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PROFESSOR: So
again, welcome back.

00:00:24.020 --> 00:00:28.190
And today's topic
is a continuation

00:00:28.190 --> 00:00:29.420
of what we did last time.

00:00:29.420 --> 00:00:31.850
We still have a little
bit of work and thinking

00:00:31.850 --> 00:00:38.600
to do concerning
polar coordinates.

00:00:38.600 --> 00:00:50.210
So we're going to talk
about polar coordinates.

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And my first job today is to
talk a little bit about area.

00:00:58.320 --> 00:01:01.760
That's something we
didn't mention last time.

00:01:01.760 --> 00:01:05.320
And since we're all
back from Thanksgiving,

00:01:05.320 --> 00:01:10.130
we can certainly talk
about it in terms of a pie.

00:01:10.130 --> 00:01:14.650
Which is the basic idea for
area in polar coordinates.

00:01:14.650 --> 00:01:21.840
Here's our pie, and
here's a slice of the pie.

00:01:21.840 --> 00:01:25.700
The slice has a piece of
arc length on it, which

00:01:25.700 --> 00:01:28.150
I'm going to call delta theta.

00:01:28.150 --> 00:01:31.540
And the area of that
shaded-in slice,

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I'm going to call delta A.

00:01:35.070 --> 00:01:38.210
And let's suppose
that the radius is a.

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Little a.

00:01:39.350 --> 00:01:45.950
So this is a pie of radius a.

00:01:45.950 --> 00:01:48.130
That's our picture.

00:01:48.130 --> 00:01:50.510
Now, it's pretty easy
to figure out what

00:01:50.510 --> 00:01:54.390
the area that slice of pie is.

00:01:54.390 --> 00:01:58.650
The total area is,
of course, pi a^2.

00:01:58.650 --> 00:02:00.030
We know that.

00:02:00.030 --> 00:02:05.190
And to get this fraction,
delta A, all we have to do

00:02:05.190 --> 00:02:10.270
is take the percentage of the
arc of the total circumference.

00:02:10.270 --> 00:02:16.720
That's (delta theta) / This is
the fraction of area-- sorry,

00:02:16.720 --> 00:02:18.590
fraction of the
total circumference,

00:02:18.590 --> 00:02:20.900
the total length around the rim.

00:02:20.900 --> 00:02:24.150
And then we multiply
that by pi a^2.

00:02:24.150 --> 00:02:27.000
And that's giving
us the total area.

00:02:27.000 --> 00:02:30.500
And if you work that out,
that's delta A is equal

00:02:30.500 --> 00:02:36.860
to, the pi's cancel and we
have 1/2 a^2 delta theta.

00:02:36.860 --> 00:02:44.990
So here's the basic formula.

00:02:44.990 --> 00:02:54.510
And now what we need to do is to
talk about a variable pie here.

00:02:54.510 --> 00:02:58.790
That would be a pie with
a kind of a wavy crust.

00:02:58.790 --> 00:03:01.520
Which is coming
around like this.

00:03:01.520 --> 00:03:04.140
So r = r(theta).

00:03:04.140 --> 00:03:07.860
The distance from
the center is varying

00:03:07.860 --> 00:03:13.420
with the place where we are, the
angle where we're shooting out.

00:03:13.420 --> 00:03:22.070
And now I want to subdivide
that into little chunks here.

00:03:22.070 --> 00:03:25.760
Now, the idea for adding
up the area, the total area

00:03:25.760 --> 00:03:29.110
of this piece
that's swept out, is

00:03:29.110 --> 00:03:33.160
to break it up into
little slices whose areas

00:03:33.160 --> 00:03:37.580
are almost easy to calculate.

00:03:37.580 --> 00:03:42.900
Namely, what we're
going to do is to take,

00:03:42.900 --> 00:03:46.000
and I'm going to
label it this way.

00:03:46.000 --> 00:03:49.520
I'm going to take these little
circular arcs, which go--

00:03:49.520 --> 00:03:55.770
So I'm going to extend
past where this goes.

00:03:55.770 --> 00:03:58.620
And then I'm going to take
each circular arc here.

00:03:58.620 --> 00:04:00.510
So here's a circular arc.

00:04:00.510 --> 00:04:04.010
And then here's
another circular arc.

00:04:04.010 --> 00:04:05.590
And here's another circular arc.

00:04:05.590 --> 00:04:08.250
It's just right on
the nose in that case.

00:04:08.250 --> 00:04:12.050
Now, in these two
cases, so basically

00:04:12.050 --> 00:04:14.910
the picture that I'm trying
to draw for you is this.

00:04:14.910 --> 00:04:17.960
I have some sector.

00:04:17.960 --> 00:04:19.445
And then I have
some circular arc.

00:04:19.445 --> 00:04:23.400
And maybe it takes
a little extra.

00:04:23.400 --> 00:04:26.780
There's a little extra area,
I'm making an error in the area.

00:04:26.780 --> 00:04:28.400
This is a little extra area.

00:04:28.400 --> 00:04:32.410
And maybe to draw
it the other way.

00:04:32.410 --> 00:04:34.530
I'm a little short on this one.

00:04:34.530 --> 00:04:37.190
And let's say on this one
I'm right on the nose.

00:04:37.190 --> 00:04:42.950
I have the same arc as
the curve of the surface.

00:04:42.950 --> 00:04:45.580
Now this is a little bit
like the step functions

00:04:45.580 --> 00:04:47.240
that we used in Riemann sums.

00:04:47.240 --> 00:04:48.940
It's practically the same.

00:04:48.940 --> 00:04:52.490
Eventually, this little band
of stuff that we're missing by,

00:04:52.490 --> 00:04:56.110
if we take very, very
narrow little slices here,

00:04:56.110 --> 00:04:58.610
is going to be negligible.

00:04:58.610 --> 00:05:01.360
It'll get closer and
closer to the curve itself.

00:05:01.360 --> 00:05:04.290
So that area will tend
to 0 in the limit.

00:05:04.290 --> 00:05:05.940
So we don't have
to worry about it.

00:05:05.940 --> 00:05:09.970
And the approximate
relationship is sitting here.

00:05:09.970 --> 00:05:12.110
Where this distance now is r.

00:05:12.110 --> 00:05:14.340
So this radius is r.

00:05:14.340 --> 00:05:18.330
And this is this delta theta.

00:05:18.330 --> 00:05:23.080
And so in the approximate case,
what we have is that delta A is

00:05:23.080 --> 00:05:28.830
approximately 1/2
r^2 delta theta.

00:05:28.830 --> 00:05:30.840
Which is practically the
same thing we had here.

00:05:30.840 --> 00:05:34.985
Except that that r is
replacing the constant there.

00:05:34.985 --> 00:05:39.290
And it's approximately
true, because r is varying.

00:05:39.290 --> 00:05:42.230
And then in the limit,
we have the exact formula

00:05:42.230 --> 00:05:43.870
for the differential.

00:05:43.870 --> 00:05:46.200
Which is this one.

00:05:46.200 --> 00:05:49.190
So this is the main
formula for area.

00:05:49.190 --> 00:05:52.000
And if you like, the total
area then is going to be

00:05:52.000 --> 00:05:56.290
the integral from some starting
place to some end place of 1/2

00:05:56.290 --> 00:05:57.070
r^2 d theta.

00:06:00.020 --> 00:06:04.210
Now, this is only useful in
the situation that we're in.

00:06:04.210 --> 00:06:07.120
Namely-- So this is the
other important formula.

00:06:07.120 --> 00:06:11.440
And this is only useful when
r is a function of theta.

00:06:11.440 --> 00:06:17.930
When this is the way in which
the region is presented to us.

00:06:17.930 --> 00:06:20.430
So that's the setup.

00:06:20.430 --> 00:06:23.310
And that's our main formula.

00:06:23.310 --> 00:06:27.884
Let's do what example.

00:06:27.884 --> 00:06:29.300
The example that
I'm going to take

00:06:29.300 --> 00:06:31.750
is the one that we did
at the end of last time,

00:06:31.750 --> 00:06:33.790
or near the end of last time.

00:06:33.790 --> 00:06:39.260
Which was this formula
here. r = 2a cos(theta).

00:06:39.260 --> 00:06:46.920
Remember, that was the same
as (x-a)^2 + y^2 = a^2.

00:06:46.920 --> 00:06:49.120
So this is what
we did last time.

