WEBVTT
00:00:00.000 --> 00:00:02.330
The following content is
provided under a Creative
00:00:02.330 --> 00:00:03.610
Commons license.
00:00:03.610 --> 00:00:05.750
Your support will help
MIT OpenCourseWare
00:00:05.750 --> 00:00:09.460
continue to offer high quality
educational resources for free.
00:00:09.460 --> 00:00:12.550
To make a donation, or to
view additional materials
00:00:12.550 --> 00:00:16.150
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:16.150 --> 00:00:21.840
at ocw.mit.edu.
00:00:21.840 --> 00:00:25.190
PROFESSOR: Well, because
our subject today
00:00:25.190 --> 00:00:28.280
is trig integrals
and substitutions,
00:00:28.280 --> 00:00:33.640
Professor Jerison called in his
substitute teacher for today.
00:00:33.640 --> 00:00:46.150
That's me.
00:00:46.150 --> 00:00:49.860
Professor Miller.
00:00:49.860 --> 00:00:52.790
And I'm going to try to tell
you about trig substitutions
00:00:52.790 --> 00:00:54.610
and trig integrals.
00:00:54.610 --> 00:00:59.650
And I'll be here tomorrow to
do more of the same, as well.
00:00:59.650 --> 00:01:02.320
So, this is about trigonometry,
and maybe first thing I'll do
00:01:02.320 --> 00:01:24.180
is remind you of some basic
things about trigonometry.
00:01:24.180 --> 00:01:27.110
So, if I have a
circle, trigonometry
00:01:27.110 --> 00:01:29.730
is all based on the
circle of radius 1
00:01:29.730 --> 00:01:32.000
and centered at the origin.
00:01:32.000 --> 00:01:34.870
And so if this is an angle
of theta, up from the x-axis,
00:01:34.870 --> 00:01:36.350
then the coordinates
of this point
00:01:36.350 --> 00:01:39.550
are cosine theta and sine theta.
00:01:39.550 --> 00:01:42.090
And so that leads right away
to some trig identities,
00:01:42.090 --> 00:01:43.330
which you know very well.
00:01:43.330 --> 00:01:46.820
But I'm going to put them up
here because we'll use them
00:01:46.820 --> 00:01:51.220
over and over again today.
00:01:51.220 --> 00:01:55.220
Remember the convention sin^2
theta secretly means (sin
00:01:55.220 --> 00:01:57.530
theta)^2.
00:01:57.530 --> 00:01:59.040
It would be more
sensible to write
00:01:59.040 --> 00:02:01.600
a parenthesis around
the sine of theta
00:02:01.600 --> 00:02:03.350
and then say you square that.
00:02:03.350 --> 00:02:06.380
But everybody in the world puts
the 2 up there over the sin,
00:02:06.380 --> 00:02:09.080
and so I'll do that too.
00:02:09.080 --> 00:02:11.995
So that follows just because
the circle has radius 1.
00:02:11.995 --> 00:02:13.870
But then there are some
other identities too,
00:02:13.870 --> 00:02:15.410
which I think you remember.
00:02:15.410 --> 00:02:19.170
I'll write them down
here. cos(2theta),
00:02:19.170 --> 00:02:22.490
there's this double angle
formula that says cos(2theta) =
00:02:22.490 --> 00:02:23.840
cos^2(theta) - sin^2(theta).
00:02:29.090 --> 00:02:31.000
And there's also the
double angle formula
00:02:31.000 --> 00:02:34.620
for the sin(2theta).
00:02:34.620 --> 00:02:38.480
Remember what that says?
00:02:38.480 --> 00:02:41.300
2 sin(theta) cos(theta).
00:02:46.437 --> 00:02:48.020
I'm going to use
these trig identities
00:02:48.020 --> 00:02:50.370
and I'm going to use them
in a slightly different way.
00:02:50.370 --> 00:02:53.090
And so I'd like to pay a little
more attention to this one
00:02:53.090 --> 00:02:57.120
and get a different way
of writing this one out.
00:02:57.120 --> 00:03:06.930
So this is actually
the half angle formula.
00:03:06.930 --> 00:03:14.170
And that says, I'm going to
try to express the cos(theta)
00:03:14.170 --> 00:03:16.400
in terms of the cos(2theta).
00:03:16.400 --> 00:03:19.590
So if I know the
cos(2theta), I want
00:03:19.590 --> 00:03:23.320
to try to express the
cos theta in terms of it.
00:03:23.320 --> 00:03:30.130
Well, I'll start with a
cos(2theta) and play with that.
00:03:30.130 --> 00:03:30.630
OK.
00:03:30.630 --> 00:03:36.600
Well, we know what this is, it's
cos^2(theta) - sin^2(theta).
00:03:36.600 --> 00:03:39.520
But we also know what the
sin^2(theta) is in terms
00:03:39.520 --> 00:03:40.250
of the cosine.
00:03:40.250 --> 00:03:44.630
So I can eliminate the
sin^2 from this picture.
00:03:44.630 --> 00:03:48.400
So this is equal to cos^2(theta)
minus the quantity 1 -
00:03:48.400 --> 00:03:50.650
cos^2(theta).
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
00:03:57.510 --> 00:03:59.940
cos^2(theta) - 1.
00:03:59.940 --> 00:04:02.240
There's this cos^2,
which gets a plus sign.
00:04:02.240 --> 00:04:04.540
Because of these
two minus signs.
00:04:04.540 --> 00:04:06.290
And there's the one
that was there before,
00:04:06.290 --> 00:04:10.270
so altogether there
are two of them.
00:04:10.270 --> 00:04:12.380
I want to isolate
what cosine is.
00:04:12.380 --> 00:04:16.070
Or rather, what cos^2 is.
00:04:16.070 --> 00:04:17.680
So let's solve for that.
00:04:17.680 --> 00:04:20.120
So I'll put the 1
on the other side.
00:04:20.120 --> 00:04:24.120
And I get 1 + cos(2theta).
00:04:24.120 --> 00:04:27.910
And then, I want to divide by
this 2, and so that puts a 2
00:04:27.910 --> 00:04:30.280
in this denominator here.
00:04:30.280 --> 00:04:33.280
So some people call that
the half angle formula.
00:04:33.280 --> 00:04:36.730
What it really is for us is
it's a way of eliminating powers
00:04:36.730 --> 00:04:38.800
from sines and cosines.
00:04:38.800 --> 00:04:41.290
I've gotten rid of this
square at the expense
00:04:41.290 --> 00:04:43.720
of putting in a 2theta here.
00:04:43.720 --> 00:04:45.260
We'll use that.
00:04:45.260 --> 00:04:51.022
And, similarly, same calculation
shows that sin^2(theta) = (1 -
00:04:51.022 --> 00:04:51.730
cos(2theta)) / 2.
00:04:55.850 --> 00:05:01.480
Same cosine, in that formula
also, but it has a minus sign.
00:05:01.480 --> 00:05:03.640
For the sin^2.
00:05:03.640 --> 00:05:07.860
OK. so that's my little
review of trig identities
00:05:07.860 --> 00:05:14.620
that we'll make use of
as this lecture goes on.
