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PROFESSOR: Well, because
our subject today

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00:00:25,190 --> 00:00:28,280
is trig integrals
and substitutions,

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00:00:28,280 --> 00:00:33,640
Professor Jerison called in his
substitute teacher for today.

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00:00:33,640 --> 00:00:46,150
That's me.

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00:00:46,150 --> 00:00:49,860
Professor Miller.

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00:00:49,860 --> 00:00:52,790
And I'm going to try to tell
you about trig substitutions

14
00:00:52,790 --> 00:00:54,610
and trig integrals.

15
00:00:54,610 --> 00:00:59,650
And I'll be here tomorrow to
do more of the same, as well.

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00:00:59,650 --> 00:01:02,320
So, this is about trigonometry,
and maybe first thing I'll do

17
00:01:02,320 --> 00:01:24,180
is remind you of some basic
things about trigonometry.

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00:01:24,180 --> 00:01:27,110
So, if I have a
circle, trigonometry

19
00:01:27,110 --> 00:01:29,730
is all based on the
circle of radius 1

20
00:01:29,730 --> 00:01:32,000
and centered at the origin.

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00:01:32,000 --> 00:01:34,870
And so if this is an angle
of theta, up from the x-axis,

22
00:01:34,870 --> 00:01:36,350
then the coordinates
of this point

23
00:01:36,350 --> 00:01:39,550
are cosine theta and sine theta.

24
00:01:39,550 --> 00:01:42,090
And so that leads right away
to some trig identities,

25
00:01:42,090 --> 00:01:43,330
which you know very well.

26
00:01:43,330 --> 00:01:46,820
But I'm going to put them up
here because we'll use them

27
00:01:46,820 --> 00:01:51,220
over and over again today.

28
00:01:51,220 --> 00:01:55,220
Remember the convention sin^2
theta secretly means (sin

29
00:01:55,220 --> 00:01:57,530
theta)^2.

30
00:01:57,530 --> 00:01:59,040
It would be more
sensible to write

31
00:01:59,040 --> 00:02:01,600
a parenthesis around
the sine of theta

32
00:02:01,600 --> 00:02:03,350
and then say you square that.

33
00:02:03,350 --> 00:02:06,380
But everybody in the world puts
the 2 up there over the sin,

34
00:02:06,380 --> 00:02:09,080
and so I'll do that too.

35
00:02:09,080 --> 00:02:11,995
So that follows just because
the circle has radius 1.

36
00:02:11,995 --> 00:02:13,870
But then there are some
other identities too,

37
00:02:13,870 --> 00:02:15,410
which I think you remember.

38
00:02:15,410 --> 00:02:19,170
I'll write them down
here. cos(2theta),

39
00:02:19,170 --> 00:02:22,490
there's this double angle
formula that says cos(2theta) =

40
00:02:22,490 --> 00:02:23,840
cos^2(theta) - sin^2(theta).

41
00:02:23,840 --> 00:02:29,090


42
00:02:29,090 --> 00:02:31,000
And there's also the
double angle formula

43
00:02:31,000 --> 00:02:34,620
for the sin(2theta).

44
00:02:34,620 --> 00:02:38,480
Remember what that says?

45
00:02:38,480 --> 00:02:41,300
2 sin(theta) cos(theta).

46
00:02:41,300 --> 00:02:46,437


47
00:02:46,437 --> 00:02:48,020
I'm going to use
these trig identities

48
00:02:48,020 --> 00:02:50,370
and I'm going to use them
in a slightly different way.

49
00:02:50,370 --> 00:02:53,090
And so I'd like to pay a little
more attention to this one

50
00:02:53,090 --> 00:02:57,120
and get a different way
of writing this one out.

51
00:02:57,120 --> 00:03:06,930
So this is actually
the half angle formula.

52
00:03:06,930 --> 00:03:14,170
And that says, I'm going to
try to express the cos(theta)

53
00:03:14,170 --> 00:03:16,400
in terms of the cos(2theta).

54
00:03:16,400 --> 00:03:19,590
So if I know the
cos(2theta), I want

55
00:03:19,590 --> 00:03:23,320
to try to express the
cos theta in terms of it.

56
00:03:23,320 --> 00:03:30,130
Well, I'll start with a
cos(2theta) and play with that.

57
00:03:30,130 --> 00:03:30,630
OK.

58
00:03:30,630 --> 00:03:36,600
Well, we know what this is, it's
cos^2(theta) - sin^2(theta).

59
00:03:36,600 --> 00:03:39,520
But we also know what the
sin^2(theta) is in terms

60
00:03:39,520 --> 00:03:40,250
of the cosine.

61
00:03:40,250 --> 00:03:44,630
So I can eliminate the
sin^2 from this picture.

62
00:03:44,630 --> 00:03:48,400
So this is equal to cos^2(theta)
minus the quantity 1 -

63
00:03:48,400 --> 00:03:50,650
cos^2(theta).

64
00:03:50,650 --> 00:03:57,510
I put in what sin^2 is in
terms of cos^2 And so that's 2

65
00:03:57,510 --> 00:03:59,940
cos^2(theta) - 1.

66
00:03:59,940 --> 00:04:02,240
There's this cos^2,
which gets a plus sign.

67
00:04:02,240 --> 00:04:04,540
Because of these
two minus signs.

68
00:04:04,540 --> 00:04:06,290
And there's the one
that was there before,

69
00:04:06,290 --> 00:04:10,270
so altogether there
are two of them.

70
00:04:10,270 --> 00:04:12,380
I want to isolate
what cosine is.

71
00:04:12,380 --> 00:04:16,070
Or rather, what cos^2 is.

72
00:04:16,070 --> 00:04:17,680
So let's solve for that.

73
00:04:17,680 --> 00:04:20,120
So I'll put the 1
on the other side.

74
00:04:20,120 --> 00:04:24,120
And I get 1 + cos(2theta).

75
00:04:24,120 --> 00:04:27,910
And then, I want to divide by
this 2, and so that puts a 2

76
00:04:27,910 --> 00:04:30,280
in this denominator here.

77
00:04:30,280 --> 00:04:33,280
So some people call that
the half angle formula.

78
00:04:33,280 --> 00:04:36,730
What it really is for us is
it's a way of eliminating powers

79
00:04:36,730 --> 00:04:38,800
from sines and cosines.

80
00:04:38,800 --> 00:04:41,290
I've gotten rid of this
square at the expense

81
00:04:41,290 --> 00:04:43,720
of putting in a 2theta here.

82
00:04:43,720 --> 00:04:45,260
We'll use that.

83
00:04:45,260 --> 00:04:51,022
And, similarly, same calculation
shows that sin^2(theta) = (1 -

84
00:04:51,022 --> 00:04:51,730
cos(2theta)) / 2.

85
00:04:51,730 --> 00:04:55,850


86
00:04:55,850 --> 00:05:01,480
Same cosine, in that formula
also, but it has a minus sign.

87
00:05:01,480 --> 00:05:03,640
For the sin^2.

88
00:05:03,640 --> 00:05:07,860
OK. so that's my little
review of trig identities

89
00:05:07,860 --> 00:05:14,620
that we'll make use of
as this lecture goes on.

90
00:05:14,620 --> 00:05:17,720
I want to talk about trig
identity-- trig integrals,

91
00:05:17,720 --> 00:05:23,500
and you know some trig
integrals, I'm sure, already.

92
00:05:23,500 --> 00:05:26,740
Like, well, let me write
the differential form first.

93
00:05:26,740 --> 00:05:30,400
You know that d sin
theta, or maybe I'll

94
00:05:30,400 --> 00:05:35,620
say d sin x, is,
let's see, that's

95
00:05:35,620 --> 00:05:39,030
the derivative of sin
x times dx, right.

