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PROFESSOR: OK.
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Now, today we get to move
on from integral formulas
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and methods of integration
back to some geometry.
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And this is more or
less going to lead
00:00:36.870 --> 00:00:39.270
into the kinds of
tools you'll be using
00:00:39.270 --> 00:00:43.120
in multivariable calculus.
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The first thing that
we're going to do today
00:00:45.420 --> 00:00:58.220
is discuss arc length.
00:00:58.220 --> 00:01:02.940
Like all of the cumulative
sums that we've worked on,
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this one has a storyline and
a picture associated to it,
00:01:06.710 --> 00:01:09.360
which involves
dividing things up.
00:01:09.360 --> 00:01:11.690
If you have a
roadway, if you like,
00:01:11.690 --> 00:01:18.290
and you have mileage markers
along the road, like this,
00:01:18.290 --> 00:01:22.490
all the way up
to, say, s_n here,
00:01:22.490 --> 00:01:28.740
then the length along the road
is described by this parameter,
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s.
00:01:30.560 --> 00:01:32.930
Which is arc length.
00:01:32.930 --> 00:01:37.680
And if we look at a graph
of this sort of thing,
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if this is the last point b,
and this is the first point a,
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then you can think in terms of
having points above x_1, x_2,
00:01:47.120 --> 00:01:50.460
x_3, etc.
00:01:50.460 --> 00:01:53.940
The same as we did
with Riemann sums.
00:01:53.940 --> 00:01:56.050
And then the way
that we're going
00:01:56.050 --> 00:02:07.540
to approximate this is by taking
the straight lines between each
00:02:07.540 --> 00:02:09.470
of these points.
00:02:09.470 --> 00:02:11.360
As things get
smaller and smaller,
00:02:11.360 --> 00:02:15.210
the straight line is going to
be fairly close to the curve.
00:02:15.210 --> 00:02:16.980
And that's the main idea.
00:02:16.980 --> 00:02:20.050
So let me just depict
one little chunk of this.
00:02:20.050 --> 00:02:20.900
Which is like this.
00:02:20.900 --> 00:02:24.530
One straight line, and here's
the curved surface there.
00:02:24.530 --> 00:02:26.830
And the distance along
the curved surface is what
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I'm calling delta s, the
change in the length between--
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so this would be s_2 - s_1
if I depicted that one.
00:02:34.920 --> 00:02:42.490
So this would be delta s
is, say s. s_i - s_(i-1),
00:02:42.490 --> 00:02:45.240
some increment there.
00:02:45.240 --> 00:02:49.670
And then I can figure
out what the length
00:02:49.670 --> 00:02:51.750
of the orange segment is.
00:02:51.750 --> 00:02:54.630
Because the horizontal
distance is delta x.
00:02:54.630 --> 00:02:57.290
And the vertical
distance is delta y.
00:02:57.290 --> 00:03:02.950
And so the formula is that the
hypotenuse is delta (delta x)^2
00:03:02.950 --> 00:03:06.470
+ (delta y)^2.
00:03:06.470 --> 00:03:09.420
Square root.
00:03:09.420 --> 00:03:11.870
And delta s is
approximately that.
00:03:11.870 --> 00:03:15.871
So what we're saying is that
(delta s)^2 is approximately
00:03:15.871 --> 00:03:16.370
this.
00:03:16.370 --> 00:03:21.870
So this is the hypotenuse.
00:03:21.870 --> 00:03:23.500
Squared.
00:03:23.500 --> 00:03:29.520
And it's very close to
the length of the curve.
00:03:29.520 --> 00:03:34.940
And the whole idea of calculus
is that in the infinitesimal,
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this is exactly correct.
00:03:44.150 --> 00:03:48.010
So that's what's going
to happen in the limit.
00:03:48.010 --> 00:03:52.000
And that is the basis for
calculating arc length.
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I'm going to rewrite that
formula on the next board.
00:03:56.060 --> 00:03:59.460
But I'm going to write it in
the more customary fashion.
00:03:59.460 --> 00:04:02.200
We've done this before,
a certain amount.
00:04:02.200 --> 00:04:03.970
But I just want to
emphasize it here
00:04:03.970 --> 00:04:09.390
because this handwriting
is a little bit peculiar.
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This ds is really all one thing.
00:04:11.710 --> 00:04:15.690
What I really mean is to put
the parenthesis around it.
00:04:15.690 --> 00:04:16.660
It's one thing.
00:04:16.660 --> 00:04:19.070
It's not d times s, it's ds.
00:04:19.070 --> 00:04:20.020
It's one thing.
00:04:20.020 --> 00:04:20.940
And we square it.
00:04:20.940 --> 00:04:22.750
But for whatever
reason people have
00:04:22.750 --> 00:04:26.112
gotten into the habit of
omitting the parentheses.
00:04:26.112 --> 00:04:28.070
So you're just going to
have to live with that.
00:04:28.070 --> 00:04:31.050
And realize that this is not d
of s^2 or anything like that.
00:04:31.050 --> 00:04:32.755
And similarly, this
is a single number,
00:04:32.755 --> 00:04:34.130
and this is a single number.
00:04:34.130 --> 00:04:36.060
Infinitesimal.
00:04:36.060 --> 00:04:39.400
So that's just the way
that this idea here
00:04:39.400 --> 00:04:42.194
gets written in our notation.
00:04:42.194 --> 00:04:43.860
And this is the first
time we're dealing
00:04:43.860 --> 00:04:46.070
with squares of infinitesimals.
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So it's just a little different.