00:05:14.620 --> 00:05:17.720
I want to talk about trig
identity-- trig integrals,
00:05:17.720 --> 00:05:23.500
and you know some trig
integrals, I'm sure, already.
00:05:23.500 --> 00:05:26.740
Like, well, let me write
the differential form first.
00:05:26.740 --> 00:05:30.400
You know that d sin
theta, or maybe I'll
00:05:30.400 --> 00:05:35.620
say d sin x, is,
let's see, that's
00:05:35.620 --> 00:05:39.030
the derivative of sin
x times dx, right.
00:05:39.030 --> 00:05:46.120
The derivative of
sin x is cos x, dx.
00:05:46.120 --> 00:05:50.080
And so if I integrate both sides
here, the integral form of this
00:05:50.080 --> 00:05:55.390
is the integral of cos x dx.
00:05:55.390 --> 00:05:59.850
Is sin x plus a constant.
00:05:59.850 --> 00:06:07.410
And in the same way,
d cos x = -sin x dx.
00:06:07.410 --> 00:06:10.320
Right, the derivative of
the cosine is minus sine.
00:06:10.320 --> 00:06:15.330
And when I integrate that, I
find the integral of sin x dx
00:06:15.330 --> 00:06:21.140
is -cos x + c.
00:06:21.140 --> 00:06:22.820
So that's our starting point.
00:06:22.820 --> 00:06:27.510
And the game today, for the
first half of the lecture,
00:06:27.510 --> 00:06:33.550
is to use that basic-- just
those basic integration
00:06:33.550 --> 00:06:37.710
formulas, together with
clever use of trig identities
00:06:37.710 --> 00:06:41.260
in order to compute more
complicated formulas involving
00:06:41.260 --> 00:06:42.820
trig functions.
00:06:42.820 --> 00:06:47.040
So the first thing,
the first topic,
00:06:47.040 --> 00:06:52.250
is to think about integrals of
the form sin^n (x) cos^n (x)
00:06:52.250 --> 00:06:52.750
dx.
00:06:56.190 --> 00:07:04.070
Where here I have in mind m and
n are non-negative integers.
00:07:04.070 --> 00:07:05.950
So let's try to integrate these.
00:07:05.950 --> 00:07:09.640
I'll show you some applications
of these pretty soon.
00:07:09.640 --> 00:07:11.250
Looking down the
road a little bit,
00:07:11.250 --> 00:07:14.050
integrals like this show
up in Fourier series
00:07:14.050 --> 00:07:16.190
and many other subjects
in mathematics.
00:07:16.190 --> 00:07:20.000
It turns out they're quite
important to be able to do.
00:07:20.000 --> 00:07:23.860
So that's why we're
doing them now.
00:07:23.860 --> 00:07:29.690
Well, so there are two
cases to think about here.
00:07:29.690 --> 00:07:32.070
When you're integrating
things like this.
00:07:32.070 --> 00:07:35.890
There's the easy case, and
let's do that one first.
00:07:35.890 --> 00:07:49.495
The easy case is when at
least one exponent is odd.
00:07:49.495 --> 00:07:50.370
That's the easy case.
00:07:50.370 --> 00:07:55.870
So, for example, suppose
that I wanted to integrate,
00:07:55.870 --> 00:08:02.560
well, let's take the case m = 1.
00:08:02.560 --> 00:08:09.530
So I'm integrating
sin^n (x) cos x dx.
00:08:09.530 --> 00:08:17.050
I'm taking-- Oh, I
could do that one.
00:08:17.050 --> 00:08:23.600
Let's see if that's
what I want to take.
00:08:23.600 --> 00:08:27.540
Yeah.
00:08:27.540 --> 00:08:30.590
My confusion is that I meant
to have this a different power.
00:08:30.590 --> 00:08:34.750
You were thinking that.
00:08:34.750 --> 00:08:36.670
So let's do this
case when m = 1.
00:08:36.670 --> 00:08:38.680
So the integral I'm
trying to do is any power
00:08:38.680 --> 00:08:41.450
of the sine times the cosine.
00:08:41.450 --> 00:08:44.520
Well, here's the trick.
00:08:44.520 --> 00:08:48.920
Recognize, use this
formula up at the top
00:08:48.920 --> 00:08:53.570
there to see cos x dx as
something that we already
00:08:53.570 --> 00:08:55.670
have on the blackboard.
00:08:55.670 --> 00:08:59.570
So, the way to exploit that
is to make a substitution.
00:08:59.570 --> 00:09:08.320
And substitution is
going to be u = sin x.
00:09:08.320 --> 00:09:09.610
And here's why.
00:09:09.610 --> 00:09:12.820
Then this integral that I'm
trying to do is the integral
00:09:12.820 --> 00:09:18.760
of u^n, that's already
a simplification.
00:09:18.760 --> 00:09:22.160
And then there's that cos x dx.
00:09:22.160 --> 00:09:25.120
When you make a substitution,
you've got to go all the way
00:09:25.120 --> 00:09:27.970
and replace everything
in the expression
00:09:27.970 --> 00:09:33.000
by things involving this new
variable that I've introduced.
00:09:33.000 --> 00:09:35.390
So I'd better get
rid of the cos x dx
00:09:35.390 --> 00:09:39.140
and rewrite it in terms
of du or in terms of u.
00:09:39.140 --> 00:09:45.290
And I can do that because du,
according to that formula,
00:09:45.290 --> 00:09:50.600
is cos x dx.
00:09:50.600 --> 00:09:53.810
Let me put a box around that.
00:09:53.810 --> 00:09:55.800
That's our substitution.
00:09:55.800 --> 00:09:57.500
When you make a
substitution, you also
00:09:57.500 --> 00:10:00.740
want to compute the
differential of the variable
00:10:00.740 --> 00:10:03.100
that you substitute in.
00:10:03.100 --> 00:10:08.800
So the cos x dx that appears
here is just, exactly, du.
00:10:08.800 --> 00:10:11.500
And I've replaced this trig
integral with something
00:10:11.500 --> 00:10:13.330
that doesn't involve
trig functions at all.
00:10:13.330 --> 00:10:14.240
This is a lot easier.
00:10:14.240 --> 00:10:17.420
We can just plug into
what we know here.
00:10:17.420 --> 00:10:23.640
This is u^(n+1) /
(n+1) plus a constant,
00:10:23.640 --> 00:10:26.620
and I've done the integral.
00:10:26.620 --> 00:10:29.580
But I'm not quite done
with the problem yet.
00:10:29.580 --> 00:10:34.310
Because to be nice to your
reader and to yourself,
00:10:34.310 --> 00:10:36.560
you should go back at
this point, probably,
00:10:36.560 --> 00:10:40.399
go back and get rid of this new
variable that you introduced.
00:10:40.399 --> 00:10:42.440
You're the one who introduced
this variable, you.
00:10:42.440 --> 00:10:45.950
Nobody except you,
really, knows what it is.