96
00:05:39,030 --> 00:05:46,120
The derivative of
sin x is cos x, dx.

97
00:05:46,120 --> 00:05:50,080
And so if I integrate both sides
here, the integral form of this

98
00:05:50,080 --> 00:05:55,390
is the integral of cos x dx.

99
00:05:55,390 --> 00:05:59,850
Is sin x plus a constant.

100
00:05:59,850 --> 00:06:07,410
And in the same way,
d cos x = -sin x dx.

101
00:06:07,410 --> 00:06:10,320
Right, the derivative of
the cosine is minus sine.

102
00:06:10,320 --> 00:06:15,330
And when I integrate that, I
find the integral of sin x dx

103
00:06:15,330 --> 00:06:21,140
is -cos x + c.

104
00:06:21,140 --> 00:06:22,820
So that's our starting point.

105
00:06:22,820 --> 00:06:27,510
And the game today, for the
first half of the lecture,

106
00:06:27,510 --> 00:06:33,550
is to use that basic-- just
those basic integration

107
00:06:33,550 --> 00:06:37,710
formulas, together with
clever use of trig identities

108
00:06:37,710 --> 00:06:41,260
in order to compute more
complicated formulas involving

109
00:06:41,260 --> 00:06:42,820
trig functions.

110
00:06:42,820 --> 00:06:47,040
So the first thing,
the first topic,

111
00:06:47,040 --> 00:06:52,250
is to think about integrals of
the form sin^n (x) cos^n (x)

112
00:06:52,250 --> 00:06:52,750
dx.

113
00:06:52,750 --> 00:06:56,190


114
00:06:56,190 --> 00:07:04,070
Where here I have in mind m and
n are non-negative integers.

115
00:07:04,070 --> 00:07:05,950
So let's try to integrate these.

116
00:07:05,950 --> 00:07:09,640
I'll show you some applications
of these pretty soon.

117
00:07:09,640 --> 00:07:11,250
Looking down the
road a little bit,

118
00:07:11,250 --> 00:07:14,050
integrals like this show
up in Fourier series

119
00:07:14,050 --> 00:07:16,190
and many other subjects
in mathematics.

120
00:07:16,190 --> 00:07:20,000
It turns out they're quite
important to be able to do.

121
00:07:20,000 --> 00:07:23,860
So that's why we're
doing them now.

122
00:07:23,860 --> 00:07:29,690
Well, so there are two
cases to think about here.

123
00:07:29,690 --> 00:07:32,070
When you're integrating
things like this.

124
00:07:32,070 --> 00:07:35,890
There's the easy case, and
let's do that one first.

125
00:07:35,890 --> 00:07:49,495
The easy case is when at
least one exponent is odd.

126
00:07:49,495 --> 00:07:50,370
That's the easy case.

127
00:07:50,370 --> 00:07:55,870
So, for example, suppose
that I wanted to integrate,

128
00:07:55,870 --> 00:08:02,560
well, let's take the case m = 1.

129
00:08:02,560 --> 00:08:09,530
So I'm integrating
sin^n (x) cos x dx.

130
00:08:09,530 --> 00:08:17,050
I'm taking-- Oh, I
could do that one.

131
00:08:17,050 --> 00:08:23,600
Let's see if that's
what I want to take.

132
00:08:23,600 --> 00:08:27,540
Yeah.

133
00:08:27,540 --> 00:08:30,590
My confusion is that I meant
to have this a different power.

134
00:08:30,590 --> 00:08:34,750
You were thinking that.

135
00:08:34,750 --> 00:08:36,670
So let's do this
case when m = 1.

136
00:08:36,670 --> 00:08:38,680
So the integral I'm
trying to do is any power

137
00:08:38,680 --> 00:08:41,450
of the sine times the cosine.

138
00:08:41,450 --> 00:08:44,520
Well, here's the trick.

139
00:08:44,520 --> 00:08:48,920
Recognize, use this
formula up at the top

140
00:08:48,920 --> 00:08:53,570
there to see cos x dx as
something that we already

141
00:08:53,570 --> 00:08:55,670
have on the blackboard.

142
00:08:55,670 --> 00:08:59,570
So, the way to exploit that
is to make a substitution.

143
00:08:59,570 --> 00:09:08,320
And substitution is
going to be u = sin x.

144
00:09:08,320 --> 00:09:09,610
And here's why.

145
00:09:09,610 --> 00:09:12,820
Then this integral that I'm
trying to do is the integral

146
00:09:12,820 --> 00:09:18,760
of u^n, that's already
a simplification.

147
00:09:18,760 --> 00:09:22,160
And then there's that cos x dx.

148
00:09:22,160 --> 00:09:25,120
When you make a substitution,
you've got to go all the way

149
00:09:25,120 --> 00:09:27,970
and replace everything
in the expression

150
00:09:27,970 --> 00:09:33,000
by things involving this new
variable that I've introduced.

151
00:09:33,000 --> 00:09:35,390
So I'd better get
rid of the cos x dx

152
00:09:35,390 --> 00:09:39,140
and rewrite it in terms
of du or in terms of u.

153
00:09:39,140 --> 00:09:45,290
And I can do that because du,
according to that formula,

154
00:09:45,290 --> 00:09:50,600
is cos x dx.

155
00:09:50,600 --> 00:09:53,810
Let me put a box around that.

156
00:09:53,810 --> 00:09:55,800
That's our substitution.

157
00:09:55,800 --> 00:09:57,500
When you make a
substitution, you also

158
00:09:57,500 --> 00:10:00,740
want to compute the
differential of the variable

159
00:10:00,740 --> 00:10:03,100
that you substitute in.

160
00:10:03,100 --> 00:10:08,800
So the cos x dx that appears
here is just, exactly, du.

161
00:10:08,800 --> 00:10:11,500
And I've replaced this trig
integral with something

162
00:10:11,500 --> 00:10:13,330
that doesn't involve
trig functions at all.

163
00:10:13,330 --> 00:10:14,240
This is a lot easier.

164
00:10:14,240 --> 00:10:17,420
We can just plug into
what we know here.

165
00:10:17,420 --> 00:10:23,640
This is u^(n+1) /
(n+1) plus a constant,

166
00:10:23,640 --> 00:10:26,620
and I've done the integral.

167
00:10:26,620 --> 00:10:29,580
But I'm not quite done
with the problem yet.

168
00:10:29,580 --> 00:10:34,310
Because to be nice to your
reader and to yourself,

169
00:10:34,310 --> 00:10:36,560
you should go back at
this point, probably,

170
00:10:36,560 --> 00:10:40,399
go back and get rid of this new
variable that you introduced.

171
00:10:40,399 --> 00:10:42,440
You're the one who introduced
this variable, you.

172
00:10:42,440 --> 00:10:45,950
Nobody except you,
really, knows what it is.

173
00:10:45,950 --> 00:10:48,080
But the rest of the
world knows what

174
00:10:48,080 --> 00:10:51,360
they asked for the first
place that involved x.

175
00:10:51,360 --> 00:10:53,460
So I have to go back
and get rid of this.

176
00:10:53,460 --> 00:10:57,830
And that's not hard to do in
this case, because u = sin x.

177
00:10:57,830 --> 00:11:04,390
And so I make this
back substitution.

178
00:11:04,390 --> 00:11:05,820
And that's what you get.

179
00:11:05,820 --> 00:11:11,040
So there's the answer.

