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PROFESSOR: Well, because our
subject today is trig integrals
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00:00:26 --> 00:00:30
substitutions, Professor
Jerison called in his
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00:00:30 --> 00:00:33
substitute teacher for today.
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That's me.
13
00:00:46 --> 00:00:49
Professor Miller.
14
00:00:49 --> 00:00:52
And I'm going to try to tell
you about trig substitutions
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and trig integrals.
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00:00:54 --> 00:00:59
And I'll be here tomorrow to
do more of the same, as well.
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00:00:59 --> 00:01:02
So, this is about trigonometry,
and maybe first thing I'll do
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00:01:02 --> 00:01:24
is remind you of some basic
things about trigonometry.
19
00:01:24 --> 00:01:27
So, if I have a circle,
trigonometry is all based on
20
00:01:27 --> 00:01:32
the circle of radius 1, and
centered at the origin.
21
00:01:32 --> 00:01:35
And so if this is an angle of
theta, up from the x-axis, then
22
00:01:35 --> 00:01:37
the coordinates of this point
are cosine theta
23
00:01:37 --> 00:01:39
and sine theta.
24
00:01:39 --> 00:01:42
And so that leads right away
to some trig identities,
25
00:01:42 --> 00:01:43
which you know very well.
26
00:01:43 --> 00:01:46
But I'm going to put them up
here because we'll use them
27
00:01:46 --> 00:01:51
over and over again today.
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00:01:51 --> 00:01:54
Remember the convention
sin^2 theta secretly
29
00:01:54 --> 00:01:55
means (sin theta)^2.
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00:01:57 --> 00:02:00
It would be more sensible to
write a parenthesis around
31
00:02:00 --> 00:02:03
the sign of theta and
then say you square that.
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00:02:03 --> 00:02:06
But everybody in the world puts
the 2 up there over the sin,
33
00:02:06 --> 00:02:09
and so I'll do that too.
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00:02:09 --> 00:02:12
So that follows just because
the circle has radius 1.
35
00:02:12 --> 00:02:14
But then there are some
other identities too, which
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00:02:14 --> 00:02:15
I think you remember.
37
00:02:15 --> 00:02:17
I'll write them down here.
38
00:02:17 --> 00:02:21
Cos 2 theta, there's this
double angle formula that
39
00:02:21 --> 00:02:29
says cos 2 theta = cos
^2 theta - sin ^2 theta.
40
00:02:29 --> 00:02:31
And there's also the
double angle formula
41
00:02:31 --> 00:02:34
for the sin 2 theta.
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00:02:34 --> 00:02:38
Remember what that says?
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00:02:38 --> 00:02:46
2 sin theta cos theta.
44
00:02:46 --> 00:02:48
I'm going to use these trig
identities and I'm going to
45
00:02:48 --> 00:02:50
use them in a slightly
different way.
46
00:02:50 --> 00:02:53
And so I'd like to pay a little
more attention to this one and
47
00:02:53 --> 00:02:57
get a different way of
writing this one out.
48
00:02:57 --> 00:03:06
So this is actually the
half angle formula.
49
00:03:06 --> 00:03:14
And that says, I'm going to try
to express the cos theta in
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00:03:14 --> 00:03:16
terms of the cos 2 theta.
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00:03:16 --> 00:03:20
So if I know the cos 2 theta,
I want to try to express the
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00:03:20 --> 00:03:23
cos theta in terms of it.
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00:03:23 --> 00:03:30
Well, I'll start with a cos
2 theta and play with that.
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OK.
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00:03:30 --> 00:03:34
Well, we know what this
is, it's cos ^2 theta
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00:03:34 --> 00:03:36
- sin ^2 theta.
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00:03:36 --> 00:03:38
But we also know what the
sin square root of theta
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00:03:38 --> 00:03:40
is in terms of the cosine.
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00:03:40 --> 00:03:44
So I can eliminate the
sin^2 from this picture.
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00:03:44 --> 00:03:48
So this is equal to the cosine
^2 theta - (the quantity
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00:03:48 --> 00:03:50
1 - cos ^2 theta).
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00:03:50 --> 00:03:53
I put in what sin^2 is
in terms of cos^2.
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And so that's 2 cos
^2 of theta - 1.
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There's this cos^2,
which gets a plus sign.
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Because of these
two minus signs.
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And there's the one that was
there before, so altogether
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there are two of them.
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I want to isolate
what cosine is.
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Or rather, what cos^2 is.
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00:04:16 --> 00:04:17
So let's solve for that.
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So I'll put the 1
on the other side.
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And I get 1 + cos 2 theta.
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And then, I want to divide by
this 2, and so that puts a
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2 in this denominator here.
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So some people call that
the half angle formula.
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00:04:33 --> 00:04:35
What it really is for us is
it's a way of eliminating
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powers from sines and cosines.
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I've gotten rid of this square
at the expense of putting
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in a 2 theta here.
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We'll use that.
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00:04:45 --> 00:04:49
And, similarly, same
calculation shows that the sin
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00:04:49 --> 00:04:55
^2 theta = 1 cos 2 theta / 2.
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00:04:55 --> 00:05:01
Same cosine, in that formula
also, but it has a minus sign.
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00:05:01 --> 00:05:01
For the sin^2.
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OK. so that's my little review
of trig identities that we'll
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make use of as this
lecture goes on.
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00:05:14 --> 00:05:16
I want to talk about
trig identity.
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00:05:16 --> 00:05:19
Trig integrals, and you
know some trig integrals,
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00:05:19 --> 00:05:23
I'm sure, already.
90
00:05:23 --> 00:05:26
Like, well, let me write the
differential form first.
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00:05:26 --> 00:05:34
You know that d sin theta, or
maybe I'll say d sin x, is,
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00:05:34 --> 00:05:39
let's see, that's the
derivative of sin x dx, right.
93
00:05:39 --> 00:05:46
The derivative of
sin x = cos x dx.
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00:05:46 --> 00:05:49
And so if I integrate both
sides here, the integral form
95
00:05:49 --> 00:05:55
of this is the integral
of the cos x dx.
96
00:05:55 --> 00:05:59
Is the sin x + a constant.
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00:05:59 --> 00:06:07
And in the same way, d
cos x = - sin x dx.
98
00:06:07 --> 00:06:10
Right, the derivative of
the cosine is - sine.
99
00:06:10 --> 00:06:12
And when I integrate that,
I find the integral of the
100
00:06:12 --> 00:06:21
sin x dx = - cos x + c.