00:10:45.950 --> 00:10:48.080
But the rest of the
world knows what
00:10:48.080 --> 00:10:51.360
they asked for the first
place that involved x.
00:10:51.360 --> 00:10:53.460
So I have to go back
and get rid of this.
00:10:53.460 --> 00:10:57.830
And that's not hard to do in
this case, because u = sin x.
00:10:57.830 --> 00:11:04.390
And so I make this
back substitution.
00:11:04.390 --> 00:11:05.820
And that's what you get.
00:11:05.820 --> 00:11:11.040
So there's the answer.
00:11:11.040 --> 00:11:15.930
OK, so the game was, I use this
odd power of the cosine here,
00:11:15.930 --> 00:11:19.180
and I could see it appearing as
the differential of the sine.
00:11:19.180 --> 00:11:22.470
So that's what made
this substitution work.
00:11:22.470 --> 00:11:25.060
Let's do another example
to see how that works out
00:11:25.060 --> 00:11:36.490
in a slightly different case.
00:11:36.490 --> 00:11:48.420
So here's another example.
00:11:48.420 --> 00:11:50.050
Now I do have an odd power.
00:11:50.050 --> 00:11:53.800
One of the exponents is odd,
so I'm in the easy case.
00:11:53.800 --> 00:11:56.310
But it's not 1.
00:11:56.310 --> 00:12:05.590
The game now is to
use this trig identity
00:12:05.590 --> 00:12:10.500
to get rid of the largest
even power that you can,
00:12:10.500 --> 00:12:13.870
from this odd power here.
00:12:13.870 --> 00:12:24.710
So use sin^2 x = 1 - cos^2 x, to
eliminate a lot of powers from
00:12:24.710 --> 00:12:26.220
that odd power.
00:12:26.220 --> 00:12:28.370
Watch what happens.
00:12:28.370 --> 00:12:31.490
So this is not really a
substitution or anything,
00:12:31.490 --> 00:12:34.320
this is just a trig identity.
00:12:34.320 --> 00:12:38.130
This sine cubed is sine
squared times the sine.
00:12:38.130 --> 00:12:41.360
And the sine squared
is 1 - cos^2 x.
00:12:41.360 --> 00:12:43.770
And then I have the
remaining sin x.
00:12:43.770 --> 00:12:48.970
And then I have cos^2 x dx.
00:12:48.970 --> 00:12:53.580
So let me rewrite that a little
bit to see how this works out.
00:12:53.580 --> 00:12:59.490
This is the integral
of cos^2 x minus,
00:12:59.490 --> 00:13:01.240
and then there's the
product of these two.
00:13:01.240 --> 00:13:09.370
That's cos^4 x times sin x dx.
00:13:09.370 --> 00:13:12.200
So now I'm really
exactly in the situation
00:13:12.200 --> 00:13:13.700
that I was in over here.
00:13:13.700 --> 00:13:17.570
I've got a single power
of a sine or cosine.
00:13:17.570 --> 00:13:20.060
It happens that
it's a sine here.
00:13:20.060 --> 00:13:21.960
But that's not going
to cause any trouble,
00:13:21.960 --> 00:13:26.290
we can go ahead and play the
same game that I did there.
00:13:26.290 --> 00:13:28.860
So, so far I've just been
using trig identities.
00:13:28.860 --> 00:13:41.770
But now I'll use a
trig substitution.
00:13:41.770 --> 00:13:45.335
And I think I want to write
these as powers of a variable.
00:13:45.335 --> 00:13:47.960
And then this is going to be the
differential of that variable.
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.
00:13:58.190 --> 00:14:04.950
There's the substitution.
00:14:04.950 --> 00:14:09.550
So when I make that
substitution, what do we get.
00:14:09.550 --> 00:14:11.710
Cosine squared becomes u^2.
00:14:15.470 --> 00:14:23.150
Cosine to the 4th becomes u^4,
and sin x dx becomes not quite
00:14:23.150 --> 00:14:27.570
du, watch for the signum,
watch for this minus sign here.
00:14:27.570 --> 00:14:32.100
It becomes -du.
00:14:32.100 --> 00:14:32.820
But that's OK.
00:14:32.820 --> 00:14:34.720
The minus sign comes outside.
00:14:34.720 --> 00:14:36.860
And I can integrate
both of these powers,
00:14:36.860 --> 00:14:43.540
so I get -u^3 / 3.
00:14:43.540 --> 00:14:48.950
And then this 4th power gives me
a 5th power, when I integrate.
00:14:48.950 --> 00:14:53.390
And don't forget the constant.
00:14:53.390 --> 00:14:55.100
Am I done?
00:14:55.100 --> 00:14:55.850
Not quite done.
00:14:55.850 --> 00:14:57.690
I have to back
substitute and get rid
00:14:57.690 --> 00:15:00.830
of my choice of variable, u,
and replace it with yours.
00:15:00.830 --> 00:15:01.330
Questions?
00:15:01.330 --> 00:15:06.049
STUDENT: [INAUDIBLE]
00:15:06.049 --> 00:15:07.340
PROFESSOR: There should indeed.
00:15:07.340 --> 00:15:10.190
I forgot this minus sign
when I came down here.
00:15:10.190 --> 00:15:12.690
So these two gang up
to give me a plus.
00:15:12.690 --> 00:15:14.960
Was that what the other
question was about, too?
00:15:14.960 --> 00:15:16.720
Thanks.
00:15:16.720 --> 00:15:18.030
So let's back substitute.
00:15:18.030 --> 00:15:23.900
And I'm going to
put that over here.
00:15:23.900 --> 00:15:27.920
And the result is, well, I just
replace the u by cosine of x.
00:15:27.920 --> 00:15:38.160
So this is - -cos^3(x) / 3 plus,
thank you, cos^5(x) / 5 + c.
00:15:38.160 --> 00:15:44.670
And there's the answer.
00:15:44.670 --> 00:15:47.210
By the way, you can remember
one of the nice things
00:15:47.210 --> 00:15:49.160
about doing an
integral is it's fairly
00:15:49.160 --> 00:15:51.280
easy to check your answer.
00:15:51.280 --> 00:15:53.680
You can always differentiate
the thing you get,
00:15:53.680 --> 00:15:56.890
and see whether you get the
right thing when you go back.
00:15:56.890 --> 00:15:59.280
It's not too hard
to use the power
00:15:59.280 --> 00:16:02.500
rules and the
differentiation rule
00:16:02.500 --> 00:16:06.080
for the cosine to get
back to this if you
00:16:06.080 --> 00:16:09.350
want to check the work.
00:16:09.350 --> 00:16:12.830
Let's do one more
example, just to handle
00:16:12.830 --> 00:16:15.980
an example of this
easy case, which you
00:16:15.980 --> 00:16:18.490
might have thought of at first.
00:16:18.490 --> 00:16:22.720
Suppose I just want to
integrate a cube. sin^3 x.
00:16:29.320 --> 00:16:32.040
No cosine in sight.
00:16:32.040 --> 00:16:35.780
But I do have an
odd power of a trig
00:16:35.780 --> 00:16:37.180
function, of a sine or cosine.