00:06:49.120 --> 00:06:53.550
We connected this
rectangular representation

00:06:53.550 --> 00:06:54.920
to that polar representation.

00:06:54.920 --> 00:07:02.130
And the picture is of a circle.

00:07:02.130 --> 00:07:11.580
Where this is the point (2a, 0).

00:07:11.580 --> 00:07:16.140
So let's figure out
what the area is.

00:07:16.140 --> 00:07:19.480
Well, first of all,
we have to figure out

00:07:19.480 --> 00:07:22.040
when we sweep out
the area, we have

00:07:22.040 --> 00:07:27.160
to realize that we only
go from -pi/2 to pi/2.

00:07:27.160 --> 00:07:30.620
So that's something we
can get from the picture.

00:07:30.620 --> 00:07:32.900
You can also get it
directly from this formula

00:07:32.900 --> 00:07:38.030
if you realize that cosine is
positive in this range here.

00:07:38.030 --> 00:07:40.430
And at the ends, it's 0.

00:07:40.430 --> 00:07:46.870
So the thing encloses
a region at these ends.

00:07:46.870 --> 00:07:54.550
So at the ends, cosine of plus
or minus pi/2 is equal to 0.

00:07:54.550 --> 00:08:02.320
That's what cinches this up
like a little sack, if you like.

00:08:02.320 --> 00:08:07.530
So the area is now going to be
the integral from -pi/2 to pi/2

00:08:07.530 --> 00:08:13.410
of 1/2 times the square of
r, that's (2a cos(theta))^2,

00:08:13.410 --> 00:08:18.111
d theta.

00:08:18.111 --> 00:08:18.610
Question.

00:08:18.610 --> 00:08:25.434
STUDENT: [INAUDIBLE]

00:08:25.434 --> 00:08:27.600
PROFESSOR: How do I know
from looking at the picture

00:08:27.600 --> 00:08:33.400
that I'm going from -pi/2
to pi/2, is the question.

00:08:33.400 --> 00:08:36.820
I do it with my whole body.

00:08:36.820 --> 00:08:39.860
I say, here I am pointing down.

00:08:39.860 --> 00:08:41.120
That's -pi/2.

00:08:41.120 --> 00:08:43.890
I sweep up, that's 0.

00:08:43.890 --> 00:08:47.710
And I get all the way
up to here. that's pi/2.

00:08:47.710 --> 00:08:48.910
So that's the way I do it.

00:08:48.910 --> 00:08:51.450
That's really the way I
do it, I'm being honest.

00:08:51.450 --> 00:08:54.520
Now if you're a machine,
you can't actually look.

00:08:54.520 --> 00:08:57.800
And you don't have a body,
so you can't point your arms.

00:08:57.800 --> 00:08:59.650
Then you would have
to go by the formulas.

00:08:59.650 --> 00:09:02.950
And you'd have to actually use
something like this formula

00:09:02.950 --> 00:09:03.900
here.

00:09:03.900 --> 00:09:07.830
The fact that this is
where the loop cinches up.

00:09:07.830 --> 00:09:10.750
This is where the
radius comes into 0.

00:09:10.750 --> 00:09:11.270
At pi/2.

00:09:11.270 --> 00:09:17.230
So you need to know that in
order to understand the range.

00:09:17.230 --> 00:09:17.970
Another question.

00:09:17.970 --> 00:09:23.089
STUDENT: [INAUDIBLE]

00:09:23.089 --> 00:09:24.630
PROFESSOR: So when
we're doing these,

00:09:24.630 --> 00:09:28.000
should we just guess that
it's going to be a loop?

00:09:28.000 --> 00:09:30.590
I'm probably going to give
you some clues as to what's

00:09:30.590 --> 00:09:31.090
going on.

00:09:31.090 --> 00:09:33.990
Because it's very hard to
figure these things out.

00:09:33.990 --> 00:09:36.920
Sometimes it'll be bounded by
one curve and another curve,

00:09:36.920 --> 00:09:39.780
and I'll say it's the thing
in between those two curves.

00:09:39.780 --> 00:09:42.710
That's the kind of
thing that I could do.

00:09:42.710 --> 00:09:47.590
Here, you really should
know this one in advance.

00:09:47.590 --> 00:09:51.270
This is by far the
most-- or this is one

00:09:51.270 --> 00:09:52.540
of the typical cases, anyway.

00:09:52.540 --> 00:09:54.690
I'm going to give you
a couple more examples.

00:09:54.690 --> 00:09:57.130
Don't get too
worked up over this.

00:09:57.130 --> 00:09:59.400
You will somehow be
able to visualize it.

00:09:59.400 --> 00:10:05.070
I'll give you some examples
to help you out with it later.

00:10:05.070 --> 00:10:06.360
So here's the situation.

00:10:06.360 --> 00:10:07.620
Here's my integral.

00:10:07.620 --> 00:10:10.450
And now we're faced
with a trig integral.

00:10:10.450 --> 00:10:12.970
Which we have to
remember how to do.

00:10:12.970 --> 00:10:14.770
Now, the trig integral
here-- so first

00:10:14.770 --> 00:10:16.760
let me factor out the constants.

00:10:16.760 --> 00:10:21.740
This is 4a 4a^2 / 2, so it's
2a^2 integral from -pi/2

00:10:21.740 --> 00:10:26.870
to pi/2 of cos^2(theta) d theta.

00:10:26.870 --> 00:10:29.510
And now you have to
remember what you're

00:10:29.510 --> 00:10:32.570
supposed to do at this point.

00:10:32.570 --> 00:10:34.310
So think, if you
haven't done it yet,

00:10:34.310 --> 00:10:37.830
this is practice you need to do.

00:10:37.830 --> 00:10:40.660
This trig integral is handled
by a double angle formula.

00:10:40.660 --> 00:10:42.520
As it happens, I'm
going to be giving you

00:10:42.520 --> 00:10:44.670
these formulas on
the review sheet.

00:10:44.670 --> 00:10:47.480
You'll see they're written
on the review sheet.

00:10:47.480 --> 00:10:49.940
At least in some form.

00:10:49.940 --> 00:10:51.940
So for example,
there's a formula,

00:10:51.940 --> 00:10:54.480
and this will be
on the exam, too.

00:10:54.480 --> 00:10:58.440
So this is the correct
formula to use here.

00:10:58.440 --> 00:11:04.240
Is that this is + (1 +
cos(2theta)) / 2 d theta.

00:11:04.240 --> 00:11:08.230
So that's the substitution that
you use for the cosine squared

00:11:08.230 --> 00:11:12.490
in order to integrate it.

00:11:12.490 --> 00:11:15.440
That serves as a little
review of trig integrals.

00:11:15.440 --> 00:11:18.830
And now, this is quite easy.

00:11:18.830 --> 00:11:27.560
This integral now is easy.

00:11:27.560 --> 00:11:28.380
Why is it easy?

00:11:28.380 --> 00:11:30.640
Well, because it's
the antiderivative

00:11:30.640 --> 00:11:32.650
of a constant, cos(2theta),
its antiderivative

00:11:32.650 --> 00:11:34.640
you're supposed to be
able to write down.

00:11:34.640 --> 00:11:36.820
So the antiderivative
of 1 is theta.

00:11:36.820 --> 00:11:39.510
And the antiderivative
of the cosine

00:11:39.510 --> 00:11:50.120
is 1/2 the sine
when it's 2 theta.

00:11:50.120 --> 00:11:52.930
And that is a ^2 a^2
(pi/2 - (-pi/2)).

00:11:55.920 --> 00:12:00.070
And the signs go away
because they're both 0.

00:12:00.070 --> 00:12:04.900
So all told we get pi a^2, which
is certainly what we would like

00:12:04.900 --> 00:12:05.400
it to be.

00:12:05.400 --> 00:12:09.482
It's the area of the circle.

00:12:09.482 --> 00:12:10.190
Another question?

00:12:10.190 --> 00:12:28.982
STUDENT: [INAUDIBLE]

00:12:28.982 --> 00:12:30.440
PROFESSOR: The
question, so I'm not

00:12:30.440 --> 00:12:31.990
sure which question
you're asking.

00:12:31.990 --> 00:12:35.360
I pivoted my arm around (0, 0).

00:12:35.360 --> 00:12:38.550
This point, this is the point
we're talking about, (0, 0),

00:12:38.550 --> 00:12:39.290
is a key point.

00:12:39.290 --> 00:12:43.310
It's where I guess you could
say I stuck my elbow there.

