.

148
00:06:55,190 --> 00:06:57,360
I want it to be bigger.

149
00:06:57,360 --> 00:06:59,020
So I don't want this to be 2.

150
00:06:59,020 --> 00:07:00,350
I want this to be 1.

151
00:07:00,350 --> 00:07:04,380
I'm going to make this 2, 0.

152
00:07:04,380 --> 00:07:06,980
I want to have a little
bigger picture.

153
00:07:06,980 --> 00:07:07,290
OK.

154
00:07:07,290 --> 00:07:14,010
So it's going to be 2, 0 at
theta equals 0 and r equal 2.

155
00:07:14,010 --> 00:07:16,070
Is it ever hit this
point again?

156
00:07:16,070 --> 00:07:16,930
Well, it is.

157
00:07:16,930 --> 00:07:18,610
And it's going to hit that
point again because it's

158
00:07:18,610 --> 00:07:20,840
periodic and I've gone
out to 4 pi.

159
00:07:20,840 --> 00:07:23,870
If I rotate all the way around,
I'm at 2 pi for theta.

160
00:07:23,870 --> 00:07:26,330
If I rotate all the way around
again I'm at 4 pi for theta,

161
00:07:26,330 --> 00:07:28,740
and my radius there
is 2, also.

162
00:07:28,740 --> 00:07:32,120
So this 2, 0 happens again
when I have this point.

163
00:07:32,120 --> 00:07:34,160
So it's going to close up.

164
00:07:34,160 --> 00:07:34,430
OK.

165
00:07:34,430 --> 00:07:35,430
And then what else happens?

166
00:07:35,430 --> 00:07:38,320
Well, as theta is rotating,
let's take theta

167
00:07:38,320 --> 00:07:40,110
between 0 and pi.

168
00:07:40,110 --> 00:07:43,560
theta rotates from 0 to
pi going like this.

169
00:07:43,560 --> 00:07:46,270
Notice what's happening
to the r value.

170
00:07:46,270 --> 00:07:49,910
The r value is going
from 2 down to 1.

171
00:07:49,910 --> 00:07:53,780
Now, I'm not going to be
totally exact, but--

172
00:07:53,780 --> 00:07:58,450
here's minus 1, OK,
in the x, y plane.

173
00:07:58,450 --> 00:08:02,070
I'm not going to be totally
exactly, but this curve is

174
00:08:02,070 --> 00:08:03,830
going to look something like--

175
00:08:03,830 --> 00:08:06,565

176
00:08:06,565 --> 00:08:07,870
there's 2--

177
00:08:07,870 --> 00:08:09,145
it's going to look
something like--

178
00:08:09,145 --> 00:08:13,700

179
00:08:13,700 --> 00:08:16,620
oops, I over-shot, we'll
make that negative 1--

180
00:08:16,620 --> 00:08:18,470
something like this.

181
00:08:18,470 --> 00:08:19,710
And what's the point?

182
00:08:19,710 --> 00:08:24,050
The point is that I start at
radius 2, and by the time I

183
00:08:24,050 --> 00:08:28,240
get to theta equals
pi I've gone down.

184
00:08:28,240 --> 00:08:29,740
And so my radius is 1.

185
00:08:29,740 --> 00:08:33,850
This has radius 1
and angle pi.

186
00:08:33,850 --> 00:08:36,200
So that represents this
part of the curve.

187
00:08:36,200 --> 00:08:38,400
That's this part of the curve.

188
00:08:38,400 --> 00:08:41,900
Now, what's nice about this
drawing is that we know this

189
00:08:41,900 --> 00:08:44,070
part of the curve and this part
of the curve should look

190
00:08:44,070 --> 00:08:45,310
exactly the same.

191
00:08:45,310 --> 00:08:48,620
So once I've drawn half of
this, I'm going to know

192
00:08:48,620 --> 00:08:49,110
everything.

193
00:08:49,110 --> 00:08:51,500
Once I've gone from
0 to 2 pi, I can

194
00:08:51,500 --> 00:08:52,930
just reflect it, basically.

195
00:08:52,930 --> 00:08:56,050
We'll see what I mean by reflect
in this case, but the

196
00:08:56,050 --> 00:08:58,690
same radii are happening
again and some sort

197
00:08:58,690 --> 00:09:00,670
of symmetric fashion.