00:16:37.180 --> 00:16:39.020
So I'm in the easy case.
00:16:39.020 --> 00:16:44.740
And the procedure that I was
suggesting says I want to take
00:16:44.740 --> 00:16:48.700
out the largest even power
that I can, from the sin^3.
00:16:48.700 --> 00:16:52.870
So I'll take that out, that's
a sin^2, and write it as 1 -
00:16:52.870 --> 00:16:53.370
cos^2.
00:16:57.020 --> 00:16:58.380
Well, now I'm very happy.
00:16:58.380 --> 00:17:00.470
Because it's just
like the situation
00:17:00.470 --> 00:17:06.017
we had somewhere
on the board here.
00:17:06.017 --> 00:17:07.850
It's just like the
situation we had up here.
00:17:07.850 --> 00:17:11.660
I've got a power of a
cosine times sin x dx.
00:17:11.660 --> 00:17:16.310
So exactly the same
substitution steps in.
00:17:16.310 --> 00:17:19.070
You get, and maybe
you can see what
00:17:19.070 --> 00:17:20.860
happens without doing the work.
00:17:20.860 --> 00:17:22.630
Shall I do the work here?
00:17:22.630 --> 00:17:24.450
I make the same substitution.
00:17:24.450 --> 00:17:30.680
And so this is (1 - u
(1 - u^2) times -du.
00:17:33.540 --> 00:17:40.050
Which is u - u^3 / 3.
00:17:40.050 --> 00:17:42.050
But then I want to put
this minus sign in place,
00:17:42.050 --> 00:17:47.700
and so that gives me -u +
u^3 / 3 plus a constant.
00:17:47.700 --> 00:17:58.090
And then I back substitute
and get cos x + cos^3 x / 3.
00:17:58.090 --> 00:17:59.350
So this is the easy case.
00:17:59.350 --> 00:18:02.190
If you have some odd
power to play with,
00:18:02.190 --> 00:18:07.990
then you can make use of it and
it's pretty straightforward.
00:18:07.990 --> 00:18:10.680
OK the harder case is when
you don't have an odd power.
00:18:10.680 --> 00:18:11.610
So what's the program?
00:18:11.610 --> 00:18:13.770
I'm going to do the
harder case, and then I'm
00:18:13.770 --> 00:18:19.340
going to show you an example of
how to integrate square roots.
00:18:19.340 --> 00:18:26.260
And do an application, using
these ideas from trigonometry.
00:18:26.260 --> 00:18:30.250
So I want to keep
this blackboard.
00:18:30.250 --> 00:18:34.470
Maybe I'll come back
and start here again.
00:18:34.470 --> 00:18:55.240
So the harder case is when
they're only even exponents.
00:18:55.240 --> 00:18:58.190
I'm still trying to
integrate the same form.
00:18:58.190 --> 00:19:00.370
But now all the
exponents are even.
00:19:00.370 --> 00:19:03.510
So we have to do some game.
00:19:03.510 --> 00:19:10.400
And here the game is use
the half angle formula.
00:19:10.400 --> 00:19:16.455
Which I just erased, very
sadly, on the board here.
00:19:16.455 --> 00:19:17.830
Maybe I'll rewrite
them over here
00:19:17.830 --> 00:19:23.550
so we have them on the board.
00:19:23.550 --> 00:19:44.270
I think I remember
what they were.
00:19:44.270 --> 00:19:46.300
So the game is I'm going
to use that half angle
00:19:46.300 --> 00:19:50.180
formula to start getting
rid of those even powers.
00:19:50.180 --> 00:19:54.510
Half angle formula written like
this, exactly, talks about-- it
00:19:54.510 --> 00:19:57.790
rewrites even powers
of sines and cosines.
00:19:57.790 --> 00:20:00.950
So let's see how that
works out in an example.
00:20:00.950 --> 00:20:08.880
How about just the cosine
squared for a start.
00:20:08.880 --> 00:20:09.700
What to do?
00:20:09.700 --> 00:20:11.970
I can't pull anything out.
00:20:11.970 --> 00:20:15.180
I could rewrite
this as 1 - sin^2,
00:20:15.180 --> 00:20:17.290
but then I'd be faced with
integrating the sin^2,
00:20:17.290 --> 00:20:19.480
which is exactly as hard.
00:20:19.480 --> 00:20:23.100
So instead, let's use
this formula here.
00:20:23.100 --> 00:20:29.180
This is really the same
as (1+cos(2theta)) / 2.
00:20:29.180 --> 00:20:32.700
And now, this is easy.
00:20:32.700 --> 00:20:34.340
It's got two parts to it.
00:20:34.340 --> 00:20:38.550
Integrating one half
gives me theta over-- Oh.
00:20:38.550 --> 00:20:42.440
Miraculously, the x
turned into a theta.
00:20:42.440 --> 00:20:44.330
Let's put it back as x.
00:20:44.330 --> 00:20:47.440
I get x/2 by integrating 1/2.
00:20:47.440 --> 00:20:50.110
So, notice that something
non-trigonometric occurs here
00:20:50.110 --> 00:20:52.710
when I do these even integrals.
00:20:52.710 --> 00:20:54.800
x/2 appears.
00:20:54.800 --> 00:20:57.880
And then the other one, OK, so
this takes a little thought.
00:20:57.880 --> 00:21:01.870
The integral of the
cosine is the sine,
00:21:01.870 --> 00:21:05.520
or is it minus the sine.
00:21:05.520 --> 00:21:11.390
Negative sine.
00:21:11.390 --> 00:21:12.634
Shall we take a vote?
00:21:12.634 --> 00:21:13.550
I think it's positive.
00:21:13.550 --> 00:21:18.310
And so you get sin(2x),
but is that right?
00:21:18.310 --> 00:21:19.420
Over 2.
00:21:19.420 --> 00:21:24.150
If I differentiate the
sin(2x), this 2 comes out.
00:21:24.150 --> 00:21:25.950
And would give me
an extra 2 here.
00:21:25.950 --> 00:21:29.580
So there's an extra 2
that I have to put in here
00:21:29.580 --> 00:21:34.670
when I integrate it.
00:21:34.670 --> 00:21:37.230
And there's the answer.
00:21:37.230 --> 00:21:39.480
This is not a substitution.
00:21:39.480 --> 00:21:41.550
I just played with
trig identities here.
00:21:41.550 --> 00:21:45.020
And then did a
simple trig integral,
00:21:45.020 --> 00:21:46.900
getting your help to
get the sign right.
00:21:46.900 --> 00:21:49.310
And thinking about what
this 2 is going to do.
00:21:49.310 --> 00:21:52.990
It produces a 2 in
the denominator.
00:21:52.990 --> 00:21:59.270
But it's not applying
any complicated thing.
00:21:59.270 --> 00:22:03.100
It's just using this identity.
00:22:03.100 --> 00:22:05.580
Let's do another example
that's a little bit harder.
00:22:05.580 --> 00:22:07.690
This time, sin^2 times cos^2.