180
00:11:11,040 --> 00:11:15,930
OK, so the game was, I use this
odd power of the cosine here,

181
00:11:15,930 --> 00:11:19,180
and I could see it appearing as
the differential of the sine.

182
00:11:19,180 --> 00:11:22,470
So that's what made
this substitution work.

183
00:11:22,470 --> 00:11:25,060
Let's do another example
to see how that works out

184
00:11:25,060 --> 00:11:36,490
in a slightly different case.

185
00:11:36,490 --> 00:11:48,420
So here's another example.

186
00:11:48,420 --> 00:11:50,050
Now I do have an odd power.

187
00:11:50,050 --> 00:11:53,800
One of the exponents is odd,
so I'm in the easy case.

188
00:11:53,800 --> 00:11:56,310
But it's not 1.

189
00:11:56,310 --> 00:12:05,590
The game now is to
use this trig identity

190
00:12:05,590 --> 00:12:10,500
to get rid of the largest
even power that you can,

191
00:12:10,500 --> 00:12:13,870
from this odd power here.

192
00:12:13,870 --> 00:12:24,710
So use sin^2 x = 1 - cos^2 x, to
eliminate a lot of powers from

193
00:12:24,710 --> 00:12:26,220
that odd power.

194
00:12:26,220 --> 00:12:28,370
Watch what happens.

195
00:12:28,370 --> 00:12:31,490
So this is not really a
substitution or anything,

196
00:12:31,490 --> 00:12:34,320
this is just a trig identity.

197
00:12:34,320 --> 00:12:38,130
This sine cubed is sine
squared times the sine.

198
00:12:38,130 --> 00:12:41,360
And the sine squared
is 1 - cos^2 x.

199
00:12:41,360 --> 00:12:43,770
And then I have the
remaining sin x.

200
00:12:43,770 --> 00:12:48,970
And then I have cos^2 x dx.

201
00:12:48,970 --> 00:12:53,580
So let me rewrite that a little
bit to see how this works out.

202
00:12:53,580 --> 00:12:59,490
This is the integral
of cos^2 x minus,

203
00:12:59,490 --> 00:13:01,240
and then there's the
product of these two.

204
00:13:01,240 --> 00:13:09,370
That's cos^4 x times sin x dx.

205
00:13:09,370 --> 00:13:12,200
So now I'm really
exactly in the situation

206
00:13:12,200 --> 00:13:13,700
that I was in over here.

207
00:13:13,700 --> 00:13:17,570
I've got a single power
of a sine or cosine.

208
00:13:17,570 --> 00:13:20,060
It happens that
it's a sine here.

209
00:13:20,060 --> 00:13:21,960
But that's not going
to cause any trouble,

210
00:13:21,960 --> 00:13:26,290
we can go ahead and play the
same game that I did there.

211
00:13:26,290 --> 00:13:28,860
So, so far I've just been
using trig identities.

212
00:13:28,860 --> 00:13:41,770
But now I'll use a
trig substitution.

213
00:13:41,770 --> 00:13:45,335
And I think I want to write
these as powers of a variable.

214
00:13:45,335 --> 00:13:47,960
And then this is going to be the
differential of that variable.

215
00:13:47,960 --> 00:13:58,190
So I'll take u to be cos x, and
that means that du = -sin x dx.

216
00:13:58,190 --> 00:14:04,950
There's the substitution.

217
00:14:04,950 --> 00:14:09,550
So when I make that
substitution, what do we get.

218
00:14:09,550 --> 00:14:11,710
Cosine squared becomes u^2.

219
00:14:11,710 --> 00:14:15,470


220
00:14:15,470 --> 00:14:23,150
Cosine to the 4th becomes u^4,
and sin x dx becomes not quite

221
00:14:23,150 --> 00:14:27,570
du, watch for the signum,
watch for this minus sign here.

222
00:14:27,570 --> 00:14:32,100
It becomes -du.

223
00:14:32,100 --> 00:14:32,820
But that's OK.

224
00:14:32,820 --> 00:14:34,720
The minus sign comes outside.

225
00:14:34,720 --> 00:14:36,860
And I can integrate
both of these powers,

226
00:14:36,860 --> 00:14:43,540
so I get -u^3 / 3.

227
00:14:43,540 --> 00:14:48,950
And then this 4th power gives me
a 5th power, when I integrate.

228
00:14:48,950 --> 00:14:53,390
And don't forget the constant.

229
00:14:53,390 --> 00:14:55,100
Am I done?

230
00:14:55,100 --> 00:14:55,850
Not quite done.

231
00:14:55,850 --> 00:14:57,690
I have to back
substitute and get rid

232
00:14:57,690 --> 00:15:00,830
of my choice of variable, u,
and replace it with yours.

233
00:15:00,830 --> 00:15:01,330
Questions?

234
00:15:01,330 --> 00:15:06,049
STUDENT: [INAUDIBLE]

235
00:15:06,049 --> 00:15:07,340
PROFESSOR: There should indeed.

236
00:15:07,340 --> 00:15:10,190
I forgot this minus sign
when I came down here.

237
00:15:10,190 --> 00:15:12,690
So these two gang up
to give me a plus.

238
00:15:12,690 --> 00:15:14,960
Was that what the other
question was about, too?

239
00:15:14,960 --> 00:15:16,720
Thanks.

240
00:15:16,720 --> 00:15:18,030
So let's back substitute.

241
00:15:18,030 --> 00:15:23,900
And I'm going to
put that over here.

242
00:15:23,900 --> 00:15:27,920
And the result is, well, I just
replace the u by cosine of x.

243
00:15:27,920 --> 00:15:38,160
So this is - -cos^3(x) / 3 plus,
thank you, cos^5(x) / 5 + c.

244
00:15:38,160 --> 00:15:44,670
And there's the answer.

245
00:15:44,670 --> 00:15:47,210
By the way, you can remember
one of the nice things

246
00:15:47,210 --> 00:15:49,160
about doing an
integral is it's fairly

247
00:15:49,160 --> 00:15:51,280
easy to check your answer.

248
00:15:51,280 --> 00:15:53,680
You can always differentiate
the thing you get,

249
00:15:53,680 --> 00:15:56,890
and see whether you get the
right thing when you go back.

250
00:15:56,890 --> 00:15:59,280
It's not too hard
to use the power

251
00:15:59,280 --> 00:16:02,500
rules and the
differentiation rule

252
00:16:02,500 --> 00:16:06,080
for the cosine to get
back to this if you

253
00:16:06,080 --> 00:16:09,350
want to check the work.

254
00:16:09,350 --> 00:16:12,830
Let's do one more
example, just to handle

255
00:16:12,830 --> 00:16:15,980
an example of this
easy case, which you

256
00:16:15,980 --> 00:16:18,490
might have thought of at first.

257
00:16:18,490 --> 00:16:22,720
Suppose I just want to
integrate a cube. sin^3 x.

258
00:16:22,720 --> 00:16:29,320


259
00:16:29,320 --> 00:16:32,040
No cosine in sight.

260
00:16:32,040 --> 00:16:35,780
But I do have an
odd power of a trig

261
00:16:35,780 --> 00:16:37,180
function, of a sine or cosine.

262
00:16:37,180 --> 00:16:39,020
So I'm in the easy case.

263
00:16:39,020 --> 00:16:44,740
And the procedure that I was
suggesting says I want to take

264
00:16:44,740 --> 00:16:48,700
out the largest even power
that I can, from the sin^3.

265
00:16:48,700 --> 00:16:52,870
So I'll take that out, that's
a sin^2, and write it as 1 -

266
00:16:52,870 --> 00:16:53,370
cos^2.

267
00:16:53,370 --> 00:16:57,020


268
00:16:57,020 --> 00:16:58,380
Well, now I'm very happy.