00:12:43.310 --> 00:12:48.367
Now, the reason is that
it's the place where r = 0.

00:12:48.367 --> 00:12:50.950
So it's more or less the center
of the universe from the point

00:12:50.950 --> 00:12:54.630
of view of this problem.

00:12:54.630 --> 00:12:57.730
So it's the reference
point and if you like,

00:12:57.730 --> 00:13:01.500
when you're doing this, it's a
little bit like a radar screen.

00:13:01.500 --> 00:13:03.050
Everything is
centered at the origin

00:13:03.050 --> 00:13:07.135
and you're taking rays
coming out from it.

00:13:07.135 --> 00:13:10.390
And seeing where
they're going to go.

00:13:10.390 --> 00:13:12.230
So for example, this
is the theta = 0 ray,

00:13:12.230 --> 00:13:15.240
this is the theta = pi/4 ray.

00:13:15.240 --> 00:13:17.930
This the theta = pi/2 ray.

00:13:17.930 --> 00:13:21.250
And indeed, if my elbow is
right at this center here,

00:13:21.250 --> 00:13:24.610
I'm pointing in those
various directions.

00:13:24.610 --> 00:13:32.650
So that's what I had in
mind when I did that.

00:13:32.650 --> 00:13:35.140
You can always get these
formulas, by the way,

00:13:35.140 --> 00:13:42.120
from the original business, x =
r cos(theta), y = r sin(theta).

00:13:42.120 --> 00:13:44.689
But it's useful to have the
geometric picture as well.

00:13:44.689 --> 00:13:46.230
In other words, if
you were a machine

00:13:46.230 --> 00:13:48.150
you'd have to rely
on these formulas.

00:13:48.150 --> 00:13:49.480
And plot things using these.

00:13:49.480 --> 00:13:57.090
Always.

00:13:57.090 --> 00:14:00.509
Now, in terms of plotting
I want to expand your brain

00:14:00.509 --> 00:14:01.050
a little bit.

00:14:01.050 --> 00:14:03.810
So we need just a little bit
more practice with plotting.

00:14:03.810 --> 00:14:05.870
In polar coordinates.

00:14:05.870 --> 00:14:09.580
And so, the first question
that I want to ask you

00:14:09.580 --> 00:14:15.370
is, what happens outside
of this range of theta?

00:14:15.370 --> 00:14:19.960
In other words, what happens
if theta's beyond pi/2?

00:14:19.960 --> 00:14:22.080
Can somebody see
what's happening

00:14:22.080 --> 00:14:23.340
to the formulas in that case?

00:14:23.340 --> 00:14:27.480
So what I'm looking at
now, let's go back to it.

00:14:27.480 --> 00:14:35.600
What I'm looking at
is this formula here.

00:14:35.600 --> 00:14:37.690
But to use the
elbow analogy here,

00:14:37.690 --> 00:14:39.280
I'm swept around like this.

00:14:39.280 --> 00:14:40.810
But now I'm going
to point this way.

00:14:40.810 --> 00:14:42.340
I'm going to point
out over there.

00:14:42.340 --> 00:14:49.450
My hand is up here in
the northwest direction.

00:14:49.450 --> 00:14:51.519
So what's going to happen?

00:14:51.519 --> 00:14:52.560
Somebody want to tell me?

00:14:52.560 --> 00:14:55.210
STUDENT: [INAUDIBLE]

00:14:55.210 --> 00:14:56.920
PROFESSOR: It goes
around itself.

00:14:56.920 --> 00:14:57.640
That's right.

00:14:57.640 --> 00:15:02.400
What happens is that when r
crosses this vertical, r = 0,

00:15:02.400 --> 00:15:05.210
when it crosses over
here it goes negative.

00:15:05.210 --> 00:15:08.540
So although my theta is
pointing me this way,

00:15:08.540 --> 00:15:10.490
the thing is going
to go backwards.

00:15:10.490 --> 00:15:12.590
And there's another clue.

00:15:12.590 --> 00:15:13.590
Which is very important.

00:15:13.590 --> 00:15:15.290
How far backwards is it going?

00:15:15.290 --> 00:15:18.370
Well, you don't actually need to
know anything but this equation

00:15:18.370 --> 00:15:23.530
here, to understand that it
has to be on the same circle.

00:15:23.530 --> 00:15:26.040
So when I'm pointing
this way, the things

00:15:26.040 --> 00:15:29.550
points backwards to
this point over there.

00:15:29.550 --> 00:15:32.012
So what happens is,
it goes around once.

00:15:32.012 --> 00:15:33.470
And then when I
point out this way,

00:15:33.470 --> 00:15:35.614
it sweeps around a second time.

00:15:35.614 --> 00:15:37.530
It just keeps on going
around the same circle.

00:15:37.530 --> 00:15:38.867
So over here it's empty.

00:15:38.867 --> 00:15:40.700
Because it's pointing
the other way and it's

00:15:40.700 --> 00:15:42.570
sweeping around the same curve.

00:15:42.570 --> 00:15:47.600
A second time.

00:15:47.600 --> 00:15:50.960
Now, if you were foolish
enough to integrate, say,

00:15:50.960 --> 00:15:54.060
from 0 to 2pi or some wider
range, what would happen

00:15:54.060 --> 00:15:55.810
is you would just
double the area.

00:15:55.810 --> 00:16:00.970
Because you would have
swept it out twice.

00:16:00.970 --> 00:16:02.890
So that's the mistake
that you'll make.

00:16:02.890 --> 00:16:05.390
Sometimes you'll count things
as negative and positive.

00:16:05.390 --> 00:16:07.392
But because there's
a square here,

00:16:07.392 --> 00:16:08.725
it's always a positive quantity.

00:16:08.725 --> 00:16:13.890
And you'll always over-count
if you go too far.

00:16:13.890 --> 00:16:15.930
So that's what happens.

00:16:15.930 --> 00:16:17.930
Again, it sweeps
out the same region.

00:16:17.930 --> 00:16:19.890
That's because these
two equations really

00:16:19.890 --> 00:16:22.050
are equivalent to each other.

00:16:22.050 --> 00:16:24.080
It's just that this one
sweeps it out twice.

00:16:24.080 --> 00:16:28.930
And this one doesn't say
how it's sweeping it out.

00:16:28.930 --> 00:16:30.380
Yeah, another question.

00:16:30.380 --> 00:16:32.046
STUDENT: Doesn't this
equation also work

00:16:32.046 --> 00:16:34.320
if you just go from 0 to pi?

00:16:34.320 --> 00:16:36.630
PROFESSOR: Does the
integration work

00:16:36.630 --> 00:16:38.900
if you just go from 0 to pi?

00:16:38.900 --> 00:16:40.730
The answer is yes.

00:16:40.730 --> 00:16:42.884
That's a very weird
object, though.

00:16:42.884 --> 00:16:44.300
Let me just show
you what that is.

00:16:44.300 --> 00:16:47.360
If you started from 0 to 2pi.

00:16:47.360 --> 00:16:50.120
So I'll illustrate it on here.

00:16:50.120 --> 00:16:53.290
The first thing that you
swept out between 0 and pi/2

00:16:53.290 --> 00:16:54.750
is this part here.

00:16:54.750 --> 00:16:56.160
That was swept out.

00:16:56.160 --> 00:17:00.190
And then, when you're going
around this next quadrant here,

00:17:00.190 --> 00:17:05.026
you're actually sweeping
out this underside here.

00:17:05.026 --> 00:17:06.900
So actually, you're
getting it because you're

00:17:06.900 --> 00:17:09.710
getting half of it on one half,
and getting the other half

00:17:09.710 --> 00:17:10.990
on the other quadrant.

00:17:10.990 --> 00:17:14.120
So it's actually giving
you the right answer.

00:17:14.120 --> 00:17:15.590
That turns out to be OK.

00:17:15.590 --> 00:17:17.740
It's a little weird way
to chop up a circle.

00:17:17.740 --> 00:17:23.706
But it's legal.

00:17:23.706 --> 00:17:25.080
But of course,
that's an accident

00:17:25.080 --> 00:17:26.260
of this particular figure.

00:17:26.260 --> 00:17:27.830
You can't count
on that happening.

00:17:27.830 --> 00:17:29.580
It's much better to
line it up exactly

00:17:29.580 --> 00:17:32.240
with what the figure does.

00:17:32.240 --> 00:17:35.630
So don't do that too often.

00:17:35.630 --> 00:17:40.640
You might run into troubles.