101
00:06:21 --> 00:06:22
So that's our starting point.
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00:06:22 --> 00:06:27
And the game today, for the
first half of the lecture, is
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00:06:27 --> 00:06:34
to use that basic, just those
basic integration formulas,
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00:06:34 --> 00:06:38
together with clever use of
trig identities in order to
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00:06:38 --> 00:06:41
compute more complicated
formulas involving
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00:06:41 --> 00:06:42
trig functions.
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So the first thing, the first
topic, is to think about
108
00:06:47 --> 00:06:52
integrals of the form
sin^n (x) cos^n(x).
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00:06:52 --> 00:06:56
110
00:06:56 --> 00:07:00
Where here I have
in mind m and n.
111
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Are non-negative integers.
112
00:07:04 --> 00:07:05
So let's try to
integrate these.
113
00:07:05 --> 00:07:09
I'll show you some applications
of these pretty soon.
114
00:07:09 --> 00:07:12
Looking down the road a little
bit, integrals like this show
115
00:07:12 --> 00:07:16
up in Fourier series and many
other subjects in mathematics.
116
00:07:16 --> 00:07:20
It turns out they're quite
important to be able to do.
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00:07:20 --> 00:07:23
So that's why we're
doing them now.
118
00:07:23 --> 00:07:29
Well, so there are two
cases to think about here.
119
00:07:29 --> 00:07:32
When you're integrating
things like this.
120
00:07:32 --> 00:07:35
There's the easy case, and
let's do that one first.
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00:07:35 --> 00:07:49
The easy case is when at
least one exponent is odd.
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00:07:49 --> 00:07:50
That's the easy case.
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00:07:50 --> 00:07:56
So, for example, suppose that
I wanted to integrate, well,
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00:07:56 --> 00:08:02
let's take the case m = 1.
125
00:08:02 --> 00:08:09
So I'm integrating
sin^n (x) cos x dx.
126
00:08:09 --> 00:08:15
I'm taking -- oh.
127
00:08:15 --> 00:08:17
I could do that one.
128
00:08:17 --> 00:08:23
Let's see if that's
what I want to take.
129
00:08:23 --> 00:08:27
Yeah.
130
00:08:27 --> 00:08:30
My confusion is that I meant to
have this a different power.
131
00:08:30 --> 00:08:34
You were thinking that.
132
00:08:34 --> 00:08:36
So let's do this
case when m = 1.
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00:08:36 --> 00:08:41
So the integral I'm trying to
do is any power of sin cos.
134
00:08:41 --> 00:08:44
Well, here's the trick.
135
00:08:44 --> 00:08:51
Recognize, use this formula up
at the top there to see cos x
136
00:08:51 --> 00:08:55
dx as something that we already
have on the blackboard.
137
00:08:55 --> 00:08:59
So, the way to exploit that
is to make a substitution.
138
00:08:59 --> 00:09:08
And substitution is
going to be u = sin x.
139
00:09:08 --> 00:09:09
And here's why.
140
00:09:09 --> 00:09:12
Then this integral that
I'm trying to do is
141
00:09:12 --> 00:09:15
the integral of u ^ n.
142
00:09:15 --> 00:09:18
That's already a
simplification.
143
00:09:18 --> 00:09:22
And then there's that cos x dx.
144
00:09:22 --> 00:09:25
When you make a substitution,
you've got to go all the way
145
00:09:25 --> 00:09:29
and replace everything in
the expression by things
146
00:09:29 --> 00:09:33
involving this new variable
that I've introduced.
147
00:09:33 --> 00:09:36
So I'd better get rid of the
cosine of x dx and rewrite it
148
00:09:36 --> 00:09:39
in terms of du or
in terms of u.
149
00:09:39 --> 00:09:44
And I can do that because
du, according to that
150
00:09:44 --> 00:09:50
formula, is the cos x dx.
151
00:09:50 --> 00:09:53
Let me put a box around that.
152
00:09:53 --> 00:09:55
That's our substitution.
153
00:09:55 --> 00:09:55
When
154
00:09:55 --> 00:09:58
you make a substitution, you
also want to compute the
155
00:09:58 --> 00:10:03
differential of the variable
that you substitute in.
156
00:10:03 --> 00:10:08
So the cos x dx that appears
here is just, exactly, du.
157
00:10:08 --> 00:10:11
And I've replaced this trig
integral with something that
158
00:10:11 --> 00:10:13
doesn't involve trig
functions at all.
159
00:10:13 --> 00:10:14
This is a lot easier.
160
00:10:14 --> 00:10:17
We can just plug into
what we know here.
161
00:10:17 --> 00:10:23
This is (u ^ (n + 1) / n
+ 1) + a constant, and
162
00:10:23 --> 00:10:26
I've done the integral.
163
00:10:26 --> 00:10:29
But I'm not quite done
with the problem yet.
164
00:10:29 --> 00:10:34
Because to be nice to your
reader and to yourself, you
165
00:10:34 --> 00:10:38
should go back at this point,
probably, go back and get rid
166
00:10:38 --> 00:10:40
of this new variable
that you introduced.
167
00:10:40 --> 00:10:42
You're the one who introduced
this variable, you.
168
00:10:42 --> 00:10:45
Nobody except you, really,
knows what it is.
169
00:10:45 --> 00:10:48
But the rest of the world
knows what they asked for the
170
00:10:48 --> 00:10:51
first place that involved x.
171
00:10:51 --> 00:10:53
So I have to go back
and get rid of this.
172
00:10:53 --> 00:10:57
And that's not hard to do in
this case, because u = sin x.
173
00:10:57 --> 00:11:04
And so I make this
back substitution.
174
00:11:04 --> 00:11:05
And that's what you get.
175
00:11:05 --> 00:11:11
So there's the answer.
176
00:11:11 --> 00:11:15
OK, so the game was, I use this
odd power of the cosine here,
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00:11:15 --> 00:11:19
and I could see it appearing as
the differential of the sine.
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00:11:19 --> 00:11:22
So that's what made this
substitution work.
179
00:11:22 --> 00:11:25
Let's do another example to
see how that works out in
180
00:11:25 --> 00:11:36
a slightly different case.
181
00:11:36 --> 00:11:48
So here's another example.
182
00:11:48 --> 00:11:50
Now I do have an odd power.
183
00:11:50 --> 00:11:53
One of the exponents is odd,
so I'm in the easy case.
184
00:11:53 --> 00:11:56
But it's not 1.