269
00:16:58,380 --> 00:17:00,470
Because it's just
like the situation

270
00:17:00,470 --> 00:17:06,017
we had somewhere
on the board here.

271
00:17:06,017 --> 00:17:07,850
It's just like the
situation we had up here.

272
00:17:07,850 --> 00:17:11,660
I've got a power of a
cosine times sin x dx.

273
00:17:11,660 --> 00:17:16,310
So exactly the same
substitution steps in.

274
00:17:16,310 --> 00:17:19,070
You get, and maybe
you can see what

275
00:17:19,070 --> 00:17:20,860
happens without doing the work.

276
00:17:20,860 --> 00:17:22,630
Shall I do the work here?

277
00:17:22,630 --> 00:17:24,450
I make the same substitution.

278
00:17:24,450 --> 00:17:30,680
And so this is (1 - u
(1 - u^2) times -du.

279
00:17:30,680 --> 00:17:33,540


280
00:17:33,540 --> 00:17:40,050
Which is u - u^3 / 3.

281
00:17:40,050 --> 00:17:42,050
But then I want to put
this minus sign in place,

282
00:17:42,050 --> 00:17:47,700
and so that gives me -u +
u^3 / 3 plus a constant.

283
00:17:47,700 --> 00:17:58,090
And then I back substitute
and get cos x + cos^3 x / 3.

284
00:17:58,090 --> 00:17:59,350
So this is the easy case.

285
00:17:59,350 --> 00:18:02,190
If you have some odd
power to play with,

286
00:18:02,190 --> 00:18:07,990
then you can make use of it and
it's pretty straightforward.

287
00:18:07,990 --> 00:18:10,680
OK the harder case is when
you don't have an odd power.

288
00:18:10,680 --> 00:18:11,610
So what's the program?

289
00:18:11,610 --> 00:18:13,770
I'm going to do the
harder case, and then I'm

290
00:18:13,770 --> 00:18:19,340
going to show you an example of
how to integrate square roots.

291
00:18:19,340 --> 00:18:26,260
And do an application, using
these ideas from trigonometry.

292
00:18:26,260 --> 00:18:30,250
So I want to keep
this blackboard.

293
00:18:30,250 --> 00:18:34,470
Maybe I'll come back
and start here again.

294
00:18:34,470 --> 00:18:55,240
So the harder case is when
they're only even exponents.

295
00:18:55,240 --> 00:18:58,190
I'm still trying to
integrate the same form.

296
00:18:58,190 --> 00:19:00,370
But now all the
exponents are even.

297
00:19:00,370 --> 00:19:03,510
So we have to do some game.

298
00:19:03,510 --> 00:19:10,400
And here the game is use
the half angle formula.

299
00:19:10,400 --> 00:19:16,455
Which I just erased, very
sadly, on the board here.

300
00:19:16,455 --> 00:19:17,830
Maybe I'll rewrite
them over here

301
00:19:17,830 --> 00:19:23,550
so we have them on the board.

302
00:19:23,550 --> 00:19:44,270
I think I remember
what they were.

303
00:19:44,270 --> 00:19:46,300
So the game is I'm going
to use that half angle

304
00:19:46,300 --> 00:19:50,180
formula to start getting
rid of those even powers.

305
00:19:50,180 --> 00:19:54,510
Half angle formula written like
this, exactly, talks about-- it

306
00:19:54,510 --> 00:19:57,790
rewrites even powers
of sines and cosines.

307
00:19:57,790 --> 00:20:00,950
So let's see how that
works out in an example.

308
00:20:00,950 --> 00:20:08,880
How about just the cosine
squared for a start.

309
00:20:08,880 --> 00:20:09,700
What to do?

310
00:20:09,700 --> 00:20:11,970
I can't pull anything out.

311
00:20:11,970 --> 00:20:15,180
I could rewrite
this as 1 - sin^2,

312
00:20:15,180 --> 00:20:17,290
but then I'd be faced with
integrating the sin^2,

313
00:20:17,290 --> 00:20:19,480
which is exactly as hard.

314
00:20:19,480 --> 00:20:23,100
So instead, let's use
this formula here.

315
00:20:23,100 --> 00:20:29,180
This is really the same
as (1+cos(2theta)) / 2.

316
00:20:29,180 --> 00:20:32,700
And now, this is easy.

317
00:20:32,700 --> 00:20:34,340
It's got two parts to it.

318
00:20:34,340 --> 00:20:38,550
Integrating one half
gives me theta over-- Oh.

319
00:20:38,550 --> 00:20:42,440
Miraculously, the x
turned into a theta.

320
00:20:42,440 --> 00:20:44,330
Let's put it back as x.

321
00:20:44,330 --> 00:20:47,440
I get x/2 by integrating 1/2.

322
00:20:47,440 --> 00:20:50,110
So, notice that something
non-trigonometric occurs here

323
00:20:50,110 --> 00:20:52,710
when I do these even integrals.

324
00:20:52,710 --> 00:20:54,800
x/2 appears.

325
00:20:54,800 --> 00:20:57,880
And then the other one, OK, so
this takes a little thought.

326
00:20:57,880 --> 00:21:01,870
The integral of the
cosine is the sine,

327
00:21:01,870 --> 00:21:05,520
or is it minus the sine.

328
00:21:05,520 --> 00:21:11,390
Negative sine.

329
00:21:11,390 --> 00:21:12,634
Shall we take a vote?

330
00:21:12,634 --> 00:21:13,550
I think it's positive.

331
00:21:13,550 --> 00:21:18,310
And so you get sin(2x),
but is that right?

332
00:21:18,310 --> 00:21:19,420
Over 2.

333
00:21:19,420 --> 00:21:24,150
If I differentiate the
sin(2x), this 2 comes out.

334
00:21:24,150 --> 00:21:25,950
And would give me
an extra 2 here.

335
00:21:25,950 --> 00:21:29,580
So there's an extra 2
that I have to put in here

336
00:21:29,580 --> 00:21:34,670
when I integrate it.

337
00:21:34,670 --> 00:21:37,230
And there's the answer.

338
00:21:37,230 --> 00:21:39,480
This is not a substitution.

339
00:21:39,480 --> 00:21:41,550
I just played with
trig identities here.

340
00:21:41,550 --> 00:21:45,020
And then did a
simple trig integral,

341
00:21:45,020 --> 00:21:46,900
getting your help to
get the sign right.

342
00:21:46,900 --> 00:21:49,310
And thinking about what
this 2 is going to do.

343
00:21:49,310 --> 00:21:52,990
It produces a 2 in
the denominator.

344
00:21:52,990 --> 00:21:59,270
But it's not applying
any complicated thing.

345
00:21:59,270 --> 00:22:03,100
It's just using this identity.

346
00:22:03,100 --> 00:22:05,580
Let's do another example
that's a little bit harder.

347
00:22:05,580 --> 00:22:07,690
This time, sin^2 times cos^2.

348
00:22:07,690 --> 00:22:35,940


349
00:22:35,940 --> 00:22:37,920
Again, no odd powers.

350
00:22:37,920 --> 00:22:40,730
I've got to work a
little bit harder.

351
00:22:40,730 --> 00:22:42,930
And what I'm going to do
is apply those identities

352
00:22:42,930 --> 00:22:44,210
up there.

353
00:22:44,210 --> 00:22:47,670
Now, what I recommend
doing in this situation

354
00:22:47,670 --> 00:22:51,370
is going over to
the side somewhere.

355
00:22:51,370 --> 00:22:55,610
And do some side work.