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But immediately the first
thing we're going to do
00:04:49.641 --> 00:04:51.750
is take the square root.
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If I take the square root,
that's the square root of dx^2
00:04:56.660 --> 00:04:58.640
+ dy^2.
00:04:58.640 --> 00:05:02.400
And this is the form in which
I always remember this formula.
00:05:02.400 --> 00:05:07.030
Let's put it in some
brightly decorated form.
00:05:07.030 --> 00:05:12.380
But there are about four,
five, six other forms
00:05:12.380 --> 00:05:16.110
that you'll derive from this,
which all mean the same thing.
00:05:16.110 --> 00:05:18.479
So this is, as I say,
the way I remember it.
00:05:18.479 --> 00:05:20.270
But there are other
ways of thinking of it.
00:05:20.270 --> 00:05:23.070
And let's just write
a couple of them down.
00:05:23.070 --> 00:05:27.380
The first one is that you
can factor out the dx.
00:05:27.380 --> 00:05:29.480
So that looks like this.
00:05:29.480 --> 00:05:34.680
1 + (dy / dx)^2.
00:05:34.680 --> 00:05:37.630
And then I factored out the dx.
00:05:37.630 --> 00:05:39.820
So this is a variant.
00:05:39.820 --> 00:05:43.190
And this is the one
which actually we'll
00:05:43.190 --> 00:05:47.490
be using in practice
right now on our examples.
00:05:47.490 --> 00:05:55.600
So the conclusion is
that the arc length,
00:05:55.600 --> 00:06:05.320
which if you like is this
total s_n - s_0, if you like,
00:06:05.320 --> 00:06:09.390
is going to be equal to
the integral from a to b
00:06:09.390 --> 00:06:16.430
of the square root
of 1 + (dy/dx)^2, dx.
00:06:20.490 --> 00:06:27.500
In practice, it's also very
often written informally
00:06:27.500 --> 00:06:29.090
as this.
00:06:29.090 --> 00:06:30.750
The integral ds.
00:06:30.750 --> 00:06:32.830
So the change in
this little variable
00:06:32.830 --> 00:06:40.009
s, and this is what you'll see
notationally in many textbooks.
00:06:40.009 --> 00:06:42.550
So that's one way of writing
it, and of course the second way
00:06:42.550 --> 00:06:45.950
of writing it which is
practically the same thing is
00:06:45.950 --> 00:06:50.040
square root of 1 + f'(x)^2, dx.
00:06:50.040 --> 00:06:52.330
Mixing in a little bit
of Newton's notation.
00:06:52.330 --> 00:06:57.250
And this is with y = f(x).
00:06:57.250 --> 00:07:03.310
So this is the formula
for arc length.
00:07:03.310 --> 00:07:05.770
And as I say, I
remember it this way.
00:07:05.770 --> 00:07:09.060
But you're going to have to
derive various variants of it.
00:07:09.060 --> 00:07:11.010
And you'll have to
use some arithmetic
00:07:11.010 --> 00:07:12.650
to get to various formulas.
00:07:12.650 --> 00:07:15.190
And there will be more later.
00:07:15.190 --> 00:07:16.030
Yeah, question.
00:07:16.030 --> 00:07:20.820
STUDENT: [INAUDIBLE]
00:07:20.820 --> 00:07:27.350
PROFESSOR: OK, the question
is, is f'(x)^2 equal to f''(x).
00:07:27.350 --> 00:07:31.020
And the answer is no.
00:07:31.020 --> 00:07:33.090
And let's just see what it is.
00:07:33.090 --> 00:07:39.490
So, for example, if f(x) = x^2,
which is an example which will
00:07:39.490 --> 00:07:47.910
come up in a few minutes,
then f'(x) = 2x and f'(x)^2 =
00:07:47.910 --> 00:07:52.380
(2x)^2, which is 4x^2.
00:07:52.380 --> 00:07:56.490
Whereas f''(x) is equal to, if
I differentiate this another
00:07:56.490 --> 00:07:58.750
time, it's equal to 2.
00:07:58.750 --> 00:08:03.440
So they don't mean
the same thing.
00:08:03.440 --> 00:08:04.700
The same thing over here.
00:08:04.700 --> 00:08:07.200
You can see this dy / dx,
this is the rate of change
00:08:07.200 --> 00:08:08.190
of y with respect to x.
00:08:08.190 --> 00:08:09.632
The quantity squared.
00:08:09.632 --> 00:08:11.340
So in other words,
this thing is supposed
00:08:11.340 --> 00:08:13.300
to mean the same as that.
00:08:13.300 --> 00:08:13.800
Yeah.
00:08:13.800 --> 00:08:19.580
Another question.
00:08:19.580 --> 00:08:25.660
STUDENT: [INAUDIBLE]
00:08:25.660 --> 00:08:27.970
PROFESSOR: So the
question is, you
00:08:27.970 --> 00:08:30.430
got a little nervous because
I left out these limits.
00:08:30.430 --> 00:08:32.120
And indeed, I did
that on purpose
00:08:32.120 --> 00:08:34.530
because I didn't want to
specify what was going on.
00:08:34.530 --> 00:08:36.590
Really, if you wrote
it in terms of ds,
00:08:36.590 --> 00:08:38.710
you'd have to write
it as starting at s_0
00:08:38.710 --> 00:08:42.180
and ending at s_n to be
consistent with the variable s.
00:08:42.180 --> 00:08:45.730
But of course, if you write it
in terms of another variable,
00:08:45.730 --> 00:08:46.940
you put that variable in.