185
00:11:56 --> 00:12:07
The game now is to use this
trig identity to get rid of the
186
00:12:07 --> 00:12:13
largest even power that you
can, from this odd power here.
187
00:12:13 --> 00:12:23
So use sin^2 x = 1 - cos^2
x, to eliminate a lot of
188
00:12:23 --> 00:12:26
powers from that odd power.
189
00:12:26 --> 00:12:28
Watch what happens.
190
00:12:28 --> 00:12:31
So this is not really a
substitution or anything, this
191
00:12:31 --> 00:12:34
is just a trig identity.
192
00:12:34 --> 00:12:38
This sin^3 = sin^2 sin.
193
00:12:38 --> 00:12:41
And sin^2 = (1 - cos^2
x) And then I have
194
00:12:41 --> 00:12:43
the remaining sin x.
195
00:12:43 --> 00:12:48
And then I have cos^2 x dx.
196
00:12:48 --> 00:12:53
So let me rewrite that a little
bit to see how this works out.
197
00:12:53 --> 00:13:00
This is the integral of (cos^2
(x) -, and then there's
198
00:13:00 --> 00:13:01
the product of these two.
199
00:13:01 --> 00:13:09
That's cos^ 4 x) sin of x dx.
200
00:13:09 --> 00:13:12
So now I'm really exactly
in the situation that
201
00:13:12 --> 00:13:13
I was in over here.
202
00:13:13 --> 00:13:17
I've got a single power
of a sine or cosine.
203
00:13:17 --> 00:13:20
It happens that
it's a sine here.
204
00:13:20 --> 00:13:22
But that's not going to cause
any trouble, we can go ahead
205
00:13:22 --> 00:13:26
and play the same game
that I did there.
206
00:13:26 --> 00:13:28
So, so far I've just been
using trig identities.
207
00:13:28 --> 00:13:41
But now I'll use a
trig substitution.
208
00:13:41 --> 00:13:45
And I think I want to write
these as powers of a variable.
209
00:13:45 --> 00:13:47
And then this is going
to be the differential
210
00:13:47 --> 00:13:47
of that variable.
211
00:13:47 --> 00:13:52
So I'll take u to be cosine
of x, and that means
212
00:13:52 --> 00:13:58
that du = - sin x, dx.
213
00:13:58 --> 00:14:04
There's the substitution.
214
00:14:04 --> 00:14:09
So when I make that
substitution, what do we get.
215
00:14:09 --> 00:14:11
Cos^2 becomes u^2.
216
00:14:11 --> 00:14:15
217
00:14:15 --> 00:14:24
Cos^ 4 becomes u ^ 4, and sin x
dx becomes not quite du, watch
218
00:14:24 --> 00:14:27
for the signum, watch for
this minus sign here.
219
00:14:27 --> 00:14:32
It becomes - du.
220
00:14:32 --> 00:14:32
But that's OK.
221
00:14:32 --> 00:14:34
The minus sign comes outside.
222
00:14:34 --> 00:14:37
And I can integrate both
of these powers, so
223
00:14:37 --> 00:14:43
I get - u ^3 / 3.
224
00:14:43 --> 00:14:46
And then this 4th power
gives me a 5th power,
225
00:14:46 --> 00:14:48
when I integrate.
226
00:14:48 --> 00:14:53
And don't forget the constant.
227
00:14:53 --> 00:14:55
Am I done?
228
00:14:55 --> 00:14:55
Not quite done.
229
00:14:55 --> 00:14:58
I have to back substitute
and get rid of my choice
230
00:14:58 --> 00:15:00
of variable, u, and
replace it with yours.
231
00:15:00 --> 00:15:01
Questions?
232
00:15:01 --> 00:15:06
STUDENT: [INAUDIBLE]
233
00:15:06 --> 00:15:07
PROFESSOR: There should indeed.
234
00:15:07 --> 00:15:10
I forgot this minus sign
when I came down here.
235
00:15:10 --> 00:15:12
So these two gang up
to give me a plus.
236
00:15:12 --> 00:15:14
Was that what the other
question was about, too?
237
00:15:14 --> 00:15:16
Thanks.
238
00:15:16 --> 00:15:18
So let's back substitute.
239
00:15:18 --> 00:15:23
And I'm going to put
that over here.
240
00:15:23 --> 00:15:27
And the result is, well, I just
replace the u by cosine of x.
241
00:15:27 --> 00:15:38
So this is - cos^3 x / 3 +,
thank you, cos^5 x / 5 + c.
242
00:15:38 --> 00:15:44
And there's the answer.
243
00:15:44 --> 00:15:47
By the way, you can remember
one of the nice things about
244
00:15:47 --> 00:15:49
doing an integral is
it's fairly easy to
245
00:15:49 --> 00:15:51
check your answer.
246
00:15:51 --> 00:15:54
You can always differentiate
the thing you get, and see
247
00:15:54 --> 00:15:56
whether you get the right
thing when you go back.
248
00:15:56 --> 00:16:01
It's not too hard to use
the power rules and the
249
00:16:01 --> 00:16:06
differentiation rule for the
cosine to get back to this if
250
00:16:06 --> 00:16:09
you want to check the work.
251
00:16:09 --> 00:16:14
Let's do one more example, just
to handle an example of this
252
00:16:14 --> 00:16:18
easy case, which you might
have thought of at first.
253
00:16:18 --> 00:16:22
Suppose I just want
to integrate a cube.
254
00:16:22 --> 00:16:29
Sin^3 x.
255
00:16:29 --> 00:16:32
No cosine in sight.
256
00:16:32 --> 00:16:36
But I do have an odd power
of a trig function,
257
00:16:36 --> 00:16:37
of a sine or cosine.
258
00:16:37 --> 00:16:39
So I'm in the easy case.
259
00:16:39 --> 00:16:44
And the procedure that I was
suggesting says I want to take
260
00:16:44 --> 00:16:47
out the largest even power
that I can, from the sin^3.
261
00:16:48 --> 00:16:51
So I'll take that out,
that's a sin^2 and
262
00:16:51 --> 00:16:53
write it as 1 - cos^2.
263
00:16:53 --> 00:16:57
264
00:16:57 --> 00:16:58
Well, now I'm very happy.
265
00:16:58 --> 00:17:05
Because it's just like the
situation we had somewhere
266
00:17:05 --> 00:17:06
on the board here.
267
00:17:06 --> 00:17:07
It's just like the
situation we had up here.
268
00:17:07 --> 00:17:11
I've got a power of
a cos sin x dx.