00:22:35.940 --> 00:22:37.920
Again, no odd powers.
00:22:37.920 --> 00:22:40.730
I've got to work a
little bit harder.
00:22:40.730 --> 00:22:42.930
And what I'm going to do
is apply those identities
00:22:42.930 --> 00:22:44.210
up there.
00:22:44.210 --> 00:22:47.670
Now, what I recommend
doing in this situation
00:22:47.670 --> 00:22:51.370
is going over to
the side somewhere.
00:22:51.370 --> 00:22:55.610
And do some side work.
00:22:55.610 --> 00:22:58.680
Because it's all just
playing with trig functions.
00:22:58.680 --> 00:23:06.280
It's not actually doing
any integrals for a while.
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
00:23:11.815 --> 00:23:14.180
to use those identities
and so let's do that.
00:23:14.180 --> 00:23:16.210
So the sine is (1
- cos(2x)) / 2.
00:23:20.350 --> 00:23:22.170
And the cosine is
(1 + cos(2x)) / 2.
00:23:27.890 --> 00:23:29.870
So I just substitute them in.
00:23:29.870 --> 00:23:31.560
And now I can multiply that out.
00:23:31.560 --> 00:23:38.030
And what I have is a
difference times a sum.
00:23:38.030 --> 00:23:40.771
So you know a formula for that.
00:23:40.771 --> 00:23:43.270
Taking the product of these two
things, well there'll be a 4
00:23:43.270 --> 00:23:44.430
in the denominator.
00:23:44.430 --> 00:23:46.490
And then in the numerator,
I get the square
00:23:46.490 --> 00:23:49.650
of this minus the
square of this.
00:23:49.650 --> 00:23:59.840
(a-b)(a+b) = a^2 -
b^2. = - So I get that.
00:23:59.840 --> 00:24:02.450
Well, I'm a little bit
happier, because at least I
00:24:02.450 --> 00:24:03.710
don't have 4.
00:24:03.710 --> 00:24:07.430
I don't have 2
different squares.
00:24:07.430 --> 00:24:09.940
I still have a square, and
want to integrate this.
00:24:09.940 --> 00:24:12.170
I'm still not in the easy case.
00:24:12.170 --> 00:24:16.970
I got myself back to
an easier hard case.
00:24:16.970 --> 00:24:18.670
But we do know what
to do about this.
00:24:18.670 --> 00:24:21.140
Because I just did it up there.
00:24:21.140 --> 00:24:24.430
And I could play into
this formula that we got.
00:24:24.430 --> 00:24:29.010
But I think it's just as easy
to continue to calculate here.
00:24:29.010 --> 00:24:32.880
Use the half angle
formula again for this,
00:24:32.880 --> 00:24:34.900
and continue on your way.
00:24:34.900 --> 00:24:37.720
So I get a 1/4 from this bit.
00:24:37.720 --> 00:24:43.260
And then minus 1/4 of cos^2(2x).
00:24:45.780 --> 00:24:51.630
And when I plug in 2x in for
theta, there in the top board,
00:24:51.630 --> 00:24:59.650
I'm going to get a 4x
on the right-hand side.
00:24:59.650 --> 00:25:02.130
So it comes out like that.
00:25:02.130 --> 00:25:04.600
And I guess I could simplify
that a little bit more.
00:25:04.600 --> 00:25:05.790
This is a 1/4.
00:25:05.790 --> 00:25:07.840
Oh, but then there's a 2 here.
00:25:07.840 --> 00:25:10.630
It's half that.
00:25:10.630 --> 00:25:12.200
So then I can simplify
a little more.
00:25:12.200 --> 00:25:16.070
It's 1/4 - 1/8, which is 1/8.
00:25:16.070 --> 00:25:19.630
And then I have 1/8 cos(4x).
00:25:25.510 --> 00:25:27.560
OK, that's my side work.
00:25:27.560 --> 00:25:30.590
I just did some trig
identities over here.
00:25:30.590 --> 00:25:32.840
And rewrote sine
squared times cosine
00:25:32.840 --> 00:25:35.930
squared as something which
involves just no powers
00:25:35.930 --> 00:25:37.940
of trig, just cosine by itself.
00:25:37.940 --> 00:25:41.330
And a constant.
00:25:41.330 --> 00:25:45.090
So I can take that and
substitute it in here.
00:25:45.090 --> 00:25:48.810
And now the integration
is pretty easy.
00:25:48.810 --> 00:25:57.810
1/8, cos(4x) / 8,
dx, which is, OK
00:25:57.810 --> 00:26:01.340
the 1/8 is going to give me x/8.
00:26:01.340 --> 00:26:06.314
The integral or cosine is
plus or minus the sine.
00:26:06.314 --> 00:26:08.230
The derivative of the
sine is plus the cosine.
00:26:08.230 --> 00:26:11.440
So it's going to be plus the--
Only there's a minus here.
00:26:11.440 --> 00:26:17.890
So it's going to be the
sine-- minus sin(4x) / 8,
00:26:17.890 --> 00:26:20.797
but then I have an additional
factor in the denominator.
00:26:20.797 --> 00:26:21.880
And what's it going to be?
00:26:21.880 --> 00:26:28.830
I have to put a 4 there.
00:26:28.830 --> 00:26:32.730
So we've done that
calculation, too.
00:26:32.730 --> 00:26:38.250
So any of these-- If you keep
doing this kind of process,
00:26:38.250 --> 00:26:45.530
these two kinds
of procedures, you
00:26:45.530 --> 00:26:48.840
can now integrate
any expression that
00:26:48.840 --> 00:26:52.040
has a power of a sine times
a power of a cosine in it,
00:26:52.040 --> 00:26:56.860
by using these ideas.
00:26:56.860 --> 00:27:01.730
Now, let's see.
00:27:01.730 --> 00:27:16.210
Oh, let me give you an alternate
method for this last one here.
00:27:16.210 --> 00:27:26.460
I know what I'll do.
00:27:26.460 --> 00:27:28.850
Let me give an alternate
method for doing, really
00:27:28.850 --> 00:27:31.670
doing the side work over there.
00:27:31.670 --> 00:27:35.920
I'm trying to deal
with sin^2 times cos^2.
00:27:35.920 --> 00:27:50.080
Well that's the
square of sin x cos x.
00:27:50.080 --> 00:27:54.240
And sin x cos x
shows up right here.
00:27:54.240 --> 00:27:55.900
In another trig identity.
00:27:55.900 --> 00:27:58.410
So we can make use of that, too.
00:27:58.410 --> 00:28:01.330
That reduces the number of
factors of sines and cosines
00:28:01.330 --> 00:28:01.930
by 1.
00:28:01.930 --> 00:28:04.190
So it's going in
the right direction.
00:28:04.190 --> 00:28:11.270
This is equal to 1/2
sin(2x), squared.
00:28:11.270 --> 00:28:17.730
Sine times cosine is
1/2-- Say this right.
00:28:17.730 --> 00:28:21.280
It's sin(2x) / 2, and then
I want to square that.