00:08:46.940 --> 00:08:49.190
So this is what we do when
we change variables, right?
00:08:49.190 --> 00:08:51.570
We have many different
choices for these limits.
00:08:51.570 --> 00:08:54.530
And this is the clue as
to which variable we use.
00:08:54.530 --> 00:08:59.060
STUDENT: [INAUDIBLE]
00:08:59.060 --> 00:09:01.720
PROFESSOR: Correct.
s_0 and s_n are not
00:09:01.720 --> 00:09:03.310
the same thing as a and b.
00:09:03.310 --> 00:09:05.290
In fact, this is x_n.
00:09:05.290 --> 00:09:07.740
And this x_0, over here.
00:09:07.740 --> 00:09:08.990
That's what a and b are.
00:09:08.990 --> 00:09:13.242
But s_0 and s_n are mileage
markers on the road.
00:09:13.242 --> 00:09:15.450
They're not the same thing
as keeping track of what's
00:09:15.450 --> 00:09:16.950
happening on the x axis.
00:09:16.950 --> 00:09:18.710
So when we measure
arc length, remember
00:09:18.710 --> 00:09:27.980
it's mileage along
the curved path.
00:09:27.980 --> 00:09:32.410
So now, I need to give
you some examples.
00:09:32.410 --> 00:09:40.820
And my first example is
going to be really basic.
00:09:40.820 --> 00:09:45.630
But I hope that it helps to
give some perspective here.
00:09:45.630 --> 00:09:48.830
So I'm going to take
the example y = m
00:09:48.830 --> 00:09:52.550
x, which is a linear
function, a straight line.
00:09:52.550 --> 00:09:58.630
And then y' would be m, and so
ds is going to be the square
00:09:58.630 --> 00:10:02.650
root of 1 + (y')^2, dx.
00:10:02.650 --> 00:10:10.840
Which is the square
root of 1 + m^2, dx.
00:10:10.840 --> 00:10:17.700
And now, the length, say,
if we go from, I don't know,
00:10:17.700 --> 00:10:27.300
let's say 0 to 10, let's say,
of the graph is going to be
00:10:27.300 --> 00:10:33.990
the integral from 0 to 10 of
the square root of 1 + m^2, dx.
00:10:33.990 --> 00:10:39.910
Which of course is just
10 square root of 1 + m^2.
00:10:39.910 --> 00:10:41.810
Not very surprising.
00:10:41.810 --> 00:10:43.360
This is a constant.
00:10:43.360 --> 00:10:46.830
It just factors out and the
integral from 0 to 10 of dx
00:10:46.830 --> 00:10:51.210
is 10.
00:10:51.210 --> 00:10:54.340
Let's just draw a
picture of this.
00:10:54.340 --> 00:10:57.530
This is something
which has slope m here.
00:10:57.530 --> 00:10:59.020
And it's going to 10.
00:10:59.020 --> 00:11:02.340
So this horizontal is 10.
00:11:02.340 --> 00:11:05.930
And the vertical is 10m.
00:11:05.930 --> 00:11:08.030
Those are the
dimensions of this.
00:11:08.030 --> 00:11:11.810
And the Pythagorean theorem
says that the hypotenuse,
00:11:11.810 --> 00:11:15.457
not surprisingly, let's draw
it in here in orange to remind
00:11:15.457 --> 00:11:18.040
ourselves that it was the same
type of orange that we had over
00:11:18.040 --> 00:11:27.160
there, this length here is the
square root of 10^2 + (10m)^2.
00:11:27.160 --> 00:11:31.400
That's the formula
for the hypotenuse.
00:11:31.400 --> 00:11:38.520
And that's exactly
the same as this.
00:11:38.520 --> 00:11:40.540
Maybe you're saying
duh, this is obvious.
00:11:40.540 --> 00:11:44.160
But the point that I'm
trying to make is this.
00:11:44.160 --> 00:11:48.220
If you can figure out these
formulas for linear functions,
00:11:48.220 --> 00:11:52.510
calculus tells you how to
do it for every function.
00:11:52.510 --> 00:11:56.370
The idea of calculus is that
this easy calculation here,
00:11:56.370 --> 00:11:58.530
which you can do
without any calculus
00:11:58.530 --> 00:12:04.360
at all, all of the tools, the
notations of differentials
00:12:04.360 --> 00:12:06.115
and limits and
integrals, is going
00:12:06.115 --> 00:12:10.040
to make you be able to
do it for any curve.
00:12:10.040 --> 00:12:12.760
Because we can break things up
into these little infinitesimal
00:12:12.760 --> 00:12:13.260
bits.
00:12:13.260 --> 00:12:15.820
This is the whole idea
of all of the methods
00:12:15.820 --> 00:12:18.620
that we had to set
up integrals here.
00:12:18.620 --> 00:12:25.770
This is the main point
of these integrals.
00:12:25.770 --> 00:12:32.480
Now, so let's do something
slightly more interesting.
00:12:32.480 --> 00:12:39.970
Our next example is
going to be the circle,
00:12:39.970 --> 00:12:41.540
so y = square root of 1-x^2.
00:12:48.590 --> 00:12:51.820
If you like, that's the
graph of a semicircle.
00:12:51.820 --> 00:12:57.040
And maybe we'll set
it up here this way.
00:12:57.040 --> 00:13:00.030
So that the semicircle
goes around like this.
00:13:00.030 --> 00:13:02.310
And we'll start
it here at x = 0.
00:13:02.310 --> 00:13:04.250
And we'll go over to a.
00:13:04.250 --> 00:13:06.560
And we'll take this little
piece of the circle.