269
00:17:11 --> 00:17:16
So exactly the same
substitution steps in.
270
00:17:16 --> 00:17:19
You get, and maybe you
can see what happens
271
00:17:19 --> 00:17:20
without doing the work.
272
00:17:20 --> 00:17:22
Shall I do the work here?
273
00:17:22 --> 00:17:24
I make the same substitution.
274
00:17:24 --> 00:17:33
And so this is (1
- u ^2 )( - du).
275
00:17:33 --> 00:17:40
Which is u - u ^3 / 3.
276
00:17:40 --> 00:17:42
But then I want to put this
minus sign in place, and so
277
00:17:42 --> 00:17:47
that gives me - u + u
^3 / 3 + a constant.
278
00:17:47 --> 00:17:58
And then I back substitute
and get cos x + cos^3 x / 3.
279
00:17:58 --> 00:17:59
So this is the easy case.
280
00:17:59 --> 00:18:02
If you have some odd power
to play with, then you can
281
00:18:02 --> 00:18:07
make use of it and it's
pretty straightforward.
282
00:18:07 --> 00:18:10
OK the harder case is when
you don't have an odd power.
283
00:18:10 --> 00:18:11
So what's the program?
284
00:18:11 --> 00:18:13
I'm going to do the harder
case, and then I'm going to
285
00:18:13 --> 00:18:19
show you an example of how
to integrate square roots.
286
00:18:19 --> 00:18:26
And do an application, using
these ideas from trigonometry.
287
00:18:26 --> 00:18:30
So I want to keep
this blackboard.
288
00:18:30 --> 00:18:34
Maybe I'll come back
and start here again.
289
00:18:34 --> 00:18:55
So the harder case is when
they're only even exponents.
290
00:18:55 --> 00:18:58
I'm still trying to
integrate the same form.
291
00:18:58 --> 00:19:00
But now all the
exponents are even.
292
00:19:00 --> 00:19:03
So we have to do some game.
293
00:19:03 --> 00:19:10
And here the game is use
the half angle formula.
294
00:19:10 --> 00:19:16
Which I just erased, very
sadly, on the board here.
295
00:19:16 --> 00:19:18
Maybe I'll rewrite them
over here so we have
296
00:19:18 --> 00:19:23
them on the board.
297
00:19:23 --> 00:19:44
I think I remember
what they were.
298
00:19:44 --> 00:19:46
So the game is I'm going to use
that half angle formula to
299
00:19:46 --> 00:19:50
start getting rid of
those even powers.
300
00:19:50 --> 00:19:54
Half angle formula written like
this, exactly, talks about, it
301
00:19:54 --> 00:19:57
rewrites, even powers
of sines and cosines.
302
00:19:57 --> 00:20:00
So let's see how that
works out in an example.
303
00:20:00 --> 00:20:08
How about just the
cos^2 for a start.
304
00:20:08 --> 00:20:09
What to do?
305
00:20:09 --> 00:20:11
I can't pull anything out.
306
00:20:11 --> 00:20:15
I could rewrite this as 1 -
sin^2, but then I'd be faced
307
00:20:15 --> 00:20:19
with integrating the sin^2,
which is exactly as hard.
308
00:20:19 --> 00:20:23
So instead, let's use
this formula here.
309
00:20:23 --> 00:20:29
This is really the same
as 1 + cos 2 theta / 2.
310
00:20:29 --> 00:20:32
And now, this is easy.
311
00:20:32 --> 00:20:34
It's got two parts to it.
312
00:20:34 --> 00:20:38
Integrating one half gives
me theta over -- oh.
313
00:20:38 --> 00:20:42
Miraculously, the x
turned into a theta.
314
00:20:42 --> 00:20:44
Let's put it back as x.
315
00:20:44 --> 00:20:47
I get x / 2 by integrating 1/2.
316
00:20:47 --> 00:20:49
So, notice that something
non-trigonometric occurs
317
00:20:49 --> 00:20:54
here when I do these even
integrals. x / 2 appears.
318
00:20:54 --> 00:20:57
And then the other one, OK, so
this takes a little thought.
319
00:20:57 --> 00:21:03
The integral of the cosine
is the sine, or is it
320
00:21:03 --> 00:21:11
minus the sine. - sine.
321
00:21:11 --> 00:21:13
Shall we take a vote?
322
00:21:13 --> 00:21:13
I think it's positive.
323
00:21:13 --> 00:21:18
And so you get the sin
2x, but is that right?
324
00:21:18 --> 00:21:19
Over 2.
325
00:21:19 --> 00:21:24
If I differentiate the sin
2x, this 2 comes out.
326
00:21:24 --> 00:21:25
And would give me
an extra 2 here.
327
00:21:25 --> 00:21:29
So there's an extra 2 that
I have to put in here
328
00:21:29 --> 00:21:34
when I integrate it.
329
00:21:34 --> 00:21:37
And there's the answer.
330
00:21:37 --> 00:21:39
This is not a substitution.
331
00:21:39 --> 00:21:41
I just played with
trig identities here.
332
00:21:41 --> 00:21:45
And then did a simple trig
integral, getting your help
333
00:21:45 --> 00:21:46
to get the sign right.
334
00:21:46 --> 00:21:49
And thinking about what
this 2 is going to do.
335
00:21:49 --> 00:21:52
It produces a 2 in
the denominator.
336
00:21:52 --> 00:21:59
But it's not applying
any complicated thing.
337
00:21:59 --> 00:22:03
It's just using this identity.
338
00:22:03 --> 00:22:05
Let's do another example
that's a little bit harder.
339
00:22:05 --> 00:22:07
This time, sin^2 cos^2.
340
00:22:07 --> 00:22:35
341
00:22:35 --> 00:22:37
Again, no odd powers.
342
00:22:37 --> 00:22:40
I've got to work a
little bit harder.
343
00:22:40 --> 00:22:42
And what I'm going to
do is apply those
344
00:22:42 --> 00:22:44
identities up there.
345
00:22:44 --> 00:22:48
Now, what I recommend doing
in this situation is going
346
00:22:48 --> 00:22:51
over to the side somewhere.
347
00:22:51 --> 00:22:55
And do some side work.
348
00:22:55 --> 00:22:58
Because it's all just playing
with trig functions.
349
00:22:58 --> 00:23:06
It's not actually doing any
integrals for a while.
350
00:23:06 --> 00:23:11
So, I guess one way to get rid
of the sin^2 and the cos^2 is
351
00:23:11 --> 00:23:14
to use those identities
and so let's do that.