356
00:22:55,610 --> 00:22:58,680
Because it's all just
playing with trig functions.

357
00:22:58,680 --> 00:23:06,280
It's not actually doing
any integrals for a while.

358
00:23:06,280 --> 00:23:11,815
So, I guess one way to get rid
of the sin^2 and the cos^2 is

359
00:23:11,815 --> 00:23:14,180
to use those identities
and so let's do that.

360
00:23:14,180 --> 00:23:16,210
So the sine is (1
- cos(2x)) / 2.

361
00:23:16,210 --> 00:23:20,350


362
00:23:20,350 --> 00:23:22,170
And the cosine is
(1 + cos(2x)) / 2.

363
00:23:22,170 --> 00:23:27,890


364
00:23:27,890 --> 00:23:29,870
So I just substitute them in.

365
00:23:29,870 --> 00:23:31,560
And now I can multiply that out.

366
00:23:31,560 --> 00:23:38,030
And what I have is a
difference times a sum.

367
00:23:38,030 --> 00:23:40,771
So you know a formula for that.

368
00:23:40,771 --> 00:23:43,270
Taking the product of these two
things, well there'll be a 4

369
00:23:43,270 --> 00:23:44,430
in the denominator.

370
00:23:44,430 --> 00:23:46,490
And then in the numerator,
I get the square

371
00:23:46,490 --> 00:23:49,650
of this minus the
square of this.

372
00:23:49,650 --> 00:23:59,840
(a-b)(a+b) = a^2 -
b^2. = - So I get that.

373
00:23:59,840 --> 00:24:02,450
Well, I'm a little bit
happier, because at least I

374
00:24:02,450 --> 00:24:03,710
don't have 4.

375
00:24:03,710 --> 00:24:07,430
I don't have 2
different squares.

376
00:24:07,430 --> 00:24:09,940
I still have a square, and
want to integrate this.

377
00:24:09,940 --> 00:24:12,170
I'm still not in the easy case.

378
00:24:12,170 --> 00:24:16,970
I got myself back to
an easier hard case.

379
00:24:16,970 --> 00:24:18,670
But we do know what
to do about this.

380
00:24:18,670 --> 00:24:21,140
Because I just did it up there.

381
00:24:21,140 --> 00:24:24,430
And I could play into
this formula that we got.

382
00:24:24,430 --> 00:24:29,010
But I think it's just as easy
to continue to calculate here.

383
00:24:29,010 --> 00:24:32,880
Use the half angle
formula again for this,

384
00:24:32,880 --> 00:24:34,900
and continue on your way.

385
00:24:34,900 --> 00:24:37,720
So I get a 1/4 from this bit.

386
00:24:37,720 --> 00:24:43,260
And then minus 1/4 of cos^2(2x).

387
00:24:43,260 --> 00:24:45,780


388
00:24:45,780 --> 00:24:51,630
And when I plug in 2x in for
theta, there in the top board,

389
00:24:51,630 --> 00:24:59,650
I'm going to get a 4x
on the right-hand side.

390
00:24:59,650 --> 00:25:02,130
So it comes out like that.

391
00:25:02,130 --> 00:25:04,600
And I guess I could simplify
that a little bit more.

392
00:25:04,600 --> 00:25:05,790
This is a 1/4.

393
00:25:05,790 --> 00:25:07,840
Oh, but then there's a 2 here.

394
00:25:07,840 --> 00:25:10,630
It's half that.

395
00:25:10,630 --> 00:25:12,200
So then I can simplify
a little more.

396
00:25:12,200 --> 00:25:16,070
It's 1/4 - 1/8, which is 1/8.

397
00:25:16,070 --> 00:25:19,630
And then I have 1/8 cos(4x).

398
00:25:19,630 --> 00:25:25,510


399
00:25:25,510 --> 00:25:27,560
OK, that's my side work.

400
00:25:27,560 --> 00:25:30,590
I just did some trig
identities over here.

401
00:25:30,590 --> 00:25:32,840
And rewrote sine
squared times cosine

402
00:25:32,840 --> 00:25:35,930
squared as something which
involves just no powers

403
00:25:35,930 --> 00:25:37,940
of trig, just cosine by itself.

404
00:25:37,940 --> 00:25:41,330
And a constant.

405
00:25:41,330 --> 00:25:45,090
So I can take that and
substitute it in here.

406
00:25:45,090 --> 00:25:48,810
And now the integration
is pretty easy.

407
00:25:48,810 --> 00:25:57,810
1/8, cos(4x) / 8,
dx, which is, OK

408
00:25:57,810 --> 00:26:01,340
the 1/8 is going to give me x/8.

409
00:26:01,340 --> 00:26:06,314
The integral or cosine is
plus or minus the sine.

410
00:26:06,314 --> 00:26:08,230
The derivative of the
sine is plus the cosine.

411
00:26:08,230 --> 00:26:11,440
So it's going to be plus the--
Only there's a minus here.

412
00:26:11,440 --> 00:26:17,890
So it's going to be the
sine-- minus sin(4x) / 8,

413
00:26:17,890 --> 00:26:20,797
but then I have an additional
factor in the denominator.

414
00:26:20,797 --> 00:26:21,880
And what's it going to be?

415
00:26:21,880 --> 00:26:28,830
I have to put a 4 there.

416
00:26:28,830 --> 00:26:32,730
So we've done that
calculation, too.

417
00:26:32,730 --> 00:26:38,250
So any of these-- If you keep
doing this kind of process,

418
00:26:38,250 --> 00:26:45,530
these two kinds
of procedures, you

419
00:26:45,530 --> 00:26:48,840
can now integrate
any expression that

420
00:26:48,840 --> 00:26:52,040
has a power of a sine times
a power of a cosine in it,

421
00:26:52,040 --> 00:26:56,860
by using these ideas.

422
00:26:56,860 --> 00:27:01,730
Now, let's see.

423
00:27:01,730 --> 00:27:16,210
Oh, let me give you an alternate
method for this last one here.

424
00:27:16,210 --> 00:27:26,460
I know what I'll do.

425
00:27:26,460 --> 00:27:28,850
Let me give an alternate
method for doing, really

426
00:27:28,850 --> 00:27:31,670
doing the side work over there.

427
00:27:31,670 --> 00:27:35,920
I'm trying to deal
with sin^2 times cos^2.

428
00:27:35,920 --> 00:27:50,080
Well that's the
square of sin x cos x.

429
00:27:50,080 --> 00:27:54,240
And sin x cos x
shows up right here.

430
00:27:54,240 --> 00:27:55,900
In another trig identity.

431
00:27:55,900 --> 00:27:58,410
So we can make use of that, too.

432
00:27:58,410 --> 00:28:01,330
That reduces the number of
factors of sines and cosines

433
00:28:01,330 --> 00:28:01,930
by 1.

434
00:28:01,930 --> 00:28:04,190
So it's going in
the right direction.

435
00:28:04,190 --> 00:28:11,270
This is equal to 1/2
sin(2x), squared.

436
00:28:11,270 --> 00:28:17,730
Sine times cosine is
1/2-- Say this right.

437
00:28:17,730 --> 00:28:21,280
It's sin(2x) / 2, and then
I want to square that.

438
00:28:21,280 --> 00:28:31,030
So what I get is sin^2(2x) / 4.

439
00:28:31,030 --> 00:28:33,930
Which is, well, I'm
not too happy yet,

440
00:28:33,930 --> 00:28:35,639
because I still
have an even power.

441
00:28:35,639 --> 00:28:37,930
Remember I'm trying to
integrate this thing in the end,

442
00:28:37,930 --> 00:28:39,060
even powers are bad.