00:28:21.280 --> 00:28:31.030
So what I get is sin^2(2x) / 4.
00:28:31.030 --> 00:28:33.930
Which is, well, I'm
not too happy yet,
00:28:33.930 --> 00:28:35.639
because I still
have an even power.
00:28:35.639 --> 00:28:37.930
Remember I'm trying to
integrate this thing in the end,
00:28:37.930 --> 00:28:39.060
even powers are bad.
00:28:39.060 --> 00:28:40.910
I try to get rid of them.
00:28:40.910 --> 00:28:46.360
By using that formula,
the half angle formula.
00:28:46.360 --> 00:28:48.950
So I can apply that
to sin x here again.
00:28:48.950 --> 00:28:52.600
I get 1/4 of (1 - cos(4x)) / 2.
00:28:57.480 --> 00:28:59.730
That's what the half angle
formula says for sin^2(2x).
00:29:02.330 --> 00:29:04.790
And that's exactly the
same as the expression
00:29:04.790 --> 00:29:08.660
that I got up here, as well.
00:29:08.660 --> 00:29:11.710
It's the same expression
that I have there.
00:29:11.710 --> 00:29:16.680
So it's the same
expression as I have here.
00:29:16.680 --> 00:29:20.000
So this is just an alternate
way to play this game of using
00:29:20.000 --> 00:29:24.940
the half angle formula.
00:29:24.940 --> 00:29:27.790
OK, let's do a little
application of these things
00:29:27.790 --> 00:29:46.060
and change the
topic a little bit.
00:29:46.060 --> 00:29:48.210
So here's the problem.
00:29:48.210 --> 00:29:56.900
So this is an
application and example
00:29:56.900 --> 00:30:07.570
of a real trig substitution.
00:30:07.570 --> 00:30:22.080
So here's the problem
I want to look at.
00:30:22.080 --> 00:30:26.210
OK, so I have a circle
whose radius is a.
00:30:26.210 --> 00:30:31.710
And I cut out from it
a sort of tab, here.
00:30:31.710 --> 00:30:36.330
This tab here.
00:30:36.330 --> 00:30:38.590
And the height of
this thing is b.
00:30:38.590 --> 00:30:42.110
So this length is a number b.
00:30:42.110 --> 00:30:47.219
And what I want to do is compute
the area of that little tab.
00:30:47.219 --> 00:30:48.010
That's the problem.
00:30:48.010 --> 00:30:50.360
So there's an arc over here.
00:30:50.360 --> 00:30:55.060
And I want to find the
area of this, for a and b,
00:30:55.060 --> 00:30:57.980
in terms of a and b.
00:30:57.980 --> 00:31:06.730
So the area, well,
I guess one way
00:31:06.730 --> 00:31:12.370
to compute the area would be
to take the integral of y dx.
00:31:12.370 --> 00:31:15.980
You've seen the idea
of splitting this up
00:31:15.980 --> 00:31:20.600
into vertical strips whose
height is given by a function
00:31:20.600 --> 00:31:21.100
y(x).
00:31:21.100 --> 00:31:22.266
And then you integrate that.
00:31:22.266 --> 00:31:24.040
That's an interpretation
for the integral.
00:31:24.040 --> 00:31:27.210
The area is given by y dx.
00:31:27.210 --> 00:31:30.327
But that's a little bit awkward,
because my formula for y
00:31:30.327 --> 00:31:31.660
is going to be a little strange.
00:31:31.660 --> 00:31:34.630
It's constant, value of b, along
here, and then at this point
00:31:34.630 --> 00:31:37.660
it becomes this
arc, of the circle.
00:31:37.660 --> 00:31:39.500
So working this
out, I could do it
00:31:39.500 --> 00:31:42.100
but it's a little awkward
because expressing y
00:31:42.100 --> 00:31:45.880
as a function of x, the
top edge of this shape,
00:31:45.880 --> 00:31:50.090
it's a little awkward, and
takes two different regions
00:31:50.090 --> 00:31:51.740
to express.
00:31:51.740 --> 00:31:58.620
So, a different way to
say it is to say x dy.
00:31:58.620 --> 00:32:00.350
Maybe that'll work
a little bit better.
00:32:00.350 --> 00:32:02.810
Or maybe it won't,
but it's worth trying.
00:32:02.810 --> 00:32:05.290
I could just as well
split this region up
00:32:05.290 --> 00:32:08.420
into horizontal strips.
00:32:08.420 --> 00:32:13.010
Whose width is dy,
and whose length is x.
00:32:13.010 --> 00:32:17.250
Now I'm thinking of
this as a function of y.
00:32:17.250 --> 00:32:20.870
This is the graph
of a function of y.
00:32:20.870 --> 00:32:24.880
And that's much better, because
the function of y is, well,
00:32:24.880 --> 00:32:28.010
it's the square root
of a^2 - y^2, isn't it.
00:32:28.010 --> 00:32:34.300
That's x x^2 + y^2 = a^2.
00:32:34.300 --> 00:32:38.640
So that's what x is.
00:32:38.640 --> 00:32:41.740
And that's what I'm
asked to integrate, then.
00:32:41.740 --> 00:32:45.580
Square root of (a^2 - y^2), dy.
00:32:45.580 --> 00:32:47.500
And I can even put in
limits of integration.
00:32:47.500 --> 00:32:49.500
Maybe I should do that,
because this is supposed
00:32:49.500 --> 00:32:50.910
to be an actual number.
00:32:50.910 --> 00:32:55.130
I guess I'm integrating it
from y = 0, that's here.
00:32:55.130 --> 00:33:00.059
To y = b, dy.
00:33:00.059 --> 00:33:01.350
So this is what I want to find.
00:33:01.350 --> 00:33:07.420
This is a integral formula
for the area of that region.
00:33:07.420 --> 00:33:08.730
And this is a new form.
00:33:08.730 --> 00:33:13.970
I don't think that
you've thought
00:33:13.970 --> 00:33:20.310
about integrating expressions
like this in this class before.
00:33:20.310 --> 00:33:23.660
So, it's a new form and I
want to show you how to do it,
00:33:23.660 --> 00:33:30.290
how it's related
to trigonometry.
00:33:30.290 --> 00:33:33.080
It's related to trigonometry
through that exact picture
00:33:33.080 --> 00:33:36.710
that we have on the blackboard.
00:33:36.710 --> 00:33:42.610
After all, this a^2 - y^2
is the formula for this arc.
00:33:42.610 --> 00:33:45.460
And so, what I
propose is that we
00:33:45.460 --> 00:33:49.700
try to exploit the
connection with the circle
00:33:49.700 --> 00:33:52.860
and introduce polar coordinates.
00:33:52.860 --> 00:34:03.499
So, here if I measure
this angle then there
00:34:03.499 --> 00:34:04.790
are various things you can say.
00:34:04.790 --> 00:34:08.076
Like the coordinates of this
point here are a cos(theta),
00:34:08.076 --> 00:34:17.670
a-- Well, I'm sorry, it's not.
00:34:17.670 --> 00:34:20.780
That's an angle, but I
want to call it theta_0.