352
00:23:14 --> 00:23:20
So the sin = 1 - cos 2x / 2.
353
00:23:20 --> 00:23:27
And the cos = 1 + cos 2x / 2.
354
00:23:27 --> 00:23:29
So I just substitute them in.
355
00:23:29 --> 00:23:31
And now I can
multiply that out.
356
00:23:31 --> 00:23:38
And what I have is a
difference times a sum.
357
00:23:38 --> 00:23:40
So you know a formula for that.
358
00:23:40 --> 00:23:42
Taking the product of these
two things, well there'll
359
00:23:42 --> 00:23:44
be a 4 in the denominator.
360
00:23:44 --> 00:23:47
And then in the numerator, I
get the square of this minus
361
00:23:47 --> 00:23:57
the square of this. (a -
b)( a + b) = a ^2 - b^2.
362
00:23:57 --> 00:23:59
So I get that.
363
00:23:59 --> 00:24:02
Well, I'm a little bit
happier, because at
364
00:24:02 --> 00:24:03
least I don't have 4.
365
00:24:03 --> 00:24:07
I don't have 2
different squares.
366
00:24:07 --> 00:24:09
I still have a square, and
want to integrate this.
367
00:24:09 --> 00:24:12
I'm still not in the easy case.
368
00:24:12 --> 00:24:16
I got myself back to
an easier hard case.
369
00:24:16 --> 00:24:18
But we do know what
to do about this.
370
00:24:18 --> 00:24:21
Because I just did it up there.
371
00:24:21 --> 00:24:24
And I could play into this
formula that we got.
372
00:24:24 --> 00:24:29
But I think it's just as easy
to continue to calculate here.
373
00:24:29 --> 00:24:33
Use the half angle formula
again for this, and
374
00:24:33 --> 00:24:34
continue on your way.
375
00:24:34 --> 00:24:37
So I get a 1/4 from this bit.
376
00:24:37 --> 00:24:45
And then - 1/4 of the cos^2 2x.
377
00:24:45 --> 00:24:50
And when I plug in 2x in for
theta, there in the top
378
00:24:50 --> 00:24:59
board, I'm going to get a
4x on the right-hand side.
379
00:24:59 --> 00:25:02
So it comes out like that.
380
00:25:02 --> 00:25:04
And I guess I could simplify
that a little bit more.
381
00:25:04 --> 00:25:05
This is a 1/4.
382
00:25:05 --> 00:25:07
Oh, but then there's a 2 here.
383
00:25:07 --> 00:25:10
It's half that.
384
00:25:10 --> 00:25:12
So then I can simplify
a little more.
385
00:25:12 --> 00:25:16
It's 1/4 - 1/8, which is 1/8.
386
00:25:16 --> 00:25:25
And then I have 1/8 cos 4x.
387
00:25:25 --> 00:25:27
OK, that's my side work.
388
00:25:27 --> 00:25:30
I just did some trig
identities over here.
389
00:25:30 --> 00:25:34
And rewrote sine squared times
cosine squared as something
390
00:25:34 --> 00:25:37
which involves just no powers
of trig, just cosine by itself.
391
00:25:37 --> 00:25:41
And a constant.
392
00:25:41 --> 00:25:45
So I can take that and
substitute it in here.
393
00:25:45 --> 00:25:53
And now the integration is
pretty easy. (1/8 - cos 4x /
394
00:25:53 --> 00:26:01
8) dx, which is, OK the 1/8
is going to give me x / 8.
395
00:26:01 --> 00:26:06
The integral or cosine is
plus or minus the sine.
396
00:26:06 --> 00:26:08
The derivative of the
sine is plus the cosine.
397
00:26:08 --> 00:26:11
So it's going to be plus the,
only there's a minus here.
398
00:26:11 --> 00:26:18
So it's going to be the sin -
the sin 4x / 8, but then I
399
00:26:18 --> 00:26:20
have an additional factor
in the denominator.
400
00:26:20 --> 00:26:21
And what's it going to be?
401
00:26:21 --> 00:26:28
I have to put a 4 there.
402
00:26:28 --> 00:26:32
So we've done that
calculation, too.
403
00:26:32 --> 00:26:38
So any of these, if you keep
doing this kind of process,
404
00:26:38 --> 00:26:48
these two kinds of procedures,
you can now integrate any
405
00:26:48 --> 00:26:50
expression that has a power of
a sine times a power of
406
00:26:50 --> 00:26:56
a cosine in it, by
using these ideas.
407
00:26:56 --> 00:27:01
Now, let's see.
408
00:27:01 --> 00:27:04
Oh, let me give you an
alternate method for
409
00:27:04 --> 00:27:16
this last one here.
410
00:27:16 --> 00:27:26
I know what I'll do.
411
00:27:26 --> 00:27:29
Let me give an alternate method
for doing, really doing the
412
00:27:29 --> 00:27:33
side work over there.
i'm trying to deal
413
00:27:33 --> 00:27:35
with sin^2 cos^2.
414
00:27:35 --> 00:27:50
Well that's the square
of the sin x cos x.
415
00:27:50 --> 00:27:54
And the sin x cos x
shows up right here.
416
00:27:54 --> 00:27:55
In another trig identity.
417
00:27:55 --> 00:27:58
So we can make use
of that, too.
418
00:27:58 --> 00:28:00
That reduces the number
of factors of sines
419
00:28:00 --> 00:28:01
and cosines by 1.
420
00:28:01 --> 00:28:04
So it's going in the
right direction.
421
00:28:04 --> 00:28:11
This is equal to 1/2
sine 2x, squared.
422
00:28:11 --> 00:28:17
Sin cos = 1/2, say this right.
423
00:28:17 --> 00:28:21
It's sin 2x / 2, and then
I want to square that.
424
00:28:21 --> 00:28:31
So what I get is
the sin^2 2x / 4.
425
00:28:31 --> 00:28:34
Which is, well, I'm not
too happy yet, because I
426
00:28:34 --> 00:28:35
still have an even power.
427
00:28:35 --> 00:28:37
Remember I'm trying to
integrate this thing in the
428
00:28:37 --> 00:28:39
end, even powers are bad.
429
00:28:39 --> 00:28:40
I try to get rid of them.
430
00:28:40 --> 00:28:47
By using that formula, the half
angle formula, so I can apply
431
00:28:47 --> 00:28:48
that to the sin x here again.