00:13:06.560 --> 00:13:08.150
So down to here.
00:13:08.150 --> 00:13:11.850
If you like.
00:13:11.850 --> 00:13:14.970
So here's the
portion of the circle
00:13:14.970 --> 00:13:17.660
that I'm going to
measure the length of.
00:13:17.660 --> 00:13:19.110
Now, we know that length.
00:13:19.110 --> 00:13:20.160
It's called arc length.
00:13:20.160 --> 00:13:21.659
And I'm going to
give it a name, I'm
00:13:21.659 --> 00:13:23.400
going to call it alpha here.
00:13:23.400 --> 00:13:39.640
So alpha's the arc
length along the circle.
00:13:39.640 --> 00:13:42.660
Now, let's figure
out what it is.
00:13:42.660 --> 00:13:45.970
First, in order to do this, I
have to figure out what y' is.
00:13:45.970 --> 00:13:47.960
Or, if you like, dy/dx.
00:13:47.960 --> 00:13:50.680
Now, that's a calculation that
we've done a number of times.
00:13:50.680 --> 00:13:52.830
And I'm going to do
it slightly faster.
00:13:52.830 --> 00:13:57.160
But you remember it gives you a
square root in the denominator.
00:13:57.160 --> 00:13:59.500
And then you have the
derivative of what's
00:13:59.500 --> 00:14:01.180
inside the square root.
00:14:01.180 --> 00:14:02.260
Which is -2x.
00:14:02.260 --> 00:14:05.590
But then there's also 1/2,
because in disguise it's really
00:14:05.590 --> 00:14:07.840
(1 - x^2)^(1/2).
00:14:07.840 --> 00:14:10.584
So we've done this
calculation enough times
00:14:10.584 --> 00:14:12.500
that I'm not going to
carry it out completely.
00:14:12.500 --> 00:14:14.290
I want you to think
about what it is.
00:14:14.290 --> 00:14:17.690
It turns out to -x up here,
because the 1/2 and the 2
00:14:17.690 --> 00:14:22.890
cancel.
00:14:22.890 --> 00:14:25.870
And now the thing that
we have to integrate
00:14:25.870 --> 00:14:30.190
is this arc length element,
as it's called, ds.
00:14:30.190 --> 00:14:38.350
And that's going to be the
square root of 1 + (y')^2, dx.
00:14:38.350 --> 00:14:41.060
And so I'm going to have to
carry out the calculation,
00:14:41.060 --> 00:14:42.970
some messy calculation here.
00:14:42.970 --> 00:14:47.020
Which is that this is 1 plus
the quantity -x over square root
00:14:47.020 --> 00:14:49.274
of 1 - x^2, squared.
00:14:49.274 --> 00:14:51.440
So I have to figure out
what's under the square root
00:14:51.440 --> 00:14:55.740
sign over here in order to
carry out this calculation.
00:14:55.740 --> 00:14:58.570
Now let's do that.
00:14:58.570 --> 00:15:03.310
This is 1 + x^2 / (1 - x^2).
00:15:03.310 --> 00:15:06.140
That's what this simplifies to.
00:15:06.140 --> 00:15:11.190
And then that's equal to, over
a common denominator, 1 - x^2.
00:15:11.190 --> 00:15:13.580
1 - x^2 + x^2.
00:15:13.580 --> 00:15:16.500
And there is a little bit
of simplification now.
00:15:16.500 --> 00:15:19.190
Because the two x^2's cancel.
00:15:19.190 --> 00:15:20.390
And we get 1/(1-x^2).
00:15:28.640 --> 00:15:35.750
So now I get to finish
off the calculation
00:15:35.750 --> 00:15:40.580
by actually figuring out
what the arc length is.
00:15:40.580 --> 00:15:42.460
And what is it?
00:15:42.460 --> 00:15:51.834
Well, this alpha is equal to
the integral from 0 to a of ds.
00:15:51.834 --> 00:15:53.750
Well, it's going to be
the square root of what
00:15:53.750 --> 00:15:55.110
I have here.
00:15:55.110 --> 00:15:57.210
This was a square,
this is just what
00:15:57.210 --> 00:15:58.710
was underneath the
square root sign.
00:15:58.710 --> 00:16:01.500
This is 1 + (y')^2.
00:16:01.500 --> 00:16:03.270
Have to take the
square root of that.
00:16:03.270 --> 00:16:08.380
So what I get here is dx over
the square root of 1 - x^2.
00:16:13.350 --> 00:16:18.590
And now, we recognize this.
00:16:18.590 --> 00:16:21.100
The antiderivative of this
is something that we know.
00:16:21.100 --> 00:16:23.790
This is the inverse sine.
00:16:23.790 --> 00:16:25.830
Evaluated at 0 and a.
00:16:25.830 --> 00:16:29.490
Which is just giving
us the inverse sine
00:16:29.490 --> 00:16:35.860
of a, because the inverse
sine of 0 is equal to 0.
00:16:35.860 --> 00:16:43.980
So alpha is equal to
the inverse sine of a.
00:16:43.980 --> 00:16:51.320
That's a very fancy way of
saying that sin(alpha) = a.
00:16:51.320 --> 00:16:54.870
That's the equivalent
statement here.
00:16:54.870 --> 00:16:59.830
And what's going on here
is something that's just
00:16:59.830 --> 00:17:01.950
a little deeper than it looks.
00:17:01.950 --> 00:17:03.120
Which is this.
00:17:03.120 --> 00:17:08.030
We've just figured out a
geometric interpretation
00:17:08.030 --> 00:17:09.350
of what's going on here.