00:34:20.780 --> 00:34:25.300
And, in general you know
that the coordinates of this
00:34:25.300 --> 00:34:31.010
point are (a cos(theta),
a sin(theta)).
00:34:31.010 --> 00:34:39.370
If the radius is a, then
the angle here is theta.
00:34:39.370 --> 00:34:45.170
So x = a cos(theta),
and y = a sin(theta),
00:34:45.170 --> 00:34:49.100
just from looking at the
geometry of the circle.
00:34:49.100 --> 00:34:52.530
So let's make that
substitution. y = a sin(theta).
00:34:56.600 --> 00:35:00.740
I'm using the picture to
suggest that maybe making
00:35:00.740 --> 00:35:02.870
the substitution is
a good thing to do.
00:35:02.870 --> 00:35:06.020
Let's follow along
and see what happens.
00:35:06.020 --> 00:35:10.310
If that's true, what we're
interested in is integrating,
00:35:10.310 --> 00:35:13.580
a^2 - y^2.
00:35:13.580 --> 00:35:18.900
Which is a^2-- We're interested
in integrating the square root
00:35:18.900 --> 00:35:20.770
of a^2 - y^2.
00:35:20.770 --> 00:35:24.130
Which is the square
root of a^2 minus this.
00:35:24.130 --> 00:35:27.100
a^2 sin^2(theta).
00:35:27.100 --> 00:35:35.870
And, well, that's
equal to a cos theta.
00:35:35.870 --> 00:35:41.710
That's just sin^2 + cos^2
= 1, all over again.
00:35:41.710 --> 00:35:42.830
It's also x.
00:35:42.830 --> 00:35:44.560
This is x.
00:35:44.560 --> 00:35:46.070
And this was x.
00:35:46.070 --> 00:35:48.960
So there are a lot of different
ways to think of this.
00:35:48.960 --> 00:35:51.650
But no matter how you say
it, the thing we're trying
00:35:51.650 --> 00:35:59.270
to integrate, a^2 - y^2 is,
under this substitution it is
00:35:59.270 --> 00:36:02.570
a cos(theta).
00:36:02.570 --> 00:36:06.700
So I'm interested in integrating
the square root of (a^2 - y^2),
00:36:06.700 --> 00:36:09.260
dy.
00:36:09.260 --> 00:36:14.330
And I'm going to make this
substitution y = a sin(theta).
00:36:17.330 --> 00:36:22.320
And so under that substitution,
I've decided that the square
00:36:22.320 --> 00:36:26.170
root of a^2 - y^2
is a cos(theta).
00:36:31.170 --> 00:36:33.240
That's this.
00:36:33.240 --> 00:36:34.690
What about the dy?
00:36:34.690 --> 00:36:38.310
Well, I'd better compute the dy.
00:36:38.310 --> 00:36:40.920
So dy, just differentiating
this expression,
00:36:40.920 --> 00:36:44.820
is a cos(theta) d theta.
00:36:44.820 --> 00:36:57.240
So let's put that in. dy
= a cos(theta) d theta.
00:36:57.240 --> 00:36:58.310
OK.
00:36:58.310 --> 00:37:03.720
Making that trig substitution,
y = a sin(theta),
00:37:03.720 --> 00:37:06.750
has replaced this integral
that has a square root in it.
00:37:06.750 --> 00:37:08.640
And no trig functions.
00:37:08.640 --> 00:37:12.630
With an integral that involves
no square roots and only trig
00:37:12.630 --> 00:37:15.140
functions.
00:37:15.140 --> 00:37:17.630
In fact, it's not too hard to
integrate this now, because
00:37:17.630 --> 00:37:19.270
of the work that we've done.
00:37:19.270 --> 00:37:20.980
The a^2 comes out.
00:37:20.980 --> 00:37:22.630
This is cos^2(theta) d theta.
00:37:26.200 --> 00:37:28.580
And maybe we've done that
example already today.
00:37:28.580 --> 00:37:35.920
I think we have.
00:37:35.920 --> 00:37:38.435
Maybe we can think it through,
but maybe the easiest thing
00:37:38.435 --> 00:37:42.550
is to look back at notes
and see what we got before.
00:37:42.550 --> 00:37:46.330
That was the first example
in the hard case that I did.
00:37:46.330 --> 00:38:03.060
And what it came out to, I used
x instead of theta at the time.
00:38:03.060 --> 00:38:05.570
So this is a good step forward.
00:38:05.570 --> 00:38:08.130
I started with this
integral that I really
00:38:08.130 --> 00:38:12.440
didn't know how to do by any
means that we've had so far.
00:38:12.440 --> 00:38:15.000
And I've replaced it
by a trig integral
00:38:15.000 --> 00:38:16.250
that we do know how to do.
00:38:16.250 --> 00:38:19.160
And now I've done
that trig integral.
00:38:19.160 --> 00:38:22.400
But we're still not quite
done, because of the problem
00:38:22.400 --> 00:38:23.830
of back substituting.
00:38:23.830 --> 00:38:27.670
I'd like to go back and
rewrite this in terms
00:38:27.670 --> 00:38:32.150
of the original variable, y.
00:38:32.150 --> 00:38:34.280
Or, I'd like to go
back and rewrite it
00:38:34.280 --> 00:38:36.390
in terms of the original
limits of integration
00:38:36.390 --> 00:38:40.050
that we had in the
original problem.
00:38:40.050 --> 00:38:42.870
In doing that, it's going
to be useful to rewrite
00:38:42.870 --> 00:38:47.150
this expression and get
rid of the sin(2theta).
00:38:47.150 --> 00:38:51.840
After all, the original
y was expressed in terms
00:38:51.840 --> 00:38:54.660
of sin(theta), not sin(2theta).
00:38:54.660 --> 00:39:04.220
So let me just do that here,
and say that this, in turn,
00:39:04.220 --> 00:39:09.507
is equal to a^2
theta / 2 plus, well,
00:39:09.507 --> 00:39:11.090
sin(2theta) = 2
sin(theta) cos(theta).
00:39:18.080 --> 00:39:20.970
And so, when there's a 4 in
the denominator, what I'll get
00:39:20.970 --> 00:39:25.390
is sin(theta) cos(theta) / 2.
00:39:32.580 --> 00:39:36.630
I did that because I'm getting
closer to the original terms
00:39:36.630 --> 00:39:39.080
that the problem started with.
00:39:39.080 --> 00:39:40.130
Which was sin(theta).
00:40:05.480 --> 00:40:08.055
So let me write down the
integral that we have now.
00:40:08.055 --> 00:40:14.680
The square root of a^2
- y^2, dy is, so far,
00:40:14.680 --> 00:40:26.810
what we know is a^2 (theta / 2 +
sin(theta) cos(theta) / 2) + c.
00:40:26.810 --> 00:40:28.460
But I want to go
back and rewrite this
00:40:28.460 --> 00:40:30.660
in terms of the original value.
00:40:30.660 --> 00:40:32.650
The original variable, y.
00:40:32.650 --> 00:40:37.430
Well, what is theta
in terms of y?