432
00:28:48 --> 00:28:57
I get 1/4 of 1 - cos of 4x / 2.
433
00:28:57 --> 00:29:02
That's what the half angle
formula says for the sin^2 2x.
434
00:29:02 --> 00:29:04
And that's exactly the same
as the expression that
435
00:29:04 --> 00:29:08
I got up here, as well.
436
00:29:08 --> 00:29:11
It's the same expression
that I have there.
437
00:29:11 --> 00:29:16
So it's the same expression
as I have here.
438
00:29:16 --> 00:29:20
So this is just an alternate
way to play this game of using
439
00:29:20 --> 00:29:24
the half angle formula.
440
00:29:24 --> 00:29:28
OK, let's do a little
application of these things and
441
00:29:28 --> 00:29:46
change the topic a little bit.
442
00:29:46 --> 00:29:48
So here's the problem.
443
00:29:48 --> 00:29:57
So this is an application
and example of a real
444
00:29:57 --> 00:30:07
trig substitution.
445
00:30:07 --> 00:30:22
So here's the problem
I want to look at.
446
00:30:22 --> 00:30:26
OK, so I have a circle
whose radius is a.
447
00:30:26 --> 00:30:31
And I cut out from it
a sort of tab, here.
448
00:30:31 --> 00:30:36
This tab here.
449
00:30:36 --> 00:30:38
And the height of
this thing is b.
450
00:30:38 --> 00:30:42
So this length is a number b.
451
00:30:42 --> 00:30:44
And what I want to do
is compute the area
452
00:30:44 --> 00:30:47
of that little tab.
453
00:30:47 --> 00:30:48
That's the problem.
454
00:30:48 --> 00:30:50
So there's an arc over here.
455
00:30:50 --> 00:30:55
And I want to find the area
of this, for a and b,
456
00:30:55 --> 00:30:57
in terms of a and b.
457
00:30:57 --> 00:31:07
So the area, well, I guess
one way to compete the
458
00:31:07 --> 00:31:12
area would be to take
the integral of y dx.
459
00:31:12 --> 00:31:16
You've seen the idea of
splitting this up into vertical
460
00:31:16 --> 00:31:21
strips whose height is
given by a function, y (x).
461
00:31:21 --> 00:31:21
And then you integrate that.
462
00:31:21 --> 00:31:24
That's an interpretation
for the integral.
463
00:31:24 --> 00:31:27
The area is given by y dx.
464
00:31:27 --> 00:31:29
But that's a little bit
awkward, because my formula
465
00:31:29 --> 00:31:31
for y is going to be
a little strange.
466
00:31:31 --> 00:31:34
Its constant value of b along
here, and then at this
467
00:31:34 --> 00:31:37
point it becomes this
arc, of the circle.
468
00:31:37 --> 00:31:40
So working this out, I could do
it but it's a little awkward
469
00:31:40 --> 00:31:44
because expressing y as a
function of x, the top edge of
470
00:31:44 --> 00:31:48
this shape, it's a little
awkward and takes two
471
00:31:48 --> 00:31:51
different regions to express.
472
00:31:51 --> 00:31:58
So, a different way to
say it is to say x dy.
473
00:31:58 --> 00:32:00
Maybe that'll work a
little bit better.
474
00:32:00 --> 00:32:02
Or maybe it won't, but
it's worth trying.
475
00:32:02 --> 00:32:06
I could just as well split
this region up into
476
00:32:06 --> 00:32:08
horizontal strips.
477
00:32:08 --> 00:32:13
Whose width is dy, and
whose length is x.
478
00:32:13 --> 00:32:17
Now I'm thinking of this
as a function of y.
479
00:32:17 --> 00:32:20
This is the graph of
a function of y.
480
00:32:20 --> 00:32:24
And that's much better, because
the function of y is, well,
481
00:32:24 --> 00:32:28
it's the square root of
a^2 - y ^2, isn't it.
482
00:32:28 --> 00:32:34
That's x ^2 + y ^2 = a ^2.
483
00:32:34 --> 00:32:38
So that's what x is.
484
00:32:38 --> 00:32:41
And that's what I'm asked
to integrate, then.
485
00:32:41 --> 00:32:45
Square root of (a
^2 - y ^2) dy.
486
00:32:45 --> 00:32:47
And I can even put in
limits of integration.
487
00:32:47 --> 00:32:49
Maybe I should do that,
because this is supposed
488
00:32:49 --> 00:32:50
to be an actual number.
489
00:32:50 --> 00:32:55
I guess I'm integrating it
from y = 0, that's here.
490
00:32:55 --> 00:33:00
To y = b, dy.
491
00:33:00 --> 00:33:01
So this is what I want to find.
492
00:33:01 --> 00:33:07
This is a integral formula
for the area of that region.
493
00:33:07 --> 00:33:08
And this is a new form.
494
00:33:08 --> 00:33:17
I don't think that you've
thought about integrating
495
00:33:17 --> 00:33:20
expressions like this
in this class before.
496
00:33:20 --> 00:33:23
So, it's a new form and I want
to show you how to do it, how
497
00:33:23 --> 00:33:30
it's related to trigonometry.
498
00:33:30 --> 00:33:33
It's related to trigonometry
through that exact picture that
499
00:33:33 --> 00:33:36
we have on the blackboard.
500
00:33:36 --> 00:33:42
After all, this a^2 - y ^2 is
the formula for this arc.
501
00:33:42 --> 00:33:47
And so, what I propose is
that we try to exploit the
502
00:33:47 --> 00:33:52
connection with the circle and
introduce polar coordinates.
503
00:33:52 --> 00:34:03
So, here if I measure this
angle then there are various
504
00:34:03 --> 00:34:04
things you can say.
505
00:34:04 --> 00:34:07
Like the coordinates of
this point here are
506
00:34:07 --> 00:34:10
(a, cosine theta, a.
507
00:34:10 --> 00:34:17
Well, I'm sorry, it's not.
508
00:34:17 --> 00:34:20
That's an angle, but I
want to call it theta 0.
509
00:34:20 --> 00:34:25
And, in general you know that
the coordinates of this point
510
00:34:25 --> 00:34:31
are (a, cosine theta,
a, sine theta).
511
00:34:31 --> 00:34:39
If the radius is a, then
the angle here is theta.
512
00:34:39 --> 00:34:45
So x is a cosine theta, and y
is a sine theta, just from
513
00:34:45 --> 00:34:49
looking at the geometry
of the circle.
514
00:34:49 --> 00:34:56
So let's make that
substitution. y = a sine theta.