00:17:40.640 --> 00:17:44.110
So I'm going to give you
a couple more examples

00:17:44.110 --> 00:17:48.970
of practice with these pictures.

00:17:48.970 --> 00:17:57.080
And maybe I'm going to get
rid of this one up here.

00:17:57.080 --> 00:18:03.850
So here's another favorite.

00:18:03.850 --> 00:18:05.140
Here's another favorite.

00:18:05.140 --> 00:18:08.000
So this, if you
like, is Example 2.

00:18:08.000 --> 00:18:09.720
I guess we had an
Example 1 up there.

00:18:09.720 --> 00:18:12.160
And now we're really
not going to try

00:18:12.160 --> 00:18:13.690
to do any more area examples.

00:18:13.690 --> 00:18:15.900
The area examples are
actually straightforward.

00:18:15.900 --> 00:18:18.830
It's really just figuring out
what the picture looks like.

00:18:18.830 --> 00:18:27.200
So this is examples of drawings.

00:18:27.200 --> 00:18:34.890
So this one is one
that's kind of fun to do.

00:18:34.890 --> 00:18:37.500
This is r = sin(2theta).

00:18:37.500 --> 00:18:40.120
Something like this
is on your homework.

00:18:40.120 --> 00:18:46.830
And so what happens
here is the following.

00:18:46.830 --> 00:18:50.850
What happens here is
that at theta = 0,

00:18:50.850 --> 00:18:53.100
that's the first place.

00:18:53.100 --> 00:18:56.470
So let's just plot
a few places here.

00:18:56.470 --> 00:18:57.960
I'm not going to plot very many.

00:18:57.960 --> 00:19:00.590
Theta = 0, I get r is 1.

00:19:00.590 --> 00:19:02.940
Whoops, I get r is 0.

00:19:02.940 --> 00:19:04.020
Sorry.

00:19:04.020 --> 00:19:09.900
And then pi/4, that's where I
get sin(pi/2), I get 1 here.

00:19:09.900 --> 00:19:10.570
For this.

00:19:10.570 --> 00:19:14.400
And then again, at
pi/2 I get sin(pi),

00:19:14.400 --> 00:19:17.090
which is back at 0 again.

00:19:17.090 --> 00:19:21.010
So it's-- And the other thing
to say is in between here

00:19:21.010 --> 00:19:21.810
it's positive.

00:19:21.810 --> 00:19:22.860
In between.

00:19:22.860 --> 00:19:26.120
So what it does is,
it starts out at 0

00:19:26.120 --> 00:19:30.290
and it goes out to the
radius 1 over here.

00:19:30.290 --> 00:19:32.640
And then it comes back.

00:19:32.640 --> 00:19:35.350
So it does something like this.

00:19:35.350 --> 00:19:39.790
It goes out, and it comes back.

00:19:39.790 --> 00:19:44.900
Now because of the symmetries
of the sine function,

00:19:44.900 --> 00:19:47.070
this is pretty much
all you need to know.

00:19:47.070 --> 00:19:51.550
It does something similar
in all of the quadrants.

00:19:51.550 --> 00:19:56.000
But in order to see what it's
doing, it's useful for you

00:19:56.000 --> 00:19:57.450
to watch me draw it.

00:19:57.450 --> 00:20:00.930
Because the order is very
important for understanding

00:20:00.930 --> 00:20:01.970
what it's doing.

00:20:01.970 --> 00:20:06.170
It's similar to this weird
business with the circle here.

00:20:06.170 --> 00:20:09.110
So watch me draw this guy.

00:20:09.110 --> 00:20:11.700
I'll draw it in red because
it usually has a name.

00:20:11.700 --> 00:20:12.870
So here it is.

00:20:12.870 --> 00:20:15.450
It does this thing.

00:20:15.450 --> 00:20:17.280
And then it does this.

00:20:17.280 --> 00:20:19.330
And then it does this.

00:20:19.330 --> 00:20:21.770
And then it does that.

00:20:21.770 --> 00:20:26.130
So it's called a four-leaf rose.

00:20:26.130 --> 00:20:30.380
I drew it in pink because
it's kind of a rose here.

00:20:30.380 --> 00:20:32.050
So it started out over here.

00:20:32.050 --> 00:20:34.800
This is Step 1.

00:20:34.800 --> 00:20:40.370
And this is the range
0 < theta < pi/4.

00:20:40.370 --> 00:20:42.200
It did this part here.

00:20:42.200 --> 00:20:46.180
And then it went 2 here.

00:20:46.180 --> 00:20:48.510
So I should draw these
in white, because they're

00:20:48.510 --> 00:20:50.580
harder to read in red.

00:20:50.580 --> 00:20:52.800
But now look at what it did.

00:20:52.800 --> 00:20:55.160
It did not make a
right angle turn.

00:20:55.160 --> 00:20:56.930
It was nice and smooth.

00:20:56.930 --> 00:20:59.500
It went around here and
then it went down here.

00:20:59.500 --> 00:21:00.900
This is 3.

00:21:00.900 --> 00:21:02.840
Back here, that's 4.

00:21:02.840 --> 00:21:05.370
And then over here, that's 5.

00:21:05.370 --> 00:21:07.540
Back up here, that's 6.

00:21:07.540 --> 00:21:09.650
And then around here, that's 7.

00:21:09.650 --> 00:21:11.030
And down here, that's 8.

00:21:11.030 --> 00:21:14.750
And then back where it
started and goes around again.

00:21:14.750 --> 00:21:17.940
And this is because
actually it's switching sign

00:21:17.940 --> 00:21:19.470
when it crosses the origin.

00:21:19.470 --> 00:21:21.620
When it was over in this
quadrant the first time,

00:21:21.620 --> 00:21:28.050
it actually was tracing
what's directly behind it.

00:21:28.050 --> 00:21:29.470
So this is kind of amusing.

00:21:29.470 --> 00:21:33.681
From this little tiny formula
you get this pretty diagram

00:21:33.681 --> 00:21:34.180
here.

00:21:34.180 --> 00:21:36.490
Anyway that's, as I
say, an old favorite.

00:21:36.490 --> 00:21:40.490
And here if you want to
do the area of one leaf,

00:21:40.490 --> 00:21:42.810
you've got to make sure
you understand that it's

00:21:42.810 --> 00:21:49.650
a small piece of the whole.

00:21:49.650 --> 00:21:52.230
OK, now I have one
last drawing example

00:21:52.230 --> 00:21:54.280
that I want to discuss with you.

00:21:54.280 --> 00:21:57.150
And it involves another
skill that we haven't quite

00:21:57.150 --> 00:21:59.550
gotten enough practice with.

00:21:59.550 --> 00:22:01.320
So I'm going to do that one.

00:22:01.320 --> 00:22:04.210
And it's also preparation
for an exercise.

00:22:04.210 --> 00:22:08.360
But one that we're going
to do after the test.

00:22:08.360 --> 00:22:15.110
So here's my last example.

00:22:15.110 --> 00:22:18.300
We're going to discuss what
happens with this function

00:22:18.300 --> 00:22:20.510
here.

00:22:20.510 --> 00:22:23.440
Sorry, that's not
legible, is it.

00:22:23.440 --> 00:22:29.790
That's a cosine. r = r
= 1/(1 + 2cos(theta)).

00:22:35.310 --> 00:22:37.470
Now, the first
thing I want to do

00:22:37.470 --> 00:22:44.400
is just take our time a little
bit and plot a few points.

00:22:44.400 --> 00:22:48.260
So here's the values of theta
and here are the values of r,

00:22:48.260 --> 00:22:49.770
and we'll see what happens.

00:22:49.770 --> 00:22:52.450
And we'll try to figure
out what it's doing.

00:22:52.450 --> 00:22:56.440
When theta = 0, cosine is 1.

00:22:56.440 --> 00:23:01.230
So r = 1/3.

00:23:01.230 --> 00:23:07.950
The denominator is
1 + 2, so it's 1/3.

00:23:07.950 --> 00:23:10.060
If theta-- I'm going
to make it easy,

00:23:10.060 --> 00:23:11.540
we're not going to do so many.

00:23:11.540 --> 00:23:16.120
I'm going to do pi/2, that's
an easy value of the cosine.

00:23:16.120 --> 00:23:18.200
That's cos(pi/2) = 0.

00:23:18.200 --> 00:23:23.070
So that value of r = 1.

00:23:23.070 --> 00:23:30.060
And now I'm going to
back up and do -pi/2.