00:17:09.350 --> 00:17:13.680
That is, that we went a
distance alpha along this arc.
00:17:13.680 --> 00:17:28.570
And now remember that
the radius here is 1.
00:17:28.570 --> 00:17:34.430
And this horizontal
distance here is a.
00:17:34.430 --> 00:17:37.450
This distance here is a.
00:17:37.450 --> 00:17:40.600
And so the geometric
interpretation of this
00:17:40.600 --> 00:17:51.000
is that this angle
is alpha radians.
00:17:51.000 --> 00:17:54.950
And sin(alpha) = a.
00:17:54.950 --> 00:17:57.570
So this is consistent
with our definition
00:17:57.570 --> 00:18:00.430
previously, our previous
geometric definition
00:18:00.430 --> 00:18:02.530
of radians.
00:18:02.530 --> 00:18:07.190
But this is really your first
true definition of radians.
00:18:07.190 --> 00:18:09.850
We never actually--
People told you
00:18:09.850 --> 00:18:12.570
that radians were the arc
length along this curve.
00:18:12.570 --> 00:18:14.770
This is the first time
you're deriving it.
00:18:14.770 --> 00:18:18.420
This is the first time you're
seeing it correctly done.
00:18:18.420 --> 00:18:20.134
And furthermore, this
is the first time
00:18:20.134 --> 00:18:21.550
you're seeing a
correct definition
00:18:21.550 --> 00:18:24.100
of the sine function.
00:18:24.100 --> 00:18:26.156
Remember we had
this crazy way, we
00:18:26.156 --> 00:18:27.530
defined the
exponential function,
00:18:27.530 --> 00:18:29.661
then we had another
way of defining the log
00:18:29.661 --> 00:18:30.660
function as an integral.
00:18:30.660 --> 00:18:32.900
Then we defined the
exponential in terms of it.
00:18:32.900 --> 00:18:34.690
Well, this is the
same sort of thing.
00:18:34.690 --> 00:18:36.700
What's really happening
here is that if you
00:18:36.700 --> 00:18:38.190
want to know what
radians are, you
00:18:38.190 --> 00:18:40.847
have to calculate this number.
00:18:40.847 --> 00:18:42.430
If you've calculated
this number, then
00:18:42.430 --> 00:18:47.930
by definition if sine is
the thing whose alpha radian
00:18:47.930 --> 00:18:49.890
amount gives you
a, then it must be
00:18:49.890 --> 00:18:52.520
that this is sine inverse of a.
00:18:52.520 --> 00:18:55.430
And so the first thing that
gets defined is the arcsine.
00:18:55.430 --> 00:18:56.930
And the next thing
that gets defined
00:18:56.930 --> 00:19:00.250
is the sine, afterwards.
00:19:00.250 --> 00:19:04.500
This is the way the
foundational approach actually
00:19:04.500 --> 00:19:06.730
works when you start
from first principles.
00:19:06.730 --> 00:19:10.200
This arc length being one
of the first principles.
00:19:10.200 --> 00:19:13.720
So now we have a solid
foundation for trig functions.
00:19:13.720 --> 00:19:15.760
And this is giving
that connection.
00:19:15.760 --> 00:19:18.260
Of course, it's consistent with
everything you already knew,
00:19:18.260 --> 00:19:22.079
so you don't have to make any
transitional thinking here.
00:19:22.079 --> 00:19:23.620
It's just that this
is the first time
00:19:23.620 --> 00:19:25.570
it's being done rigorously.
00:19:25.570 --> 00:19:36.380
Because you only
now have arc length.
00:19:36.380 --> 00:19:41.220
So these are examples, as I say,
that maybe you already know.
00:19:41.220 --> 00:19:44.820
And maybe we'll do one that
we don't know quite as well.
00:19:44.820 --> 00:19:49.300
Let's find the
length of a parabola.
00:19:49.300 --> 00:19:59.180
This is Example 3.
00:19:59.180 --> 00:20:00.660
Now, that was what
I was suggesting
00:20:00.660 --> 00:20:03.140
we were going to do earlier.
00:20:03.140 --> 00:20:06.730
So this is the function y x^2.
00:20:06.730 --> 00:20:09.800
y' = 2x.
00:20:09.800 --> 00:20:20.120
And so ds is equal to the
square root of 1 + (2x)^2, dx.
00:20:20.120 --> 00:20:24.600
And now I can figure out what
a piece of a parabola is.
00:20:24.600 --> 00:20:28.220
So I'll draw the piece
of parabola up to a,
00:20:28.220 --> 00:20:30.840
let's say, starting from 0.
00:20:30.840 --> 00:20:32.680
So that's the chunk.
00:20:32.680 --> 00:20:45.450
And then its arc length,
between 0 and a of this curve,
00:20:45.450 --> 00:21:02.400
is the integral from 0 to a of
square root of 1 + 4x^2, dx.
00:21:02.400 --> 00:21:08.490
OK, now if you like, this is
the answer to the question.
00:21:08.490 --> 00:21:11.040
But people hate
looking at answers
00:21:11.040 --> 00:21:13.800
when they're integrals
if they can be evaluated.
00:21:13.800 --> 00:21:16.510
So one of the reasons why we
went through all this rigmarole
00:21:16.510 --> 00:21:18.800
of calculating these
things is to show you
00:21:18.800 --> 00:21:22.120
that we can actually evaluate
a bunch of these functions
00:21:22.120 --> 00:21:23.340
here more explicitly.