515
00:34:56 --> 00:35:00
I'm using the picture to
suggest that maybe making
516
00:35:00 --> 00:35:02
the substitution is
a good thing to do.
517
00:35:02 --> 00:35:06
Let's follow along and
see what happens.
518
00:35:06 --> 00:35:08
If that's true, what
we're interested in is
519
00:35:08 --> 00:35:13
integrating a^2 - y ^2.
520
00:35:13 --> 00:35:18
Which is a ^2, we're interested
in integrating the square
521
00:35:18 --> 00:35:20
root of (a ^2 - y^2).
522
00:35:20 --> 00:35:23
Which is the square root
of a ^2 minus this.
523
00:35:23 --> 00:35:27
a ^2 sin ^2 theta.
524
00:35:27 --> 00:35:35
And, well, that's
= a cos theta.
525
00:35:35 --> 00:35:41
That's just sin^2 + cos^2
= 1, all over again.
526
00:35:41 --> 00:35:42
It's also x.
527
00:35:42 --> 00:35:44
This is x.
528
00:35:44 --> 00:35:46
And this was x.
529
00:35:46 --> 00:35:48
So there are a lot of different
ways to think of this.
530
00:35:48 --> 00:35:51
But no matter how you say it,
the thing we're trying to
531
00:35:51 --> 00:35:57
integrate, a squared, a ^2 - y
^2 is, under this substitution
532
00:35:57 --> 00:36:02
it is a cos theta.
533
00:36:02 --> 00:36:04
So I'm interested in
integrating the square
534
00:36:04 --> 00:36:09
root of (a ^2 - y ^2) dy.
535
00:36:09 --> 00:36:17
And I'm going to make this
substitution y = a sin theta.
536
00:36:17 --> 00:36:22
And so under that substitution,
I've decided that the
537
00:36:22 --> 00:36:31
square root of a ^2 -
y ^2 = a cos theta.
538
00:36:31 --> 00:36:33
That's this.
539
00:36:33 --> 00:36:34
What about the dy?
540
00:36:34 --> 00:36:38
Well, I'd better
compute the dy.
541
00:36:38 --> 00:36:41
So dy just differentiating
this expression, is
542
00:36:41 --> 00:36:44
a cos theta d theta.
543
00:36:44 --> 00:36:57
So let's put that in. dy
= a cos theta d theta.
544
00:36:57 --> 00:36:58
OK.
545
00:36:58 --> 00:37:04
Making that trig substitution,
y = a sin theta has replaced
546
00:37:04 --> 00:37:06
this integral that has
a square root in it.
547
00:37:06 --> 00:37:08
And no trig functions.
548
00:37:08 --> 00:37:12
With an integral that
involves no square roots
549
00:37:12 --> 00:37:15
and only trig functions.
550
00:37:15 --> 00:37:17
In fact, it's not too hard to
integrate this now, because
551
00:37:17 --> 00:37:19
of the work that we've done.
552
00:37:19 --> 00:37:20
The a ^2 comes out.
553
00:37:20 --> 00:37:26
This is cos^2 theta. d theta.
554
00:37:26 --> 00:37:28
And maybe we've done that
example already today.
555
00:37:28 --> 00:37:35
I think we have.
556
00:37:35 --> 00:37:38
Maybe we can think it through,
but maybe the easiest thing is
557
00:37:38 --> 00:37:42
to look back at notes and
see what we got before.
558
00:37:42 --> 00:37:46
That was the first example in
the hard case that I did.
559
00:37:46 --> 00:38:03
And what it came out to, I used
x instead of theta at the time.
560
00:38:03 --> 00:38:05
So this is a good step forward.
561
00:38:05 --> 00:38:08
I started with this integral
that I really didn't know
562
00:38:08 --> 00:38:12
how to do by any means
that we've had so far.
563
00:38:12 --> 00:38:15
And I've replaced it by
a trig integral that
564
00:38:15 --> 00:38:16
we do know how to do.
565
00:38:16 --> 00:38:19
And now I've done
that trig integral.
566
00:38:19 --> 00:38:22
But we're still not quite
done, because of the problem
567
00:38:22 --> 00:38:23
of back substituting.
568
00:38:23 --> 00:38:29
I'd like to go back and
rewrite this in terms of
569
00:38:29 --> 00:38:32
the original variable, y.
570
00:38:32 --> 00:38:34
Or, I'd like to go back and
rewrite it in terms of the
571
00:38:34 --> 00:38:37
original limits of integration
that we had in the
572
00:38:37 --> 00:38:40
original problem.
573
00:38:40 --> 00:38:43
In doing that, it's going to
be useful to rewrite this
574
00:38:43 --> 00:38:47
expression and get rid
of the sin 2 theta.
575
00:38:47 --> 00:38:52
After all, the original y was
expressed in terms of the sin
576
00:38:52 --> 00:38:54
theta, not the sin 2 theta.
577
00:38:54 --> 00:39:04
So let me just do that here,
and say that this, in turn,
578
00:39:04 --> 00:39:14
is equal to a^2 theta / 2 +,
well, the sin 2 theta =
579
00:39:14 --> 00:39:18
2 sin theta cos theta.
580
00:39:18 --> 00:39:20
And so, when there's a 4 in the
denominator, what I'll get
581
00:39:20 --> 00:39:32
is sin theta cos theta / 2.
582
00:39:32 --> 00:39:36
I did that because I'm getting
closer to the original terms
583
00:39:36 --> 00:39:39
that the problem started with.
584
00:39:39 --> 00:40:05
Which was sin theta.
585
00:40:05 --> 00:40:08
So let me write down the
integral that we have now.
586
00:40:08 --> 00:40:15
The square root of a ^2 - y ^2
dy is, so far, what we know
587
00:40:15 --> 00:40:26
is a ^2 (theta / 2 + sin
theta cos theta / 2) + c.
588
00:40:26 --> 00:40:28
But I want to go back and
rewrite this in terms
589
00:40:28 --> 00:40:30
of the original value.
590
00:40:30 --> 00:40:32
The original variable, y.
591
00:40:32 --> 00:40:37
Well, what is theta
in terms of y?
592
00:40:37 --> 00:40:40
Let's see. y in terms of
theta was given like this.
593
00:40:40 --> 00:40:44
So what is theta in terms of y?
594
00:40:44 --> 00:40:44
Ah.
595
00:40:44 --> 00:40:48
So here the fearsome arc
sin rears its head, right?