00:23:30.060 --> 00:23:33.390
-pi/2, again, cosine is 0.

00:23:33.390 --> 00:23:39.500
And r = 1.

00:23:39.500 --> 00:23:42.930
So now I'd like to just plot
those points anyway, and see

00:23:42.930 --> 00:23:47.410
what's going on with
this expression here.

00:23:47.410 --> 00:23:49.910
The first one is a
rectangular-- I'm

00:23:49.910 --> 00:23:53.450
going to write the
rectangular coordinates here,

00:23:53.450 --> 00:23:56.010
not the polar coordinates.

00:23:56.010 --> 00:24:01.670
The rectangular coordinates here
are 1/3 out at the horizontal,

00:24:01.670 --> 00:24:04.220
so it's (1/3, 0).

00:24:04.220 --> 00:24:07.300
The polar coordinates
is (1/3, 0),

00:24:07.300 --> 00:24:10.890
but the rectangular
coordinate is also that.

00:24:10.890 --> 00:24:14.610
And over here, at pi/2,
the distance is 1.

00:24:14.610 --> 00:24:18.940
So this is the point (0,
1) in x-y coordinates.

00:24:18.940 --> 00:24:26.940
And then down here at,
-pi/2, it's (0, -1).

00:24:26.940 --> 00:24:28.817
Let me just emphasize.

00:24:28.817 --> 00:24:30.650
You should be able to
think of this visually

00:24:30.650 --> 00:24:33.590
if you can crank your
arm around and think it.

00:24:33.590 --> 00:24:37.630
Or if you're right-handed
you'll bend that way, no.

00:24:37.630 --> 00:24:38.270
Anyway.

00:24:38.270 --> 00:24:40.250
Or you'll have to
use-- But this also

00:24:40.250 --> 00:24:48.890
works using this formulas x =
r cos(theta), y = r sin(theta).

00:24:48.890 --> 00:24:53.010
Notice that in this case, r
was 1 but the cosine was 0.

00:24:53.010 --> 00:24:57.100
So you plug in theta = -pi/2.

00:24:57.100 --> 00:24:58.990
And r = 1.

00:24:58.990 --> 00:25:01.520
And lo and behold,
you get 0 here.

00:25:01.520 --> 00:25:03.640
And here you get
-1 here you get 1.

00:25:03.640 --> 00:25:05.830
So this is -1.

00:25:05.830 --> 00:25:07.750
So this is an example.

00:25:07.750 --> 00:25:11.960
I did it purely visually
or sort of organically.

00:25:11.960 --> 00:25:16.190
But you can also do it by
plugging in the numbers.

00:25:16.190 --> 00:25:21.130
Now in between, the
denominator is positive.

00:25:21.130 --> 00:25:22.810
And it's something in between.

00:25:22.810 --> 00:25:26.600
It's going to sweep around
something like this.

00:25:26.600 --> 00:25:29.200
That's what happens in between.

00:25:29.200 --> 00:25:32.090
As theta increases
from -pi/2 to pi/2.

00:25:34.600 --> 00:25:36.430
And now something
interesting happens

00:25:36.430 --> 00:25:38.260
with this particular
function, which

00:25:38.260 --> 00:25:40.450
is that we notice that
the denominator is

00:25:40.450 --> 00:25:43.280
0 at a certain place.

00:25:43.280 --> 00:25:48.610
Namely, if I solve
2 cos(theta) = -1,

00:25:48.610 --> 00:25:51.290
then the denominator
is going to be 0 there.

00:25:51.290 --> 00:25:57.470
That's cos(theta) =
-1/2, so theta is equal

00:25:57.470 --> 00:26:01.240
to, it turns out,
plus or minus 2pi/3.

00:26:01.240 --> 00:26:03.800
Those are the values here.

00:26:03.800 --> 00:26:09.050
So when we're out here
somewhere, in these directions,

00:26:09.050 --> 00:26:10.120
there's nothing.

00:26:10.120 --> 00:26:14.220
It's going infinitely far out.

00:26:14.220 --> 00:26:20.120
Those ways.

00:26:20.120 --> 00:26:22.500
OK that's about as
much as we'll be

00:26:22.500 --> 00:26:24.740
able to figure out
of this diagram

00:26:24.740 --> 00:26:26.980
without doing some
analytic work.

00:26:26.980 --> 00:26:31.970
And that's the other little
piece that I want to explain.

00:26:31.970 --> 00:26:34.700
Namely, going backwards
from polar coordinates

00:26:34.700 --> 00:26:36.740
to rectangular coordinates.

00:26:36.740 --> 00:26:38.930
Which is one thing
that we haven't done.

00:26:38.930 --> 00:26:40.870
So let's do that.

00:26:40.870 --> 00:26:48.680
So what is the
rectangular equation?

00:26:48.680 --> 00:26:59.690
That means the (x, y)
equation for this r = 1 /

00:26:59.690 --> 00:27:00.440
(1 + 2cos(theta)).

00:27:03.400 --> 00:27:06.320
And let's see what it is.

00:27:06.320 --> 00:27:08.850
Well, first I'm going to
clear the denominator here.

00:27:08.850 --> 00:27:15.200
This is r + 2r cos(theta) = 1.

00:27:15.200 --> 00:27:23.020
And now I'm going to rewrite
it as r = 1 - 2r cos(theta).

00:27:23.020 --> 00:27:25.280
And the reason for that
is that in a minute

00:27:25.280 --> 00:27:27.300
I'll explain to you why.

00:27:27.300 --> 00:27:29.620
This is 1 - 2x.

00:27:29.620 --> 00:27:32.810
And this guy, I'm
going to square now.

00:27:32.810 --> 00:27:38.010
I'm going to make
this r^2 = (1 - 2x)^2.

00:27:38.010 --> 00:27:41.030
And now, with an r^2, I
can plug in x^2 + y^2.

00:27:49.680 --> 00:27:53.381
So this is a
standard thing to do.

00:27:53.381 --> 00:27:54.880
And it's basically
what you're going

00:27:54.880 --> 00:27:57.870
to do any time you're faced
with an equation like this.

00:27:57.870 --> 00:27:59.470
Is try to work it out.

00:27:59.470 --> 00:28:04.180
And, in these situations
where you have 1 /

00:28:04.180 --> 00:28:06.990
(a + b cos(theta)),
or sin(theta),

00:28:06.990 --> 00:28:12.400
you'll always come out with some
quadratic expression like this.

00:28:12.400 --> 00:28:14.590
Now, I'm going to combine terms.

00:28:14.590 --> 00:28:19.921
So here I have -3x^2 + y^2, and
put everything on the the left

00:28:19.921 --> 00:28:20.420
side.

00:28:20.420 --> 00:28:24.390
So that's this.

00:28:24.390 --> 00:28:28.140
And we recognize, well
you're supposed to recognize,

00:28:28.140 --> 00:28:36.580
that this is what's
known as a hyperbola.

00:28:36.580 --> 00:28:39.620
If the signs are the
same, it's an ellipse.

00:28:39.620 --> 00:28:42.390
If the the signs are
opposite it's a hyperbola.

00:28:42.390 --> 00:28:44.920
And in between, if one of the
coefficients on the quadratic

00:28:44.920 --> 00:28:49.210
is 0, it's a parabola.

00:28:49.210 --> 00:28:54.844
So now we see that the picture
that we drew there is actually,

00:28:54.844 --> 00:28:56.510
turns out it's going
to have asymptotes,

00:28:56.510 --> 00:29:01.920
it's going to be a hyperbola.

00:29:01.920 --> 00:29:06.650
So now, let me ask you the last
little mind-bending question

00:29:06.650 --> 00:29:08.020
that I want to ask.

00:29:08.020 --> 00:29:12.060
Which is, what happens-- So
now I'm using my right arm,

00:29:12.060 --> 00:29:12.770
I guess.

00:29:12.770 --> 00:29:14.190
But my elbow's at
the origin here.

00:29:14.190 --> 00:29:18.290
What happens if I pass
outside, to the range where

00:29:18.290 --> 00:29:20.280
this denominator is negative.

00:29:20.280 --> 00:29:24.210
It crossed 0 and it
went to negative.

00:29:24.210 --> 00:29:28.050
It's sweeping out
something over here.

00:29:28.050 --> 00:29:32.420
Is it sweeping out
the same curve?

00:29:32.420 --> 00:29:34.320
Anybody have any
idea what it's doing?