00:21:23.340 --> 00:21:28.180
It doesn't help a lot, but
there is an explicit calculation
00:21:28.180 --> 00:21:28.680
of this.
00:21:28.680 --> 00:21:30.560
So remember how
you would do this.
00:21:30.560 --> 00:21:33.040
So this is just a
little bit of review.
00:21:33.040 --> 00:21:35.420
What we did in techniques
of integration.
00:21:35.420 --> 00:21:39.000
The first step is what?
00:21:39.000 --> 00:21:40.700
A substitution.
00:21:40.700 --> 00:21:43.950
It's a trig substitution.
00:21:43.950 --> 00:21:45.030
And what is it?
00:21:45.030 --> 00:21:47.250
STUDENT: [INAUDIBLE]
00:21:47.250 --> 00:21:50.270
PROFESSOR: So x equals
something tan(theta).
00:21:50.270 --> 00:21:54.587
I claim that it's 1/2 tan,
and I'm going to call it u.
00:21:54.587 --> 00:21:56.420
Because I'm going to
use theta for something
00:21:56.420 --> 00:21:58.161
else in a couple of days.
00:21:58.161 --> 00:21:58.660
OK?
00:21:58.660 --> 00:22:01.420
So this is the substitution.
00:22:01.420 --> 00:22:10.620
And then of course dx
= 1/2 sec^2 u du, etc.
00:22:10.620 --> 00:22:12.750
So what happens if you do this?
00:22:12.750 --> 00:22:15.160
I'll write down the
answer, but I'm not
00:22:15.160 --> 00:22:16.330
going to carry this out.
00:22:16.330 --> 00:22:19.090
Because every one of
these is horrendous.
00:22:19.090 --> 00:22:22.190
But I think I worked it out.
00:22:22.190 --> 00:22:23.370
Let's see if I'm lucky.
00:22:23.370 --> 00:22:24.300
Oh yeah.
00:22:24.300 --> 00:22:26.380
I think this is what it is.
00:22:26.380 --> 00:22:40.979
It's a 1/4 ln(2x + square root
of (1+4x^2) + 1/2 x square root
00:22:40.979 --> 00:22:41.520
of (1+4x^2)).
00:22:46.540 --> 00:22:52.190
Evaluated at a and 0.
00:22:52.190 --> 00:22:53.090
So yick.
00:22:53.090 --> 00:22:53.810
I mean, you know.
00:22:53.810 --> 00:22:55.720
STUDENT: [INAUDIBLE]
00:22:55.720 --> 00:22:58.850
PROFESSOR: Why I
did I make it 1/2?
00:22:58.850 --> 00:23:00.989
Because it turns out that
when you differentiate.
00:23:00.989 --> 00:23:02.780
So the question is,
why is there 1/2 there?
00:23:02.780 --> 00:23:05.734
If you differentiate it without
the 1/2, you get this term
00:23:05.734 --> 00:23:07.650
and it looks like it's
going to be just right.
00:23:07.650 --> 00:23:10.191
But then if you differentiate
this one you get another thing.
00:23:10.191 --> 00:23:12.170
And it all mixes together.
00:23:12.170 --> 00:23:13.660
And it turns out
that there's more.
00:23:13.660 --> 00:23:15.280
So it turns out that it's 1/2.
00:23:15.280 --> 00:23:18.750
Differentiate it and check.
00:23:18.750 --> 00:23:21.530
So this just an incredibly
long calculation.
00:23:21.530 --> 00:23:24.220
It would take fifteen minutes
or something like that.
00:23:24.220 --> 00:23:26.220
But the point is, you
do know in principle
00:23:26.220 --> 00:23:27.840
how to do these things.
00:23:27.840 --> 00:23:43.455
STUDENT: [INAUDIBLE]
00:23:43.455 --> 00:23:45.330
PROFESSOR: Oh, he was
talking about this 1/2,
00:23:45.330 --> 00:23:47.100
not this crazy 1/2 here.
00:23:47.100 --> 00:23:47.600
Sorry.
00:23:47.600 --> 00:23:48.637
STUDENT: [INAUDIBLE]
00:23:48.637 --> 00:23:49.470
PROFESSOR: Yeah, OK.
00:23:49.470 --> 00:23:50.740
So sorry about that.
00:23:50.740 --> 00:23:53.080
Thank you for helping.
00:23:53.080 --> 00:23:56.270
This factor of 1/2 here comes
about because when you square
00:23:56.270 --> 00:23:58.690
x, you don't get tan^2.
00:23:58.690 --> 00:24:02.510
When you square 2x, you
get (4x)^2 and that matches
00:24:02.510 --> 00:24:03.930
perfectly with this thing.
00:24:03.930 --> 00:24:07.300
And that's why you
need this factor here.
00:24:07.300 --> 00:24:07.800
Yeah.
00:24:07.800 --> 00:24:09.216
Another question,
way in the back.
00:24:09.216 --> 00:24:18.190
STUDENT: [INAUDIBLE]
00:24:18.190 --> 00:24:20.780
PROFESSOR: The question is,
when you do this substitution,
00:24:20.780 --> 00:24:25.040
doesn't the limit
from 0 to a change.
00:24:25.040 --> 00:24:27.020
And the answer is,
absolutely yes.
00:24:27.020 --> 00:24:30.230
The limits in terms
of u are not the same
00:24:30.230 --> 00:24:31.800
as the limits in terms of a.
00:24:31.800 --> 00:24:34.870
But if I then translate back
to the x variables, which
00:24:34.870 --> 00:24:40.820
I've done here in this bottom
formula, of x = 0 and x = a,
00:24:40.820 --> 00:24:44.740
it goes back to those in
the original variables.