00:40:37.430 --> 00:40:40.760
Let's see. y in terms of
theta was given like this.
00:40:40.760 --> 00:40:44.120
So what is theta in terms of y?
00:40:44.120 --> 00:40:44.950
Ah.
00:40:44.950 --> 00:40:48.710
So here the fearsome arcsine
rears its head, right?
00:40:48.710 --> 00:40:53.680
Theta is the angle so
that y = a sin(theta).
00:40:53.680 --> 00:40:58.230
So that means that theta is
the arcsine, or sine inverse,
00:40:58.230 --> 00:41:00.750
of y/a.
00:41:07.450 --> 00:41:12.630
So that's the first
thing that shows up here.
00:41:12.630 --> 00:41:18.290
arcsin(y/a), all over 2.
00:41:18.290 --> 00:41:19.360
That's this term.
00:41:19.360 --> 00:41:24.530
Theta is arcsin(y/a) / 2.
00:41:24.530 --> 00:41:26.630
What about the other side, here?
00:41:26.630 --> 00:41:30.620
Well sine and cosine, we knew
what they were in terms of y
00:41:30.620 --> 00:41:37.830
and in terms of x, if you like.
00:41:37.830 --> 00:41:40.300
Maybe I'll put the
a^2 inside here.
00:41:40.300 --> 00:41:42.710
That makes it a
little bit nicer.
00:41:42.710 --> 00:41:49.310
Plus, and the other term is
a^2 sin(theta) cos(theta).
00:41:49.310 --> 00:41:52.610
So the a sin(theta) is just y.
00:41:52.610 --> 00:41:55.140
Maybe I'll write this (a
sin(theta)) (a cos(theta))
00:41:55.140 --> 00:41:57.550
/ 2 + c.
00:42:02.330 --> 00:42:03.840
And so I get the same thing.
00:42:03.840 --> 00:42:06.204
And now here a
sin(theta), that's y.
00:42:06.204 --> 00:42:07.370
And what's the a cos(theta)?
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.
00:42:22.010 --> 00:42:28.140
And so there I've
rewritten everything, back
00:42:28.140 --> 00:42:31.250
in terms of the
original variable, y.
00:42:31.250 --> 00:42:36.060
And there's an answer.
00:42:36.060 --> 00:42:41.070
So I've done this indefinite
integration of a form--
00:42:41.070 --> 00:42:44.810
of this quadratic, this square
root of something which is
00:42:44.810 --> 00:42:46.940
a constant minus y^2.
00:42:46.940 --> 00:42:50.640
Whenever you see that, the thing
to think of is trigonometry.
00:42:50.640 --> 00:42:54.310
That's going to play into
the sin^2 + cos^2 identity.
00:42:54.310 --> 00:42:56.880
And the way to exploit it
is to make the substitution
00:42:56.880 --> 00:43:01.100
y = a sin(theta).
00:43:01.100 --> 00:43:04.340
You could also make a
substitution y = a cos(theta),
00:43:04.340 --> 00:43:05.540
if you wanted to.
00:43:05.540 --> 00:43:12.290
And the result would come out
to exactly the same in the end.
00:43:12.290 --> 00:43:14.430
I'm still not quite done
with the original problem
00:43:14.430 --> 00:43:23.990
that I had, because
the original problem
00:43:23.990 --> 00:43:25.810
asked for a definite integral.
00:43:25.810 --> 00:43:33.000
So let's just go back
and finish that as well.
00:43:33.000 --> 00:43:37.910
So the area was
the integral from 0
00:43:37.910 --> 00:43:45.740
to b of this square root.
00:43:45.740 --> 00:43:48.220
So I just want to evaluate
the right-hand side here.
00:43:48.220 --> 00:43:50.880
The answer that we came up
with, this indefinite integral.
00:43:50.880 --> 00:43:53.374
I want to evaluate
it at 0 and at b.
00:43:53.374 --> 00:43:54.040
Well, let's see.
00:43:54.040 --> 00:44:13.470
When I evaluate it at b, I get
a^2 arcsin(b/a) / 2 plus y,
00:44:13.470 --> 00:44:18.900
which is b, times the
square root of a^2 - b^2,
00:44:18.900 --> 00:44:23.050
putting y = b, divided by 2.
00:44:23.050 --> 00:44:26.300
So I've plugged in y
= b into that formula,
00:44:26.300 --> 00:44:27.290
this is what I get.
00:44:27.290 --> 00:44:31.280
Then when I plug in y = 0,
well the, sine of 0 is 0,
00:44:31.280 --> 00:44:34.040
so the arcsine of 0 is 0.
00:44:34.040 --> 00:44:35.720
So this term goes away.
00:44:35.720 --> 00:44:38.930
And when y = 0,
this term is 0 also.
00:44:38.930 --> 00:44:43.210
And so I don't get any
subtracted terms at all.
00:44:43.210 --> 00:44:45.600
So there's an
expression for this.
00:44:45.600 --> 00:44:52.360
Notice that this arcsin(b/a),
that's exactly this angle.
00:44:52.360 --> 00:45:00.550
arcsin(b/a), it's the angle
that you get when y = b.
00:45:00.550 --> 00:45:06.760
So this theta is
the arcsin(b/a).
00:45:09.990 --> 00:45:15.870
Put this over here.
00:45:15.870 --> 00:45:17.090
That is theta_0.
00:45:17.090 --> 00:45:21.560
That is the angle
that the corner makes.
00:45:21.560 --> 00:45:28.460
So I could rewrite this as a
a^2 theta_0 / 2 plus b times
00:45:28.460 --> 00:45:34.722
the square root of
a^2 - b^2, over 2.
00:45:34.722 --> 00:45:36.430
Let's just think about
this for a minute.
00:45:36.430 --> 00:45:40.320
I have these two terms in
the sum, is that reasonable?
00:45:40.320 --> 00:45:44.080
The first term is a^2.
00:45:44.080 --> 00:45:50.370
That's the radius squared
times this angle, times 1/2.
00:45:50.370 --> 00:45:54.280
Well, I think that is exactly
the area of this sector.
00:45:54.280 --> 00:46:03.170
a^2 theta / 2 is the formula
for the area of the sector.
00:46:03.170 --> 00:46:07.070
And this one, this is
the vertical elevation.
00:46:07.070 --> 00:46:09.500
This is the horizontal.
00:46:09.500 --> 00:46:14.550
a^2 - b^2 is this distance.
00:46:14.550 --> 00:46:16.980
Square root of a^2 - b^2.
00:46:16.980 --> 00:46:20.420
So the right-hand term is b
times the square root of a^2 -
00:46:20.420 --> 00:46:31.930
b^2 divided by 2, that's
the area of that triangle.
00:46:31.930 --> 00:46:34.280
So using a little
bit of geometry
00:46:34.280 --> 00:46:39.890
gives you the same answer as
all of this elaborate calculus.
00:46:39.890 --> 00:46:41.610
Maybe that's enough
cause for celebration
00:46:41.610 --> 00:46:43.620
for us to quit for today.