00:29:34.320 --> 00:29:34.820
Yeah.

00:29:34.820 --> 00:29:37.460
STUDENT: [INAUDIBLE]

00:29:37.460 --> 00:29:38.590
PROFESSOR: Yeah, exactly.

00:29:38.590 --> 00:29:39.090
Good answer.

00:29:39.090 --> 00:29:44.220
It's the other branch
of the hyperbola.

00:29:44.220 --> 00:29:46.390
So what's actually
happening is in disguise,

00:29:46.390 --> 00:29:48.015
there's another branch
of the hyperbola

00:29:48.015 --> 00:29:50.700
which is being swept up by
the other piece of this thing.

00:29:50.700 --> 00:29:54.220
Now, that is consistent with
these algebraic equations.

00:29:54.220 --> 00:29:56.500
The algebraic equation
that I got here

00:29:56.500 --> 00:30:00.900
doesn't say which branch
of the hyperbola I've got.

00:30:00.900 --> 00:30:05.930
It's actually got two branches.

00:30:05.930 --> 00:30:07.960
And the curve really
was, in disguise,

00:30:07.960 --> 00:30:12.130
capturing both of them.

00:30:12.130 --> 00:30:14.250
I want to make
the connection now

00:30:14.250 --> 00:30:18.770
with the basic
formula for area here.

00:30:18.770 --> 00:30:22.920
Because this is a really
beautiful connection.

00:30:22.920 --> 00:30:26.520
And I want to make that
connection in connection

00:30:26.520 --> 00:30:29.110
also with this example.

00:30:29.110 --> 00:30:32.750
The hyperbolas, as
you probably know,

00:30:32.750 --> 00:30:37.770
are the trajectories of comets.

00:30:37.770 --> 00:30:42.290
And ellipses, which is what you
would get if maybe you put 1/2

00:30:42.290 --> 00:30:44.570
here instead of a 2,
would be the trajectories

00:30:44.570 --> 00:30:47.450
of planets or asteroids.

00:30:47.450 --> 00:30:51.770
But there's actually
something much more important,

00:30:51.770 --> 00:30:53.410
physically that goes on.

00:30:53.410 --> 00:30:56.820
That's special about this
particular representation

00:30:56.820 --> 00:30:58.900
of the hyperbola.

00:30:58.900 --> 00:31:01.610
And what happens when you
get the ellipses as well.

00:31:01.610 --> 00:31:06.860
Which is that in
this case, r = 0

00:31:06.860 --> 00:31:17.520
is the focus of the hyperbola.

00:31:17.520 --> 00:31:21.390
And what that means
is that it's actually

00:31:21.390 --> 00:31:28.910
the place where the sun is.

00:31:28.910 --> 00:31:31.220
So this is the right
representation,

00:31:31.220 --> 00:31:32.670
if you want the
center of gravity

00:31:32.670 --> 00:31:36.720
in the center of your picture.

00:31:36.720 --> 00:31:38.540
And pretty much any other.

00:31:38.540 --> 00:31:40.050
I mean, you can't
tell that at all

00:31:40.050 --> 00:31:43.150
from the algebraic
equations here.

00:31:43.150 --> 00:31:46.340
So this hyperbola is
going to be the trajectory

00:31:46.340 --> 00:31:51.900
of some comet going by here.

00:31:51.900 --> 00:31:57.240
And this formula
here is actually

00:31:57.240 --> 00:32:05.040
a rather central
formula in astronomy.

00:32:05.040 --> 00:32:17.100
Namely, there's something
called Kepler's Law.

00:32:17.100 --> 00:32:23.820
Which says that the rate
of change of area which

00:32:23.820 --> 00:32:28.674
is swept out is constant.

00:32:28.674 --> 00:32:31.090
The rate of change of area
relative to the center of mass,

00:32:31.090 --> 00:32:33.640
relative to the sun.

00:32:33.640 --> 00:32:37.000
So in equal areas,
this is amount of area.

00:32:37.000 --> 00:32:43.570
So this tells you now that when
a comet goes around the sun

00:32:43.570 --> 00:32:46.500
like this, its speed varies.

00:32:46.500 --> 00:32:49.180
And it's speed varies according
to a very specific rule.

00:32:49.180 --> 00:32:52.220
Namely, this one here.

00:32:52.220 --> 00:32:54.960
And this rule was
observed by Kepler.

00:32:54.960 --> 00:32:57.910
But if you have this
connection here,

00:32:57.910 --> 00:32:59.400
we also have something else.

00:32:59.400 --> 00:33:08.390
We also know that dA/dt
= 1/2 r^2 d theta / dt.

00:33:08.390 --> 00:33:13.387
So that's this formula here,
formally dividing by t.

00:33:13.387 --> 00:33:14.970
That's the rate of
change with respect

00:33:14.970 --> 00:33:18.040
to this time parameter, which
is the honest to goodness time.

00:33:18.040 --> 00:33:20.820
Real physical time.

00:33:20.820 --> 00:33:34.920
And that means, this
quantity here is constant.

00:33:34.920 --> 00:33:37.810
And this is one of
the key insights

00:33:37.810 --> 00:33:41.540
that physicists had,
long after Kepler

00:33:41.540 --> 00:33:43.760
made his physical
observations, they realized

00:33:43.760 --> 00:33:48.280
that he had managed to get the
best physics experiment of all,

00:33:48.280 --> 00:33:50.790
because it's a
frictionless setup.

00:33:50.790 --> 00:33:53.320
Outer space, there's no air.

00:33:53.320 --> 00:33:54.710
Nothing is going on.

00:33:54.710 --> 00:33:59.260
This is what's known
nowadays as conservation

00:33:59.260 --> 00:34:10.030
of angular momentum.

00:34:10.030 --> 00:34:13.720
This is the expression
for angular momentum.

00:34:13.720 --> 00:34:16.810
And what Kepler was
observing, it turns out,

00:34:16.810 --> 00:34:18.924
is what we see all
the time in real life.

00:34:18.924 --> 00:34:21.090
Which is when you start
something spinning around it

00:34:21.090 --> 00:34:23.570
continues to spin at
roughly the same rate.

00:34:23.570 --> 00:34:26.517
Or, if you're an
ice skater and you

00:34:26.517 --> 00:34:28.600
get yourself scrunched
together a little bit more,

00:34:28.600 --> 00:34:30.330
you can spin faster.

00:34:30.330 --> 00:34:32.850
And there's an exact
quantitative rule

00:34:32.850 --> 00:34:33.740
that does that.

00:34:33.740 --> 00:34:38.240
And it's exactly this
polar formula here.

00:34:38.240 --> 00:34:39.730
So that's a neat thing.

00:34:39.730 --> 00:34:43.280
And we will do a little
exercise on this rate of change

00:34:43.280 --> 00:34:45.780
after the exam.

00:34:45.780 --> 00:34:52.030
So that's it for generalities
and a little pep talk

00:34:52.030 --> 00:34:54.940
on what's coming
up to you when you

00:34:54.940 --> 00:34:57.820
learn a little more physics.

00:34:57.820 --> 00:35:04.660
Right now we need to
talk about the exam.

00:35:04.660 --> 00:35:14.010
So first of all, let me tell
you what the topics are.

00:35:14.010 --> 00:35:17.500
They're the same as
last year's test.

00:35:17.500 --> 00:35:19.990
Which you can take a look at.

00:35:19.990 --> 00:35:24.060
And let's see.

00:35:24.060 --> 00:35:25.920
So what did we do?

00:35:25.920 --> 00:35:29.960
One of the main
topics of this unit

00:35:29.960 --> 00:35:35.660
were techniques of integration.

00:35:35.660 --> 00:35:40.390
And there are three,
which we will test.

00:35:40.390 --> 00:35:46.660
One is trig substitution.

00:35:46.660 --> 00:35:53.950
One is integration by parts.

00:35:53.950 --> 00:36:02.000
And one is partial fractions.

00:36:02.000 --> 00:36:06.020
So that's more than half
of the exam, right there.

00:36:06.020 --> 00:36:16.210
The other half of the
exam is parametric curves.

00:36:16.210 --> 00:36:18.420
Arc length.

00:36:18.420 --> 00:36:20.520
These are all interrelated.

00:36:20.520 --> 00:36:33.140
And area of surfaces
of revolution.

00:36:33.140 --> 00:36:35.150
Those are the only kind
that we can handle.

00:36:35.150 --> 00:36:38.730
Just as we did with volume
of surfaces of revolution.