198
00:09:00,670 --> 00:09:00,920
OK.

199
00:09:00,920 --> 00:09:02,240
So we've got 0 to pi.

200
00:09:02,240 --> 00:09:04,650
Now what happens between
pi and 2 pi?

201
00:09:04,650 --> 00:09:08,290
Notice pi to 2 pi in the theta
direction on the x, y plane is

202
00:09:08,290 --> 00:09:09,445
I start in this direction--

203
00:09:09,445 --> 00:09:12,150
I don't know if you can
see that, let me come

204
00:09:12,150 --> 00:09:13,250
over to this side--

205
00:09:13,250 --> 00:09:16,630
I start in this direction
for pi, and I'm

206
00:09:16,630 --> 00:09:17,620
going to rotate down.

207
00:09:17,620 --> 00:09:21,490
This is 3 pi over 2,
and this is 2 pi.

208
00:09:21,490 --> 00:09:22,470
Those are my angles.

209
00:09:22,470 --> 00:09:24,310
So what are my radii?

210
00:09:24,310 --> 00:09:30,740
Well, I start at radius 1 and
I'm going to radius 0.

211
00:09:30,740 --> 00:09:35,470
And so what happens is I'm
coming through this negative 1

212
00:09:35,470 --> 00:09:40,140
and I'm coming around, and
then by the time I get--

213
00:09:40,140 --> 00:09:42,310
it's going to be something like
this-- by the time I get

214
00:09:42,310 --> 00:09:45,260
to 2 pi, my radius is 0.

215
00:09:45,260 --> 00:09:48,430
Now just to make sure this curve
makes sense to you, you

216
00:09:48,430 --> 00:09:53,440
could pick a place, an angle,
maybe like right here.

217
00:09:53,440 --> 00:09:55,430
I don't know if that's easy to
see, but maybe that angle

218
00:09:55,430 --> 00:09:56,210
right there.

219
00:09:56,210 --> 00:09:59,480
That angle is between 3
pi over 2 and 2 pi.

220
00:09:59,480 --> 00:10:01,280
Is there positive
radius there?

221
00:10:01,280 --> 00:10:02,900
Yeah, there is positive
radius there.

222
00:10:02,900 --> 00:10:06,540
So in fact, this curve does come
into this fourth quadrant

223
00:10:06,540 --> 00:10:09,090
here and then curve back in.

224
00:10:09,090 --> 00:10:09,410
OK?

225
00:10:09,410 --> 00:10:11,550
It does curve back in.

226
00:10:11,550 --> 00:10:11,850
All right.

227
00:10:11,850 --> 00:10:16,020
So then what happens between
2 pi and 3 pi?

228
00:10:16,020 --> 00:10:16,150
OK.

229
00:10:16,150 --> 00:10:18,310
Hopefully this picture
is clear so far.

230
00:10:18,310 --> 00:10:20,240
I'm going to come back
to the other side.

231
00:10:20,240 --> 00:10:20,460
OK.

232
00:10:20,460 --> 00:10:22,390
What happens between
2 pi and 3 pi?

233
00:10:22,390 --> 00:10:24,020
2 pi, we're here.

234
00:10:24,020 --> 00:10:26,720
And 3 pi, we're back
over here again.

235
00:10:26,720 --> 00:10:30,490
And notice that the radius is
going to be doing the same

236
00:10:30,490 --> 00:10:33,200
kind of growth that it did
between 2 pi and 3 pi as it

237
00:10:33,200 --> 00:10:36,632
did decay from pi to 2 pi.

238
00:10:36,632 --> 00:10:37,500
OK?

239
00:10:37,500 --> 00:10:40,680
Because the radii now, there's
a symmetry with how the radii

240
00:10:40,680 --> 00:10:46,340
behave. So from 2 pi to 3 pi, I
start off with radius 0 and

241
00:10:46,340 --> 00:10:48,120
I have a small radius.

242
00:10:48,120 --> 00:10:53,870
And then once I get to 3 pi over
here, I'm going to have

243
00:10:53,870 --> 00:10:55,070
radius 1 again.