443
00:28:39,060 --> 00:28:40,910
I try to get rid of them.

444
00:28:40,910 --> 00:28:46,360
By using that formula,
the half angle formula.

445
00:28:46,360 --> 00:28:48,950
So I can apply that
to sin x here again.

446
00:28:48,950 --> 00:28:52,600
I get 1/4 of (1 - cos(4x)) / 2.

447
00:28:52,600 --> 00:28:57,480


448
00:28:57,480 --> 00:28:59,730
That's what the half angle
formula says for sin^2(2x).

449
00:28:59,730 --> 00:29:02,330


450
00:29:02,330 --> 00:29:04,790
And that's exactly the
same as the expression

451
00:29:04,790 --> 00:29:08,660
that I got up here, as well.

452
00:29:08,660 --> 00:29:11,710
It's the same expression
that I have there.

453
00:29:11,710 --> 00:29:16,680
So it's the same
expression as I have here.

454
00:29:16,680 --> 00:29:20,000
So this is just an alternate
way to play this game of using

455
00:29:20,000 --> 00:29:24,940
the half angle formula.

456
00:29:24,940 --> 00:29:27,790
OK, let's do a little
application of these things

457
00:29:27,790 --> 00:29:46,060
and change the
topic a little bit.

458
00:29:46,060 --> 00:29:48,210
So here's the problem.

459
00:29:48,210 --> 00:29:56,900
So this is an
application and example

460
00:29:56,900 --> 00:30:07,570
of a real trig substitution.

461
00:30:07,570 --> 00:30:22,080
So here's the problem
I want to look at.

462
00:30:22,080 --> 00:30:26,210
OK, so I have a circle
whose radius is a.

463
00:30:26,210 --> 00:30:31,710
And I cut out from it
a sort of tab, here.

464
00:30:31,710 --> 00:30:36,330
This tab here.

465
00:30:36,330 --> 00:30:38,590
And the height of
this thing is b.

466
00:30:38,590 --> 00:30:42,110
So this length is a number b.

467
00:30:42,110 --> 00:30:47,219
And what I want to do is compute
the area of that little tab.

468
00:30:47,219 --> 00:30:48,010
That's the problem.

469
00:30:48,010 --> 00:30:50,360
So there's an arc over here.

470
00:30:50,360 --> 00:30:55,060
And I want to find the
area of this, for a and b,

471
00:30:55,060 --> 00:30:57,980
in terms of a and b.

472
00:30:57,980 --> 00:31:06,730
So the area, well,
I guess one way

473
00:31:06,730 --> 00:31:12,370
to compute the area would be
to take the integral of y dx.

474
00:31:12,370 --> 00:31:15,980
You've seen the idea
of splitting this up

475
00:31:15,980 --> 00:31:20,600
into vertical strips whose
height is given by a function

476
00:31:20,600 --> 00:31:21,100
y(x).

477
00:31:21,100 --> 00:31:22,266
And then you integrate that.

478
00:31:22,266 --> 00:31:24,040
That's an interpretation
for the integral.

479
00:31:24,040 --> 00:31:27,210
The area is given by y dx.

480
00:31:27,210 --> 00:31:30,327
But that's a little bit awkward,
because my formula for y

481
00:31:30,327 --> 00:31:31,660
is going to be a little strange.

482
00:31:31,660 --> 00:31:34,630
It's constant, value of b, along
here, and then at this point

483
00:31:34,630 --> 00:31:37,660
it becomes this
arc, of the circle.

484
00:31:37,660 --> 00:31:39,500
So working this
out, I could do it

485
00:31:39,500 --> 00:31:42,100
but it's a little awkward
because expressing y

486
00:31:42,100 --> 00:31:45,880
as a function of x, the
top edge of this shape,

487
00:31:45,880 --> 00:31:50,090
it's a little awkward, and
takes two different regions

488
00:31:50,090 --> 00:31:51,740
to express.

489
00:31:51,740 --> 00:31:58,620
So, a different way to
say it is to say x dy.

490
00:31:58,620 --> 00:32:00,350
Maybe that'll work
a little bit better.

491
00:32:00,350 --> 00:32:02,810
Or maybe it won't,
but it's worth trying.

492
00:32:02,810 --> 00:32:05,290
I could just as well
split this region up

493
00:32:05,290 --> 00:32:08,420
into horizontal strips.

494
00:32:08,420 --> 00:32:13,010
Whose width is dy,
and whose length is x.

495
00:32:13,010 --> 00:32:17,250
Now I'm thinking of
this as a function of y.

496
00:32:17,250 --> 00:32:20,870
This is the graph
of a function of y.

497
00:32:20,870 --> 00:32:24,880
And that's much better, because
the function of y is, well,

498
00:32:24,880 --> 00:32:28,010
it's the square root
of a^2 - y^2, isn't it.

499
00:32:28,010 --> 00:32:34,300
That's x x^2 + y^2 = a^2.

500
00:32:34,300 --> 00:32:38,640
So that's what x is.

501
00:32:38,640 --> 00:32:41,740
And that's what I'm
asked to integrate, then.

502
00:32:41,740 --> 00:32:45,580
Square root of (a^2 - y^2), dy.

503
00:32:45,580 --> 00:32:47,500
And I can even put in
limits of integration.

504
00:32:47,500 --> 00:32:49,500
Maybe I should do that,
because this is supposed

505
00:32:49,500 --> 00:32:50,910
to be an actual number.

506
00:32:50,910 --> 00:32:55,130
I guess I'm integrating it
from y = 0, that's here.

507
00:32:55,130 --> 00:33:00,059
To y = b, dy.

508
00:33:00,059 --> 00:33:01,350
So this is what I want to find.

509
00:33:01,350 --> 00:33:07,420
This is a integral formula
for the area of that region.

510
00:33:07,420 --> 00:33:08,730
And this is a new form.

511
00:33:08,730 --> 00:33:13,970
I don't think that
you've thought

512
00:33:13,970 --> 00:33:20,310
about integrating expressions
like this in this class before.

513
00:33:20,310 --> 00:33:23,660
So, it's a new form and I
want to show you how to do it,

514
00:33:23,660 --> 00:33:30,290
how it's related
to trigonometry.

515
00:33:30,290 --> 00:33:33,080
It's related to trigonometry
through that exact picture

516
00:33:33,080 --> 00:33:36,710
that we have on the blackboard.

517
00:33:36,710 --> 00:33:42,610
After all, this a^2 - y^2
is the formula for this arc.

518
00:33:42,610 --> 00:33:45,460
And so, what I
propose is that we

519
00:33:45,460 --> 00:33:49,700
try to exploit the
connection with the circle

520
00:33:49,700 --> 00:33:52,860
and introduce polar coordinates.

521
00:33:52,860 --> 00:34:03,499
So, here if I measure
this angle then there

522
00:34:03,499 --> 00:34:04,790
are various things you can say.

523
00:34:04,790 --> 00:34:08,076
Like the coordinates of this
point here are a cos(theta),

524
00:34:08,076 --> 00:34:17,670
a-- Well, I'm sorry, it's not.

525
00:34:17,670 --> 00:34:20,780
That's an angle, but I
want to call it theta_0.

526
00:34:20,780 --> 00:34:25,300
And, in general you know
that the coordinates of this

527
00:34:25,300 --> 00:34:31,010
point are (a cos(theta),
a sin(theta)).

528
00:34:31,010 --> 00:34:39,370
If the radius is a, then
the angle here is theta.

529
00:34:39,370 --> 00:34:45,170
So x = a cos(theta),
and y = a sin(theta),

530
00:34:45,170 --> 00:34:49,100
just from looking at the
geometry of the circle.