00:36:38.730 --> 00:36:47.590
And then there's a final topic,
which is polar coordinates.

00:36:47.590 --> 00:36:57.260
And area in polar
coordinates, including area.

00:36:57.260 --> 00:36:57.810
That's it.

00:36:57.810 --> 00:37:00.270
That's what's on the test,
there are six problems.

00:37:00.270 --> 00:37:02.200
They're very similar.

00:37:02.200 --> 00:37:04.040
Well, they're not
actually that similar.

00:37:04.040 --> 00:37:07.610
But they're somewhat
similar to last year's.

00:37:07.610 --> 00:37:13.860
I'd say the test is similar.

00:37:13.860 --> 00:37:16.960
Maybe a tiny bit more
difficult. We'll see.

00:37:16.960 --> 00:37:18.241
We'll see.

00:37:18.241 --> 00:37:18.740
Yeah.

00:37:18.740 --> 00:37:23.420
STUDENT: [INAUDIBLE]

00:37:23.420 --> 00:37:26.410
PROFESSOR: The question
was, we didn't do arc length

00:37:26.410 --> 00:37:28.310
in polar coordinates, did we?

00:37:28.310 --> 00:37:30.954
And the answer is
no, we did not.

00:37:30.954 --> 00:37:32.870
We did not do arc length
in polar coordinates.

00:37:32.870 --> 00:37:35.840
When I give you an exercise,
I'm going to ask you about,

00:37:35.840 --> 00:37:37.850
if you know the speed
of a comet here, what's

00:37:37.850 --> 00:37:39.540
the speed of the comet there.

00:37:39.540 --> 00:37:41.710
And we'll have to know
about arc length for that.

00:37:41.710 --> 00:37:48.534
But we're not doing
it on this exam.

00:37:48.534 --> 00:37:49.200
Other questions.

00:37:49.200 --> 00:37:59.880
STUDENT: [INAUDIBLE]

00:37:59.880 --> 00:38:05.650
PROFESSOR: The question is,
will I expect you to know r

00:38:05.650 --> 00:38:10.220
equals--- so let's see if I
can formulate this question.

00:38:10.220 --> 00:38:13.430
It's related to this
four-leaf rose here.

00:38:13.430 --> 00:38:16.020
So the question is, suppose
I gave you something

00:38:16.020 --> 00:38:21.220
that looked like this.

00:38:21.220 --> 00:38:24.170
Would I expect you to be
able to know what it is.

00:38:24.170 --> 00:38:27.300
I think the answer, the
fair answer to give you,

00:38:27.300 --> 00:38:33.440
is if it's this complicated,
I only have two possibilities.

00:38:33.440 --> 00:38:37.190
I can give you a long
time to sketch this out.

00:38:37.190 --> 00:38:38.510
And think about what it does.

00:38:38.510 --> 00:38:40.580
Or I can tell you
that it happens

00:38:40.580 --> 00:38:42.600
to be a three-leaf rose.

00:38:42.600 --> 00:38:47.530
And then you have some
clue as to what it's doing.

00:38:47.530 --> 00:38:49.620
It doesn't have six.

00:38:49.620 --> 00:38:53.770
Because of some weird thing,
having to do with repetitions.

00:38:53.770 --> 00:38:57.790
But the odds and evens
work differently.

00:38:57.790 --> 00:39:02.540
So, in fact I would
have to tell you

00:39:02.540 --> 00:39:04.100
what the picture
looks like, if it's

00:39:04.100 --> 00:39:07.070
going to be this complicated.

00:39:07.070 --> 00:39:10.860
Similarly, so this is an
important point to make,

00:39:10.860 --> 00:39:13.500
when we come to techniques
of integration, any integral

00:39:13.500 --> 00:39:17.982
that you have, I'm not going to
tell you which of these three

00:39:17.982 --> 00:39:19.440
techniques to use
on the ones which

00:39:19.440 --> 00:39:20.810
are straightforward integrals.

00:39:20.810 --> 00:39:22.050
But if it's an
integral that I think

00:39:22.050 --> 00:39:23.760
you're going to get
stuck on, either I'm

00:39:23.760 --> 00:39:26.330
going to give you a hint,
tell you how to do it.

00:39:26.330 --> 00:39:29.280
Or I'm going to tell
you, don't do it.

00:39:29.280 --> 00:39:31.680
If I tell you don't do
it, don't try to do it.

00:39:31.680 --> 00:39:33.050
It may be impossible.

00:39:33.050 --> 00:39:35.950
And even if it's possible,
it's going to be very long.

00:39:35.950 --> 00:39:38.010
Like an hour.

00:39:38.010 --> 00:39:42.750
So don't do it
unless I tell you to.

00:39:42.750 --> 00:39:44.990
On the other hand,
all of these setups

00:39:44.990 --> 00:39:47.720
in this second
half of this unit,

00:39:47.720 --> 00:39:50.290
they involve somehow
setting something up.

00:39:50.290 --> 00:39:55.230
And they're basically
three issues.

00:39:55.230 --> 00:39:57.310
One is what the integrand is.

00:39:57.310 --> 00:40:02.440
One is what the lower limit
is, what is the upper limit.

00:40:02.440 --> 00:40:05.510
They're just three
things, three inputs,

00:40:05.510 --> 00:40:06.710
to setting up an integral.

00:40:06.710 --> 00:40:10.590
All integrals, this is going to
be the setup for all of them.

00:40:10.590 --> 00:40:13.260
And then the second
step is evaluating.

00:40:13.260 --> 00:40:17.070
Which really is what we
did in the first half here.

00:40:17.070 --> 00:40:19.856
And, unfortunately, we don't
have infinitely many techniques

00:40:19.856 --> 00:40:21.230
and indeed there's
some integrals

00:40:21.230 --> 00:40:23.790
that can't be evaluated
and some that are too long.

00:40:23.790 --> 00:40:25.575
So we'll just try
to avoid those.

00:40:25.575 --> 00:40:32.400
I'm not trying to give you
ones which are hopelessly long.

00:40:32.400 --> 00:40:33.680
Alright, other questions.

00:40:33.680 --> 00:40:34.180
Yes.

00:40:34.180 --> 00:40:47.620
STUDENT: [INAUDIBLE]

00:40:47.620 --> 00:40:50.610
PROFESSOR: The question is, will
the percentages be the same.

00:40:50.610 --> 00:40:54.834
And the answer is, no.

00:40:54.834 --> 00:40:55.750
I'll tell you exactly.

00:40:55.750 --> 00:40:57.200
This is 55 points.

00:40:57.200 --> 00:40:59.340
Unless I change
the point values.

00:40:59.340 --> 00:41:07.780
This is 55, and this is 45.

00:41:07.780 --> 00:41:11.020
That's what it came out to be.

00:41:11.020 --> 00:41:13.554
You are going to want to
know about all of the things

00:41:13.554 --> 00:41:14.720
that I've written down here.

00:41:14.720 --> 00:41:16.386
You're definitely
going to want to know,

00:41:16.386 --> 00:41:18.650
for example, surfaces
of revolution.

00:41:18.650 --> 00:41:20.710
How to set those up.

00:41:20.710 --> 00:41:24.341
Yes. there was another
question I saw.

00:41:24.341 --> 00:41:24.840
Yes.

00:41:24.840 --> 00:41:50.770
STUDENT: [INAUDIBLE]

00:41:50.770 --> 00:41:55.010
PROFESSOR: So if you have a
partial fraction with something

00:41:55.010 --> 00:42:04.110
like (x+2)^2 and maybe
an x and maybe an x+1,

00:42:04.110 --> 00:42:15.460
and you're interested in what
happens with this denominator

00:42:15.460 --> 00:42:16.120
here?

00:42:16.120 --> 00:42:23.600
So what's going to
happen is, you're

00:42:23.600 --> 00:42:32.180
going to need a coefficient
for each degree of this.

00:42:32.180 --> 00:42:40.400
So altogether, the setup
is going to be this.

00:42:40.400 --> 00:42:43.080
Plus one for x.

00:42:43.080 --> 00:42:46.440
And one for x+1.

00:42:46.440 --> 00:42:50.770
This is the setup.

00:42:50.770 --> 00:42:51.850
So you need--

00:42:51.850 --> 00:42:58.890
STUDENT: [INAUDIBLE]

00:42:58.890 --> 00:43:05.310
PROFESSOR: So if I change this
to being a 3 here, then I need,

00:43:05.310 --> 00:43:10.490
I guess I'll have to
call it E, (x+2)^3.