244
00:10:55,070 --> 00:10:56,940
I'm going to be at radius 1,
which is going to correspond

245
00:10:56,940 --> 00:10:57,840
to this point.

246
00:10:57,840 --> 00:11:00,580
So I'm going to have exactly
the same picture, which is

247
00:11:00,580 --> 00:11:03,265
dangerous because I probably
would do it wrong.

248
00:11:03,265 --> 00:11:06,840
But I'll try and draw it this
way and then talk about it.

249
00:11:06,840 --> 00:11:09,480
Hopefully that looks
about the same.

250
00:11:09,480 --> 00:11:13,570
So this curve coming through
here was from pi to 2 pi.

251
00:11:13,570 --> 00:11:17,750
This curve coming through here
was from to 2 pi to 3 pi.

252
00:11:17,750 --> 00:11:20,240
And then to finish it off, 3
pi to 4 pi is going to look

253
00:11:20,240 --> 00:11:22,310
like this curve here.

254
00:11:22,310 --> 00:11:24,260
So I come through--

255
00:11:24,260 --> 00:11:27,440
ooh, this is where it starts to
get really dangerous, but

256
00:11:27,440 --> 00:11:29,980
let's say that's
pretty close--

257
00:11:29,980 --> 00:11:33,160
so there's my 3 pi to 4 pi.

258
00:11:33,160 --> 00:11:36,410
It's again, the same growth
the way it was decaying

259
00:11:36,410 --> 00:11:38,260
between 0 and pi.

260
00:11:38,260 --> 00:11:43,940
So this is your picture in the
x, y plane of the curve, r

261
00:11:43,940 --> 00:11:47,150
equals 1 plus cosine
theta over 2.

262
00:11:47,150 --> 00:11:50,900
Now, we haven't calculated
any area problems, yet.

263
00:11:50,900 --> 00:11:54,210
So what I'd like us to do is
I'm going to shade two

264
00:11:54,210 --> 00:12:00,430
regions, and I want us to just
write down the integral form

265
00:12:00,430 --> 00:12:04,070
to find the area for each
of these regions.

266
00:12:04,070 --> 00:12:09,600
So the first region of interest
is the pink region.

267
00:12:09,600 --> 00:12:12,750
I'm going to ask us to find the
area of the pink region,

268
00:12:12,750 --> 00:12:16,420
and then I'm going to ask us to
find the area of everything

269
00:12:16,420 --> 00:12:17,670
else, the blue region.

270
00:12:17,670 --> 00:12:20,450

271
00:12:20,450 --> 00:12:24,560
So let's think about
how to do that.

272
00:12:24,560 --> 00:12:26,980
And I think I'm going to have to
come over to the other side

273
00:12:26,980 --> 00:12:28,890
to write down the integrals,
but I'll be

274
00:12:28,890 --> 00:12:30,440
coming back and forth.

275
00:12:30,440 --> 00:12:35,700
So just to remind you, what you
saw in lecture was that dA

276
00:12:35,700 --> 00:12:40,550
is equal to r squared
over 2 d theta.

277
00:12:40,550 --> 00:12:42,740
That's what dA is.

278
00:12:42,740 --> 00:12:46,450
And so this is going to be an
integral in theta, and we know

279
00:12:46,450 --> 00:12:48,650
what r is as a function
of theta.

280
00:12:48,650 --> 00:12:52,890
And so if I want to find--
actually, I should even use my

281
00:12:52,890 --> 00:12:54,330
colors appropriately.

282
00:12:54,330 --> 00:12:59,890
I should say, the pink
area is going to

283
00:12:59,890 --> 00:13:01,860
be equal to an integral.

284
00:13:01,860 --> 00:13:03,780
And I'm going to write
the r squared.

285
00:13:03,780 --> 00:13:05,010
I know what r squared is.

286
00:13:05,010 --> 00:13:10,320
It's 1 plus cosine theta
over 2 squared, and

287
00:13:10,320 --> 00:13:12,560
then an over 2 d theta.

288
00:13:12,560 --> 00:13:14,810
And then what is important
about this?

289
00:13:14,810 --> 00:13:15,910
It's our bounds.

290
00:13:15,910 --> 00:13:16,090
Right?

291
00:13:16,090 --> 00:13:17,800
Our bounds are what's
important.