531
00:34:49,100 --> 00:34:52,530
So let's make that
substitution. y = a sin(theta).

532
00:34:52,530 --> 00:34:56,600


533
00:34:56,600 --> 00:35:00,740
I'm using the picture to
suggest that maybe making

534
00:35:00,740 --> 00:35:02,870
the substitution is
a good thing to do.

535
00:35:02,870 --> 00:35:06,020
Let's follow along
and see what happens.

536
00:35:06,020 --> 00:35:10,310
If that's true, what we're
interested in is integrating,

537
00:35:10,310 --> 00:35:13,580
a^2 - y^2.

538
00:35:13,580 --> 00:35:18,900
Which is a^2-- We're interested
in integrating the square root

539
00:35:18,900 --> 00:35:20,770
of a^2 - y^2.

540
00:35:20,770 --> 00:35:24,130
Which is the square
root of a^2 minus this.

541
00:35:24,130 --> 00:35:27,100
a^2 sin^2(theta).

542
00:35:27,100 --> 00:35:35,870
And, well, that's
equal to a cos theta.

543
00:35:35,870 --> 00:35:41,710
That's just sin^2 + cos^2
= 1, all over again.

544
00:35:41,710 --> 00:35:42,830
It's also x.

545
00:35:42,830 --> 00:35:44,560
This is x.

546
00:35:44,560 --> 00:35:46,070
And this was x.

547
00:35:46,070 --> 00:35:48,960
So there are a lot of different
ways to think of this.

548
00:35:48,960 --> 00:35:51,650
But no matter how you say
it, the thing we're trying

549
00:35:51,650 --> 00:35:59,270
to integrate, a^2 - y^2 is,
under this substitution it is

550
00:35:59,270 --> 00:36:02,570
a cos(theta).

551
00:36:02,570 --> 00:36:06,700
So I'm interested in integrating
the square root of (a^2 - y^2),

552
00:36:06,700 --> 00:36:09,260
dy.

553
00:36:09,260 --> 00:36:14,330
And I'm going to make this
substitution y = a sin(theta).

554
00:36:14,330 --> 00:36:17,330


555
00:36:17,330 --> 00:36:22,320
And so under that substitution,
I've decided that the square

556
00:36:22,320 --> 00:36:26,170
root of a^2 - y^2
is a cos(theta).

557
00:36:26,170 --> 00:36:31,170


558
00:36:31,170 --> 00:36:33,240
That's this.

559
00:36:33,240 --> 00:36:34,690
What about the dy?

560
00:36:34,690 --> 00:36:38,310
Well, I'd better compute the dy.

561
00:36:38,310 --> 00:36:40,920
So dy, just differentiating
this expression,

562
00:36:40,920 --> 00:36:44,820
is a cos(theta) d theta.

563
00:36:44,820 --> 00:36:57,240
So let's put that in. dy
= a cos(theta) d theta.

564
00:36:57,240 --> 00:36:58,310
OK.

565
00:36:58,310 --> 00:37:03,720
Making that trig substitution,
y = a sin(theta),

566
00:37:03,720 --> 00:37:06,750
has replaced this integral
that has a square root in it.

567
00:37:06,750 --> 00:37:08,640
And no trig functions.

568
00:37:08,640 --> 00:37:12,630
With an integral that involves
no square roots and only trig

569
00:37:12,630 --> 00:37:15,140
functions.

570
00:37:15,140 --> 00:37:17,630
In fact, it's not too hard to
integrate this now, because

571
00:37:17,630 --> 00:37:19,270
of the work that we've done.

572
00:37:19,270 --> 00:37:20,980
The a^2 comes out.

573
00:37:20,980 --> 00:37:22,630
This is cos^2(theta) d theta.

574
00:37:22,630 --> 00:37:26,200


575
00:37:26,200 --> 00:37:28,580
And maybe we've done that
example already today.

576
00:37:28,580 --> 00:37:35,920
I think we have.

577
00:37:35,920 --> 00:37:38,435
Maybe we can think it through,
but maybe the easiest thing

578
00:37:38,435 --> 00:37:42,550
is to look back at notes
and see what we got before.

579
00:37:42,550 --> 00:37:46,330
That was the first example
in the hard case that I did.

580
00:37:46,330 --> 00:38:03,060
And what it came out to, I used
x instead of theta at the time.

581
00:38:03,060 --> 00:38:05,570
So this is a good step forward.

582
00:38:05,570 --> 00:38:08,130
I started with this
integral that I really

583
00:38:08,130 --> 00:38:12,440
didn't know how to do by any
means that we've had so far.

584
00:38:12,440 --> 00:38:15,000
And I've replaced it
by a trig integral

585
00:38:15,000 --> 00:38:16,250
that we do know how to do.

586
00:38:16,250 --> 00:38:19,160
And now I've done
that trig integral.

587
00:38:19,160 --> 00:38:22,400
But we're still not quite
done, because of the problem

588
00:38:22,400 --> 00:38:23,830
of back substituting.

589
00:38:23,830 --> 00:38:27,670
I'd like to go back and
rewrite this in terms

590
00:38:27,670 --> 00:38:32,150
of the original variable, y.

591
00:38:32,150 --> 00:38:34,280
Or, I'd like to go
back and rewrite it

592
00:38:34,280 --> 00:38:36,390
in terms of the original
limits of integration

593
00:38:36,390 --> 00:38:40,050
that we had in the
original problem.

594
00:38:40,050 --> 00:38:42,870
In doing that, it's going
to be useful to rewrite

595
00:38:42,870 --> 00:38:47,150
this expression and get
rid of the sin(2theta).

596
00:38:47,150 --> 00:38:51,840
After all, the original
y was expressed in terms

597
00:38:51,840 --> 00:38:54,660
of sin(theta), not sin(2theta).

598
00:38:54,660 --> 00:39:04,220
So let me just do that here,
and say that this, in turn,

599
00:39:04,220 --> 00:39:09,507
is equal to a^2
theta / 2 plus, well,

600
00:39:09,507 --> 00:39:11,090
sin(2theta) = 2
sin(theta) cos(theta).

601
00:39:11,090 --> 00:39:18,080


602
00:39:18,080 --> 00:39:20,970
And so, when there's a 4 in
the denominator, what I'll get

603
00:39:20,970 --> 00:39:25,390
is sin(theta) cos(theta) / 2.

604
00:39:25,390 --> 00:39:32,580


605
00:39:32,580 --> 00:39:36,630
I did that because I'm getting
closer to the original terms

606
00:39:36,630 --> 00:39:39,080
that the problem started with.

607
00:39:39,080 --> 00:39:40,130
Which was sin(theta).

608
00:39:40,130 --> 00:40:05,480


609
00:40:05,480 --> 00:40:08,055
So let me write down the
integral that we have now.

610
00:40:08,055 --> 00:40:14,680
The square root of a^2
- y^2, dy is, so far,

611
00:40:14,680 --> 00:40:26,810
what we know is a^2 (theta / 2 +
sin(theta) cos(theta) / 2) + c.

612
00:40:26,810 --> 00:40:28,460
But I want to go
back and rewrite this

613
00:40:28,460 --> 00:40:30,660
in terms of the original value.

614
00:40:30,660 --> 00:40:32,650
The original variable, y.

615
00:40:32,650 --> 00:40:37,430
Well, what is theta
in terms of y?

616
00:40:37,430 --> 00:40:40,760
Let's see. y in terms of
theta was given like this.

617
00:40:40,760 --> 00:40:44,120
So what is theta in terms of y?

618
00:40:44,120 --> 00:40:44,950
Ah.