00:24:44.740 --> 00:24:46.840
So if I write things in
the original variables,
00:24:46.840 --> 00:24:48.980
I have the original limits.
00:24:48.980 --> 00:24:51.962
If I use the u variables, I
would have to change limits.
00:24:51.962 --> 00:24:53.670
But I'm not carrying
out the integration,
00:24:53.670 --> 00:24:55.060
because I don't want to.
00:24:55.060 --> 00:25:00.630
So I brought it back
to the x formula.
00:25:00.630 --> 00:25:07.080
Other questions.
00:25:07.080 --> 00:25:11.580
OK, so now we're ready to launch
into three-space a little bit
00:25:11.580 --> 00:25:14.250
here.
00:25:14.250 --> 00:25:41.310
We're going to talk
about surface area.
00:25:41.310 --> 00:25:43.750
You're going to be
doing a lot with surface
00:25:43.750 --> 00:25:48.550
area in multivariable calculus.
00:25:48.550 --> 00:25:50.990
It's one of the
really fun things.
00:25:50.990 --> 00:25:55.430
And just remember, when
it gets complicated,
00:25:55.430 --> 00:25:57.926
that the simplest things
are the most important.
00:25:57.926 --> 00:25:59.300
And the simple
things are, if you
00:25:59.300 --> 00:26:01.820
can handle things
for linear functions,
00:26:01.820 --> 00:26:02.920
you know all the rest.
00:26:02.920 --> 00:26:04.544
So there's going to
be some complicated
00:26:04.544 --> 00:26:06.390
stuff but it'll
really only involve
00:26:06.390 --> 00:26:09.480
what's happening on planes.
00:26:09.480 --> 00:26:11.680
So let's start
with surface area.
00:26:11.680 --> 00:26:15.615
And the example that
I'd like to give
00:26:15.615 --> 00:26:20.110
- this is the only type of
example that we'll have -
00:26:20.110 --> 00:26:28.560
is the surface of rotation.
00:26:28.560 --> 00:26:31.990
And as long as we have
our parabola there,
00:26:31.990 --> 00:26:33.300
we'll use that one.
00:26:33.300 --> 00:26:51.920
So we have y = x^2,
rotated around the x-axis.
00:26:51.920 --> 00:26:54.480
So let's take a look at
what this looks like.
00:26:54.480 --> 00:26:57.910
It's the parabola, which
is going like that.
00:26:57.910 --> 00:27:01.850
And then it's being
spun around the x-axis.
00:27:01.850 --> 00:27:08.390
So some kind of shape like
this with little circles.
00:27:08.390 --> 00:27:17.880
It's some kind of
trumpet shape, right?
00:27:17.880 --> 00:27:20.170
And that's the shape
that we're-- Now, again,
00:27:20.170 --> 00:27:20.980
it's the surface.
00:27:20.980 --> 00:27:27.770
It's just the metal of the
trumpet, not the insides.
00:27:27.770 --> 00:27:33.280
Now, the principle for figuring
out what the formula for area
00:27:33.280 --> 00:27:36.470
is, is not that
different from what we
00:27:36.470 --> 00:27:38.530
did for surfaces of revolution.
00:27:38.530 --> 00:27:42.610
But it just requires a little
bit of thought and imagination.
00:27:42.610 --> 00:27:50.360
We have a little chunk
of arc length along here.
00:27:50.360 --> 00:27:55.580
And we're going to spin
that around this axis.
00:27:55.580 --> 00:28:01.610
Now, if this were a horizontal
piece of arc length,
00:28:01.610 --> 00:28:04.500
then it would spin
around just like a shell.
00:28:04.500 --> 00:28:07.550
It would just be a surface.
00:28:07.550 --> 00:28:11.670
But if it's tilted,
if it's tilted,
00:28:11.670 --> 00:28:15.030
then there's more surface area
proportional to the amount
00:28:15.030 --> 00:28:17.040
that it's tilted.
00:28:17.040 --> 00:28:19.310
So it's proportional to
the length of the segment
00:28:19.310 --> 00:28:22.890
that you spin around.
00:28:22.890 --> 00:28:29.430
So the total is going to be ds,
that's one of the factors here.
00:28:29.430 --> 00:28:32.220
Maybe I'll write that second.
00:28:32.220 --> 00:28:33.560
That's one of the dimensions.
00:28:33.560 --> 00:28:36.660
And then the other dimension
is the circumference.
00:28:36.660 --> 00:28:43.180
Which is 2 pi, in this case, y.
00:28:43.180 --> 00:28:46.300
So that's the end
of the calculation.
00:28:46.300 --> 00:28:55.990
This is the area
element of surface area.
00:28:55.990 --> 00:28:59.860
Now, when you get to 18.02,
and maybe even before that,
00:28:59.860 --> 00:29:03.000
you'll also see some people
referring to this area element
00:29:03.000 --> 00:29:09.110
when it's a curvy surface
like this with a notation dS.
00:29:09.110 --> 00:29:10.610
That's a little
confusing because we
00:29:10.610 --> 00:29:12.070
have a lower case s here.
00:29:12.070 --> 00:29:15.450
We're not going to
use it right now.
00:29:15.450 --> 00:29:17.620
But the lower case s
is usually arc length.
00:29:17.620 --> 00:29:23.950
The upper case S is
usually surface area.
00:29:23.950 --> 00:29:25.650
So.
00:29:25.650 --> 00:29:32.510
Also used for dA.