292
00:13:17,800 --> 00:13:21,760
And so let's go back and let's
look at our picture.

293
00:13:21,760 --> 00:13:25,680
What are the bounds on theta
for the pink region?

294
00:13:25,680 --> 00:13:28,540
So where does that pink
region start and stop?

295
00:13:28,540 --> 00:13:32,530
And maybe we even need to look
at this top graph, also.

296
00:13:32,530 --> 00:13:36,750
So if we think about it, we went
from 0 to pi to get this

297
00:13:36,750 --> 00:13:38,120
outer curve.

298
00:13:38,120 --> 00:13:39,440
So how do we get the
inner curve?

299
00:13:39,440 --> 00:13:44,070
The inner curve started at theta
equals pi, went to here,

300
00:13:44,070 --> 00:13:45,390
went to here--

301
00:13:45,390 --> 00:13:47,900
that was theta equals 3 pi.

302
00:13:47,900 --> 00:13:52,160
So it went all the way
from pi to 3 pi.

303
00:13:52,160 --> 00:13:55,410
Now, if you're paying good
attention, you can say, well,

304
00:13:55,410 --> 00:13:59,550
Christine, we know that this
region is totally symmetric.

305
00:13:59,550 --> 00:14:02,860
So why don't I just take the
area from pi to 2 pi and

306
00:14:02,860 --> 00:14:04,470
multiply it by 2?

307
00:14:04,470 --> 00:14:05,340
And you can.

308
00:14:05,340 --> 00:14:06,910
You can do it either way.

309
00:14:06,910 --> 00:14:09,900
So you can either take the
integral from all the way from

310
00:14:09,900 --> 00:14:13,880
pi to 3 pi, which corresponds
to starting at this angle,

311
00:14:13,880 --> 00:14:16,880
going all the way around, and
coming back to there, which

312
00:14:16,880 --> 00:14:18,240
takes you all the way
around this curve.

313
00:14:18,240 --> 00:14:21,780
Or you can go from pi to 2 pi
and multiply that by 2.

314
00:14:21,780 --> 00:14:25,470
So let me come back and let
me write that down.

315
00:14:25,470 --> 00:14:31,970
It's either pi to 3 pi, this,
or you just write it as

316
00:14:31,970 --> 00:14:34,310
integral from pi to 2 pi.

317
00:14:34,310 --> 00:14:38,430
And if I multiply this by
2, the 2 drops out.

318
00:14:38,430 --> 00:14:40,477
The 2 in the denominator
drops out.

319
00:14:40,477 --> 00:14:44,680

320
00:14:44,680 --> 00:14:46,720
I'm not going to solve this
problem for you, but I do want

321
00:14:46,720 --> 00:14:48,850
to point out the kinds of terms
you would have. You

322
00:14:48,850 --> 00:14:51,060
would have a constant term
when you square this.

323
00:14:51,060 --> 00:14:54,370
You would have a term that was
2 cosine theta over 2, which

324
00:14:54,370 --> 00:14:56,980
is easy to integrate by
a u-substitution.

325
00:14:56,980 --> 00:14:59,990
And then you would have a cosine
squared theta over 2,

326
00:14:59,990 --> 00:15:03,160
which you'd want to use the
double angle formula or the

327
00:15:03,160 --> 00:15:06,310
half angle formula that you've
seen used to manipulate these

328
00:15:06,310 --> 00:15:08,040
integrals that involve
just a cosine

329
00:15:08,040 --> 00:15:09,680
squared or a sine squared.

330
00:15:09,680 --> 00:15:11,600
So that would be
your strategy.

331
00:15:11,600 --> 00:15:13,175
OK, now let's look
at the blue area.

332
00:15:13,175 --> 00:15:14,425
OK.

333
00:15:14,425 --> 00:15:19,090

334
00:15:19,090 --> 00:15:22,110
So to find the blue area, again,
I know all that matters

335
00:15:22,110 --> 00:15:24,520
is really the bounds.

336
00:15:24,520 --> 00:15:26,515
We're going to see we have to
do a little extra work.

337
00:15:26,515 --> 00:15:29,600

338
00:15:29,600 --> 00:15:31,000
But this is our first setup.