619
00:40:44,950 --> 00:40:48,710
So here the fearsome arcsine
rears its head, right?

620
00:40:48,710 --> 00:40:53,680
Theta is the angle so
that y = a sin(theta).

621
00:40:53,680 --> 00:40:58,230
So that means that theta is
the arcsine, or sine inverse,

622
00:40:58,230 --> 00:41:00,750
of y/a.

623
00:41:00,750 --> 00:41:07,450


624
00:41:07,450 --> 00:41:12,630
So that's the first
thing that shows up here.

625
00:41:12,630 --> 00:41:18,290
arcsin(y/a), all over 2.

626
00:41:18,290 --> 00:41:19,360
That's this term.

627
00:41:19,360 --> 00:41:24,530
Theta is arcsin(y/a) / 2.

628
00:41:24,530 --> 00:41:26,630
What about the other side, here?

629
00:41:26,630 --> 00:41:30,620
Well sine and cosine, we knew
what they were in terms of y

630
00:41:30,620 --> 00:41:37,830
and in terms of x, if you like.

631
00:41:37,830 --> 00:41:40,300
Maybe I'll put the
a^2 inside here.

632
00:41:40,300 --> 00:41:42,710
That makes it a
little bit nicer.

633
00:41:42,710 --> 00:41:49,310
Plus, and the other term is
a^2 sin(theta) cos(theta).

634
00:41:49,310 --> 00:41:52,610
So the a sin(theta) is just y.

635
00:41:52,610 --> 00:41:55,140
Maybe I'll write this (a
sin(theta)) (a cos(theta))

636
00:41:55,140 --> 00:41:57,550
/ 2 + c.

637
00:41:57,550 --> 00:42:02,330


638
00:42:02,330 --> 00:42:03,840
And so I get the same thing.

639
00:42:03,840 --> 00:42:06,204
And now here a
sin(theta), that's y.

640
00:42:06,204 --> 00:42:07,370
And what's the a cos(theta)?

641
00:42:07,370 --> 00:42:12,490


642
00:42:12,490 --> 00:42:16,440
It's x, or, if you like, it's
the square root of a^2 - y^2.

643
00:42:16,440 --> 00:42:22,010


644
00:42:22,010 --> 00:42:28,140
And so there I've
rewritten everything, back

645
00:42:28,140 --> 00:42:31,250
in terms of the
original variable, y.

646
00:42:31,250 --> 00:42:36,060
And there's an answer.

647
00:42:36,060 --> 00:42:41,070
So I've done this indefinite
integration of a form--

648
00:42:41,070 --> 00:42:44,810
of this quadratic, this square
root of something which is

649
00:42:44,810 --> 00:42:46,940
a constant minus y^2.

650
00:42:46,940 --> 00:42:50,640
Whenever you see that, the thing
to think of is trigonometry.

651
00:42:50,640 --> 00:42:54,310
That's going to play into
the sin^2 + cos^2 identity.

652
00:42:54,310 --> 00:42:56,880
And the way to exploit it
is to make the substitution

653
00:42:56,880 --> 00:43:01,100
y = a sin(theta).

654
00:43:01,100 --> 00:43:04,340
You could also make a
substitution y = a cos(theta),

655
00:43:04,340 --> 00:43:05,540
if you wanted to.

656
00:43:05,540 --> 00:43:12,290
And the result would come out
to exactly the same in the end.

657
00:43:12,290 --> 00:43:14,430
I'm still not quite done
with the original problem

658
00:43:14,430 --> 00:43:23,990
that I had, because
the original problem

659
00:43:23,990 --> 00:43:25,810
asked for a definite integral.

660
00:43:25,810 --> 00:43:33,000
So let's just go back
and finish that as well.

661
00:43:33,000 --> 00:43:37,910
So the area was
the integral from 0

662
00:43:37,910 --> 00:43:45,740
to b of this square root.

663
00:43:45,740 --> 00:43:48,220
So I just want to evaluate
the right-hand side here.

664
00:43:48,220 --> 00:43:50,880
The answer that we came up
with, this indefinite integral.

665
00:43:50,880 --> 00:43:53,374
I want to evaluate
it at 0 and at b.

666
00:43:53,374 --> 00:43:54,040
Well, let's see.

667
00:43:54,040 --> 00:44:13,470
When I evaluate it at b, I get
a^2 arcsin(b/a) / 2 plus y,

668
00:44:13,470 --> 00:44:18,900
which is b, times the
square root of a^2 - b^2,

669
00:44:18,900 --> 00:44:23,050
putting y = b, divided by 2.

670
00:44:23,050 --> 00:44:26,300
So I've plugged in y
= b into that formula,

671
00:44:26,300 --> 00:44:27,290
this is what I get.

672
00:44:27,290 --> 00:44:31,280
Then when I plug in y = 0,
well the, sine of 0 is 0,

673
00:44:31,280 --> 00:44:34,040
so the arcsine of 0 is 0.

674
00:44:34,040 --> 00:44:35,720
So this term goes away.

675
00:44:35,720 --> 00:44:38,930
And when y = 0,
this term is 0 also.

676
00:44:38,930 --> 00:44:43,210
And so I don't get any
subtracted terms at all.

677
00:44:43,210 --> 00:44:45,600
So there's an
expression for this.

678
00:44:45,600 --> 00:44:52,360
Notice that this arcsin(b/a),
that's exactly this angle.

679
00:44:52,360 --> 00:45:00,550
arcsin(b/a), it's the angle
that you get when y = b.

680
00:45:00,550 --> 00:45:06,760
So this theta is
the arcsin(b/a).

681
00:45:06,760 --> 00:45:09,990


682
00:45:09,990 --> 00:45:15,870
Put this over here.

683
00:45:15,870 --> 00:45:17,090
That is theta_0.

684
00:45:17,090 --> 00:45:21,560
That is the angle
that the corner makes.

685
00:45:21,560 --> 00:45:28,460
So I could rewrite this as a
a^2 theta_0 / 2 plus b times

686
00:45:28,460 --> 00:45:34,722
the square root of
a^2 - b^2, over 2.

687
00:45:34,722 --> 00:45:36,430
Let's just think about
this for a minute.

688
00:45:36,430 --> 00:45:40,320
I have these two terms in
the sum, is that reasonable?

689
00:45:40,320 --> 00:45:44,080
The first term is a^2.

690
00:45:44,080 --> 00:45:50,370
That's the radius squared
times this angle, times 1/2.

691
00:45:50,370 --> 00:45:54,280
Well, I think that is exactly
the area of this sector.

692
00:45:54,280 --> 00:46:03,170
a^2 theta / 2 is the formula
for the area of the sector.

693
00:46:03,170 --> 00:46:07,070
And this one, this is
the vertical elevation.

694
00:46:07,070 --> 00:46:09,500
This is the horizontal.

695
00:46:09,500 --> 00:46:14,550
a^2 - b^2 is this distance.

696
00:46:14,550 --> 00:46:16,980
Square root of a^2 - b^2.

697
00:46:16,980 --> 00:46:20,420
So the right-hand term is b
times the square root of a^2 -

698
00:46:20,420 --> 00:46:31,930
b^2 divided by 2, that's
the area of that triangle.

699
00:46:31,930 --> 00:46:34,280
So using a little
bit of geometry

700
00:46:34,280 --> 00:46:39,890
gives you the same answer as
all of this elaborate calculus.

701
00:46:39,890 --> 00:46:41,610
Maybe that's enough
cause for celebration

702
00:46:41,610 --> 00:46:43,620
for us to quit for today.

703
00:46:43,620 --> 00:46:44,205