596
00:40:48 --> 00:40:53
Theta is the angle so
that y = a sin theta.
597
00:40:53 --> 00:40:56
So that means that theta
is the arc sign, or
598
00:40:56 --> 00:41:07
sine inverse, of y / a.
599
00:41:07 --> 00:41:12
So that's the first thing
that shows up here.
600
00:41:12 --> 00:41:17
Arc sin (y / a).
601
00:41:17 --> 00:41:18
All over 2.
602
00:41:18 --> 00:41:19
That's this term.
603
00:41:19 --> 00:41:24
Theta is the arc
sin ( y /a) / 2.
604
00:41:24 --> 00:41:26
What about the
other side, here?
605
00:41:26 --> 00:41:30
Well sine and cosine, we knew
what they were in terms of y
606
00:41:30 --> 00:41:37
and in terms of x, if you like.
607
00:41:37 --> 00:41:40
Maybe I'll put the
a ^2 inside here.
608
00:41:40 --> 00:41:42
That makes it a
little bit nicer.
609
00:41:42 --> 00:41:49
Plus, and the other term is
a ^2 (sin theta cos theta).
610
00:41:49 --> 00:41:52
So the a sin theta is just y.
611
00:41:52 --> 00:42:02
Maybe I'll write this (a sin
theta)( a cos theta) / 2 + c.
612
00:42:02 --> 00:42:03
And so I get the same thing.
613
00:42:03 --> 00:42:06
And now here a sin
theta, that's y.
614
00:42:06 --> 00:42:12
And what's the a cos theta?
615
00:42:12 --> 00:42:22
It's x, or, if you like, it's
the square root of a ^2 - y ^2.
616
00:42:22 --> 00:42:28
And so there I've rewritten
everything, back in terms of
617
00:42:28 --> 00:42:31
the original variable, y.
618
00:42:31 --> 00:42:36
And there's an answer.
619
00:42:36 --> 00:42:41
So I've done this indefinite
integration of a form of this
620
00:42:41 --> 00:42:44
quadratic, this square root of
something which is
621
00:42:44 --> 00:42:46
a constant - y ^2.
622
00:42:46 --> 00:42:49
Whenever you see that,
the thing to think
623
00:42:49 --> 00:42:50
of is trigonometry.
624
00:42:50 --> 00:42:54
That's going to play into the
sin^ 2 + cos^2 identity.
625
00:42:54 --> 00:42:56
And the way to exploit it is
to make the substitution
626
00:42:56 --> 00:43:01
y = a sin theta.
627
00:43:01 --> 00:43:03
You could also make a
substitution y = a cos
628
00:43:03 --> 00:43:05
theta, if you wanted to.
629
00:43:05 --> 00:43:12
And the result would come out
to exactly the same in the end.
630
00:43:12 --> 00:43:14
I'm still not quite done with
the original problem that I
631
00:43:14 --> 00:43:24
had, because the original
problem asked for a
632
00:43:24 --> 00:43:25
definite integral.
633
00:43:25 --> 00:43:33
So let's just go back and
finish that as well.
634
00:43:33 --> 00:43:38
So the area was the
integral from 0 to b
635
00:43:38 --> 00:43:45
of this square root.
636
00:43:45 --> 00:43:48
So I just want to evaluate
the right-hand side here.
637
00:43:48 --> 00:43:50
The answer that we came up
with, this indefinite integral.
638
00:43:50 --> 00:43:53
I want to evaluate
it at 0 and at b.
639
00:43:53 --> 00:43:54
Well, let's see.
640
00:43:54 --> 00:44:13
When I evaluate it at b, I get
a ^2 ( arc sin (b / a) / 2 + y,
641
00:44:13 --> 00:44:19
which is b times the square
root of a ^2 - b ^2, putting
642
00:44:19 --> 00:44:23
y = b, divided by 2.
643
00:44:23 --> 00:44:26
So I've plugged in y =
b into that formula,
644
00:44:26 --> 00:44:27
this is what I get.
645
00:44:27 --> 00:44:31
Then when I plug in y = 0,
well the, sine of 0 is 0,
646
00:44:31 --> 00:44:34
so the arc sine of 0 is 0.
647
00:44:34 --> 00:44:35
So this term goes away.
648
00:44:35 --> 00:44:38
And when y = 0, this
term is 0 also.
649
00:44:38 --> 00:44:43
And so I don't get any
subtracted terms at all.
650
00:44:43 --> 00:44:45
So there's an
expression for this.
651
00:44:45 --> 00:44:52
Notice that this arc sin ( b /
a), that's exactly this angle.
652
00:44:52 --> 00:45:00
The arc sin ( b / a), it's the
angle that you get when y = b.
653
00:45:00 --> 00:45:09
So this theta is the
arc sin (b / a).
654
00:45:09 --> 00:45:15
Put this over here.
655
00:45:15 --> 00:45:17
That is theta 0.
656
00:45:17 --> 00:45:21
That is the angle that
the corner makes.
657
00:45:21 --> 00:45:28
So I could rewrite this as a
^2 theta 0 / 2 + b times the
658
00:45:28 --> 00:45:34
square root of a ^2 - b ^2 / 2.
659
00:45:34 --> 00:45:36
Let's just think about
this for a minute.
660
00:45:36 --> 00:45:40
I have these two terms in the
sum, is that reasonable?
661
00:45:40 --> 00:45:44
The first term is a ^2.
662
00:45:44 --> 00:45:50
That's the radius squared
times this angle, times 1/2.
663
00:45:50 --> 00:45:54
Well, I think that is exactly
the area of this sector. a ^2
664
00:45:54 --> 00:46:03
theta / 2 is the formula for
the area of the sector.
665
00:46:03 --> 00:46:07
And this one, this is
the vertical elevation.
666
00:46:07 --> 00:46:14
This is the horizontal. a ^2
- b ^2 is this distance.
667
00:46:14 --> 00:46:16
Square root of a ^2 - b ^2.
668
00:46:16 --> 00:46:20
So the right-hand term is b
times the square root of a
669
00:46:20 --> 00:46:31
^2 - b ^2 / 2, that's the
area of that triangle.
670
00:46:31 --> 00:46:35
So using a little bit of
geometry gives you the same
671
00:46:35 --> 00:46:39
answer as all of this
elaborate calculus.
672
00:46:39 --> 00:46:41
Maybe that's enough cause
for celebration for
673
00:46:41 --> 00:46:43
us to quit for today.
674
00:46:43 --> 00:46:44