339
00:15:31,000 --> 00:15:33,090
And now let's go look
at the bounds.

340
00:15:33,090 --> 00:15:33,360
OK.

341
00:15:33,360 --> 00:15:35,880
So we go back to the curve.

342
00:15:35,880 --> 00:15:37,020
All right.

343
00:15:37,020 --> 00:15:38,000
What is the blue area?

344
00:15:38,000 --> 00:15:41,730
Well, the blue area, that's
a little harder.

345
00:15:41,730 --> 00:15:43,680
So let's see what happens.

346
00:15:43,680 --> 00:15:49,490
If I were to take theta from
0 to pi, what would happen?

347
00:15:49,490 --> 00:15:52,370
I would not only pick up the
blue area, but I'd pick up

348
00:15:52,370 --> 00:15:54,100
this pink stuff inside.

349
00:15:54,100 --> 00:15:55,640
But I don't want
the pink stuff.

350
00:15:55,640 --> 00:15:57,290
I just want the blue stuff.

351
00:15:57,290 --> 00:16:00,020
So what am I going to have to do
to find the blue area just,

352
00:16:00,020 --> 00:16:01,980
say, from 0 up to pi?

353
00:16:01,980 --> 00:16:05,080
I'm going to have to find the
area from 0 to pi, and then

354
00:16:05,080 --> 00:16:06,520
I'm going to have to
subtract off the

355
00:16:06,520 --> 00:16:07,722
area of this component.

356
00:16:07,722 --> 00:16:09,210
OK?

357
00:16:09,210 --> 00:16:10,660
But we know, actually,
how to find the

358
00:16:10,660 --> 00:16:12,490
area of this component.

359
00:16:12,490 --> 00:16:12,790
OK.

360
00:16:12,790 --> 00:16:14,120
So hopefully this makes sense.

361
00:16:14,120 --> 00:16:16,350
Because let's think about, when
I'm finding area, I'm

362
00:16:16,350 --> 00:16:18,420
going from the origin and I'm
coming out and I have the

363
00:16:18,420 --> 00:16:20,020
radius out there.

364
00:16:20,020 --> 00:16:24,930
So when I integrate this dA from
0 to pi for theta, I'm

365
00:16:24,930 --> 00:16:28,800
picking up pieces taht come out,
little sectors that come

366
00:16:28,800 --> 00:16:31,880
out like this between
pi over 2 and pi.

367
00:16:31,880 --> 00:16:34,960
So I'm getting more area than I
want if I just let theta go

368
00:16:34,960 --> 00:16:36,740
between 0 and pi.

369
00:16:36,740 --> 00:16:39,670
So I have to calculate all of
it, and then I have to take

370
00:16:39,670 --> 00:16:41,970
away the extra stuff.

371
00:16:41,970 --> 00:16:42,250
OK.

372
00:16:42,250 --> 00:16:45,410
So the blue area is actually
the bigger area subtracting

373
00:16:45,410 --> 00:16:47,180
the smaller area.

374
00:16:47,180 --> 00:16:48,890
And so how am I going
to write this?

375
00:16:48,890 --> 00:16:51,380
If we come back over, I'm just
going to take 2 times this

376
00:16:51,380 --> 00:16:52,440
whole thing.

377
00:16:52,440 --> 00:16:54,220
So I'm going to take
2 times this.

378
00:16:54,220 --> 00:16:57,580
And I know I have to integrate
it from 0 to pi.

379
00:16:57,580 --> 00:17:00,850
And I'm taking 2 times because
it's symmetric, remember.

380
00:17:00,850 --> 00:17:04,310
And then I'm going to subtract
track off this one that's 2

381
00:17:04,310 --> 00:17:06,360
times the thing from
pi to 2 pi.

382
00:17:06,360 --> 00:17:08,610
Now, you might say, well,
Christine, the pink stuff I'm

383
00:17:08,610 --> 00:17:13,780
interested in between 0 and pi
is actually theta between not

384
00:17:13,780 --> 00:17:15,830
pi and 2 pi, but
2 pi and 3 pi.

385
00:17:15,830 --> 00:17:17,880
But again, there's all this
symmetry in the problem.

386
00:17:17,880 --> 00:17:20,270
So it doesn't really matter.