00:43:10.490 --> 00:43:11.460
I need that.

00:43:11.460 --> 00:43:13.000
Now, it gets harder and harder.

00:43:13.000 --> 00:43:14.500
The more repeated
roots there are,

00:43:14.500 --> 00:43:17.440
the more repeated factors
there are, the harder it is.

00:43:17.440 --> 00:43:22.080
Because the ones you can pick
off by the cover-up method are,

00:43:22.080 --> 00:43:24.020
is just the top one here.

00:43:24.020 --> 00:43:25.240
And these two.

00:43:25.240 --> 00:43:28.340
So C, D, and E you can get.

00:43:28.340 --> 00:43:30.170
But B and A you're
going to have to do

00:43:30.170 --> 00:43:33.740
by either plugging in or some
other, more elaborate, algebra.

00:43:33.740 --> 00:43:36.650
So the more of these lower
terms there are, the worse off

00:43:36.650 --> 00:43:37.180
you are.

00:43:37.180 --> 00:43:48.170
STUDENT: [INAUDIBLE]

00:43:48.170 --> 00:43:57.990
PROFESSOR: The question is,
does this x^3 + 21 affect this

00:43:57.990 --> 00:43:59.220
setup.

00:43:59.220 --> 00:44:03.290
And the answer is almost no.

00:44:03.290 --> 00:44:04.790
That is, not at all.

00:44:04.790 --> 00:44:06.510
It's the same setup exactly.

00:44:06.510 --> 00:44:08.260
But, there's one thing.

00:44:08.260 --> 00:44:11.510
If the degree gets
too big, then you've

00:44:11.510 --> 00:44:18.270
got to use long division
first to knock it down.

00:44:18.270 --> 00:44:20.910
I'll give you an example
of this type of practice.

00:44:20.910 --> 00:44:22.220
Unless there are more question.

00:44:22.220 --> 00:44:22.440
Yes.

00:44:22.440 --> 00:44:23.410
STUDENT: [INAUDIBLE]

00:44:23.410 --> 00:44:26.944
PROFESSOR: Are you
going to have to know

00:44:26.944 --> 00:44:29.580
how to do reduction formulas?

00:44:29.580 --> 00:44:33.160
Anything that's a little out of
the ordinary like a reduction

00:44:33.160 --> 00:44:35.581
formula, I will have
to coach you to do.

00:44:35.581 --> 00:44:37.080
So, in other words,
what you'll have

00:44:37.080 --> 00:44:40.280
to be able to do in that
situation is follow directions.

00:44:40.280 --> 00:44:42.230
If I tell you OK,
you're faced with this,

00:44:42.230 --> 00:44:44.770
then do an integration by parts.

00:44:44.770 --> 00:44:48.530
And do that, then get
the reduction formula.

00:44:48.530 --> 00:44:49.410
STUDENT: [INAUDIBLE]

00:44:49.410 --> 00:44:50.810
PROFESSOR: Yeah.

00:44:50.810 --> 00:44:56.460
OK, so the question had do with
the partial fractions method.

00:44:56.460 --> 00:45:03.250
And what happens if
you have a quadratic.

00:45:03.250 --> 00:45:11.970
So, for instance,
if it were this,

00:45:11.970 --> 00:45:13.200
this one's too disgusting.

00:45:13.200 --> 00:45:17.220
I'm going to just do
it with two of them.

00:45:17.220 --> 00:45:20.990
So the parts with x
and x+1 are the same.

00:45:20.990 --> 00:45:23.760
But now you have
linear factors here.

00:45:23.760 --> 00:45:27.510
(Ax + B) / (x^2 + 2).

00:45:27.510 --> 00:45:33.310
And A-- maybe I'll call them
1, and A_2 x + (A_2 x + B_2) /

00:45:33.310 --> 00:45:43.790
(x^2 + 2)^2 + C / x + D / (x+1).

00:45:43.790 --> 00:45:47.060
This is the way it works.

00:45:47.060 --> 00:45:52.100
OK, I'm going to give you
one more quick example

00:45:52.100 --> 00:45:59.720
of an integration technique
just to liven things up.

00:45:59.720 --> 00:46:02.700
Let's see.

00:46:02.700 --> 00:46:06.330
So here's a somewhat
tricky example.

00:46:06.330 --> 00:46:08.080
This is just a little
trickier than I

00:46:08.080 --> 00:46:09.640
would give you on a test.

00:46:09.640 --> 00:46:14.040
But it's the same principle, and
I may do this on a final exam.

00:46:14.040 --> 00:46:20.300
So suppose you're faced
with this integral.

00:46:20.300 --> 00:46:24.250
What are you going to do?

00:46:24.250 --> 00:46:26.140
Integration by parts, great.

00:46:26.140 --> 00:46:29.360
That's right, that's
because this guy is

00:46:29.360 --> 00:46:33.820
begging to be differentiated,
to be made simpler.

00:46:33.820 --> 00:46:37.290
So that means that I
want this one to be u,

00:46:37.290 --> 00:46:40.190
and I want this one to be v'.

00:46:40.190 --> 00:46:42.120
And I want to use
integration by parts.

00:46:42.120 --> 00:46:48.400
And then u ' = 1 /
(1+x^2), and v = x^2 / 2.

00:46:51.360 --> 00:46:58.200
So the answer is now, x^2
/ x^2/2 tan^(-1) x minus

00:46:58.200 --> 00:47:00.160
the integral of this guy.

00:47:00.160 --> 00:47:02.530
Which is going to be x^2 / 2.

00:47:02.530 --> 00:47:04.040
And then I have
1 / (1 + x^2) dx.

00:47:09.200 --> 00:47:13.300
Now, you are not
done at this point.

00:47:13.300 --> 00:47:16.780
You're still in
slightly hot water.

00:47:16.780 --> 00:47:19.010
You're in tepid water, anyway.

00:47:19.010 --> 00:47:21.470
So what is it that
you have to do here?

00:47:21.470 --> 00:47:24.280
You're faced with
this integral, which

00:47:24.280 --> 00:47:30.777
I'll put on the next board.

00:47:30.777 --> 00:47:32.360
It's a lot simpler
than the other one,

00:47:32.360 --> 00:47:35.970
but as I say you're not
quite out of the woods.

00:47:35.970 --> 00:47:38.570
You're faced with
the integral of 1/2--

00:47:38.570 --> 00:47:42.410
-1/2 x^2 / (1 + x^2) dx. dx.

00:47:42.410 --> 00:47:50.350
STUDENT: [INAUDIBLE] PROFESSOR:
Trig substitution actually,

00:47:50.350 --> 00:47:52.914
interestingly, will work.

00:47:52.914 --> 00:47:54.580
But that wasn't what
I wanted you to do.

00:47:54.580 --> 00:47:56.200
I wanted you to, yeah, go ahead.

00:47:56.200 --> 00:47:57.430
STUDENT: [INAUDIBLE]

00:47:57.430 --> 00:47:59.450
PROFESSOR: Add and subtract
1 to the numerator.

00:47:59.450 --> 00:48:02.330
So now, that's the
correct answer.

00:48:02.330 --> 00:48:04.850
This is the case where the
numerator and the denominator

00:48:04.850 --> 00:48:06.200
are tied.

00:48:06.200 --> 00:48:08.270
And so you have to
use long division.

00:48:08.270 --> 00:48:10.690
But a shortcut is
just to observe

00:48:10.690 --> 00:48:13.300
that the result of
long division is

00:48:13.300 --> 00:48:20.920
the same thing as doing this.

00:48:20.920 --> 00:48:26.360
And then noticing that
this is 1 - 1 / (1 + x^2).

00:48:26.360 --> 00:48:30.040
So this is the same as long
division, in this case.

00:48:30.040 --> 00:48:34.970
Because when you divide in, it
goes in with a quotient of 1.

00:48:34.970 --> 00:48:43.509
And so this guy turns out to
be -1/2 the integral of 1 -

00:48:43.509 --> 00:48:44.050
1/(1+x^2) dx.

00:48:47.100 --> 00:48:55.000
Which is 1/2 x -
1/2 tan^(-1) x + c.

00:48:55.000 --> 00:49:02.230
So this is one extra step that
you may be faced with someday

00:49:02.230 --> 00:49:02.900
in your life.

00:49:02.900 --> 00:49:05.620
And just keep that in mind.