387
00:17:20,270 --> 00:17:23,360
But if you're a stickler, I
guess I'll even write it this

388
00:17:23,360 --> 00:17:24,790
way just to make sure.

389
00:17:24,790 --> 00:17:27,200
So I'll write it as 2
pi to 3 pi so that

390
00:17:27,200 --> 00:17:28,129
everyone's very happy.

391
00:17:28,129 --> 00:17:29,379
OK?

392
00:17:29,379 --> 00:17:34,920

393
00:17:34,920 --> 00:17:38,240
So again, what do we do?

394
00:17:38,240 --> 00:17:39,460
These 2's divide out.

395
00:17:39,460 --> 00:17:43,370
So from 0 to pi of r squared d
theta, that's going to give me

396
00:17:43,370 --> 00:17:46,890
the area of the blue
plus the pink.

397
00:17:46,890 --> 00:17:51,010
And then 2 pi to 3 pi of 1 plus
cosine over theta over 2

398
00:17:51,010 --> 00:17:54,510
squared d theta is going to give
me the area of the pink.

399
00:17:54,510 --> 00:17:58,600
So the blue plus the pink is
over here, and then the pink

400
00:17:58,600 --> 00:17:59,320
is over here.

401
00:17:59,320 --> 00:18:02,280
So when I subtract that off
I just get the blue.

402
00:18:02,280 --> 00:18:03,030
OK.

403
00:18:03,030 --> 00:18:06,220
Let me, again, just go back one
more time and point out

404
00:18:06,220 --> 00:18:08,190
what we did at the very
beginning to remind us what

405
00:18:08,190 --> 00:18:10,540
was happening.

406
00:18:10,540 --> 00:18:11,540
And then I will finish.

407
00:18:11,540 --> 00:18:13,700
So let's come back over here.

408
00:18:13,700 --> 00:18:18,190
So the idea was to graph this
curve that was in r is a

409
00:18:18,190 --> 00:18:19,620
function of theta.

410
00:18:19,620 --> 00:18:21,620
And I was supposed to understand
what that looked

411
00:18:21,620 --> 00:18:24,140
like in the x, y plane.

412
00:18:24,140 --> 00:18:30,680
And so my trick was to take the
relationship between r and

413
00:18:30,680 --> 00:18:33,050
theta and graph that explicitly

414
00:18:33,050 --> 00:18:35,160
in an r theta plane--

415
00:18:35,160 --> 00:18:37,370
so I let theta vary in the
horizontal direction and r

416
00:18:37,370 --> 00:18:39,090
vary in the vertical
direction--

417
00:18:39,090 --> 00:18:41,160
and I can do that very easily.

418
00:18:41,160 --> 00:18:44,830
And then translate that
into the x, y plane.

419
00:18:44,830 --> 00:18:49,380
So my curve, again, went the
big part, the little part

420
00:18:49,380 --> 00:18:52,390
here, little part here,
big part here.

421
00:18:52,390 --> 00:18:53,760
That was the order.

422
00:18:53,760 --> 00:18:58,045
So if you need arrows on
it, this was the order.

423
00:18:58,045 --> 00:19:00,820

424
00:19:00,820 --> 00:19:04,890
And then once I had that, the
problem was about areas.

425
00:19:04,890 --> 00:19:08,340
And there was a lot of symmetry
in this problem, but

426
00:19:08,340 --> 00:19:13,560
the main point I wanted to show
was just knowing where

427
00:19:13,560 --> 00:19:16,570
your theta starts and stops is
not enough to determine an

428
00:19:16,570 --> 00:19:20,520
area of a region if that region
is excluding some part

429
00:19:20,520 --> 00:19:22,780
that would be potentially
counted twice.

430
00:19:22,780 --> 00:19:25,720
So that was the reason I wanted
you to calculate not

431
00:19:25,720 --> 00:19:30,190
just the pink area, but also see
that the blue is not from

432
00:19:30,190 --> 00:19:33,510
theta from 0 to pi, but you
have to subtract off this

433
00:19:33,510 --> 00:19:36,230
extra stuff that you counted.

434
00:19:36,230 --> 00:19:39,270
And I guess that is
where I will stop.

435
00:19:39,270 --> 00:19:39,446