00:29:32.510 --> 00:29:33.730
The area element.
00:29:33.730 --> 00:29:39.780
Because this is a
curved area element.
00:29:39.780 --> 00:29:47.630
So let's figure
out this example.
00:29:47.630 --> 00:29:54.090
So in the example-- ...is equal
to x ^2 then the situation is,
00:29:54.090 --> 00:30:01.050
we have the surface area is
equal to the integral from,
00:30:01.050 --> 00:30:03.770
I don't know, 0 to a if those
are the limits that we wanted
00:30:03.770 --> 00:30:05.040
to choose.
00:30:05.040 --> 00:30:10.470
Of 2 pi x^2, right?
00:30:10.470 --> 00:30:11.160
Because y = x^2.
00:30:11.160 --> 00:30:17.960
Times the square
root of 1 + 4x^2, dx.
00:30:17.960 --> 00:30:20.300
Remember we had this from
our previous example.
00:30:20.300 --> 00:30:32.850
This was ds from previous.
00:30:32.850 --> 00:30:41.930
And this, of course, is 2 pi y.
00:30:41.930 --> 00:30:49.150
Now again, the calculation of
this integral is kind of long.
00:30:49.150 --> 00:30:52.060
And I'm going to omit it.
00:30:52.060 --> 00:30:54.300
But let me just point
out that it follows
00:30:54.300 --> 00:30:56.350
from the same substitution.
00:30:56.350 --> 00:31:05.150
Namely, x = 1/2 tan u is going
to work for this integral.
00:31:05.150 --> 00:31:06.200
It's kind of a mess.
00:31:06.200 --> 00:31:08.590
There's a tan squared here
and the secant squared.
00:31:08.590 --> 00:31:10.170
There's another
secant and so on.
00:31:10.170 --> 00:31:12.640
So it's one of
these trig integrals
00:31:12.640 --> 00:31:19.670
that then takes a while to do.
00:31:19.670 --> 00:31:22.940
So that just is going to
trail off into nothing.
00:31:22.940 --> 00:31:25.540
And the reason is that
what's important here
00:31:25.540 --> 00:31:27.250
is more the method.
00:31:27.250 --> 00:31:29.650
And the setup of the integrals.
00:31:29.650 --> 00:31:32.902
The actual computation, in
fact, you could go to a program
00:31:32.902 --> 00:31:34.610
and you could plug in
something like this
00:31:34.610 --> 00:31:37.300
and you would spit out
an answer immediately.
00:31:37.300 --> 00:31:41.100
So really what we just want is
for you to have enough control
00:31:41.100 --> 00:31:43.640
to see that it's an integral
that's a manageable one.
00:31:43.640 --> 00:31:45.630
And also to know that
if you plugged it in,
00:31:45.630 --> 00:31:50.970
you would get an answer.
00:31:50.970 --> 00:31:53.320
When I actually do carry
out a calculation, though,
00:31:53.320 --> 00:31:57.740
what I want to do
is to do something
00:31:57.740 --> 00:32:00.190
that has an answer
that you can remember.
00:32:00.190 --> 00:32:02.090
And that's a nice answer.
00:32:02.090 --> 00:32:05.940
So that turns out to be
the example of the surface
00:32:05.940 --> 00:32:07.580
area of a sphere.
00:32:07.580 --> 00:32:10.110
So it's analogous
to this 2 here.
00:32:10.110 --> 00:32:15.170
And maybe I should
remember this result here.
00:32:15.170 --> 00:32:24.390
Which was that the arc length
element was given by this.
00:32:24.390 --> 00:32:38.700
So we'll save that for a second.
00:32:38.700 --> 00:32:41.710
So we're going to do
this surface area now.
00:32:41.710 --> 00:32:43.930
So if you like, this
is another example.
00:32:43.930 --> 00:32:54.820
The surface area of a sphere.
00:32:54.820 --> 00:32:59.670
This is a good example,
and one, as I say,
00:32:59.670 --> 00:33:01.160
that has a really nice answer.
00:33:01.160 --> 00:33:07.130
So it's worth doing.
00:33:07.130 --> 00:33:09.770
So first of all, I'm not going
to set it up quite the way
00:33:09.770 --> 00:33:11.794
I did in Example 2.
00:33:11.794 --> 00:33:13.710
Instead, I'm going to
take the general sphere,
00:33:13.710 --> 00:33:18.790
because I'd like to watch
the dependence on the radius.
00:33:18.790 --> 00:33:22.160
So here this is going
to be the radius.
00:33:22.160 --> 00:33:27.250
It's going to be radius a.
00:33:27.250 --> 00:33:30.440
And now, if I carry out
the same calculations
00:33:30.440 --> 00:33:33.840
as before, if you think
about it for a second,
00:33:33.840 --> 00:33:42.322
you're going to get this
result. And then, the rest
00:33:42.322 --> 00:33:43.780
of the arithmetic,
which is sitting
00:33:43.780 --> 00:33:47.750
up there in the case,
a = 1, will give us
00:33:47.750 --> 00:33:53.120
that ds is equal to what?
00:33:53.120 --> 00:33:56.670
Well, maybe I'll
just carry it out.
00:33:56.670 --> 00:33:58.510
Because that's always nice.
00:33:58.510 --> 00:34:03.630
So we have 1 +
x^2 / (a^2 - x^2).
00:34:03.630 --> 00:34:07.060
That's 1 + (y')^2.
00:34:07.060 --> 00:34:09.630
And now I put this over
a common denominator.
00:34:09.630 --> 00:34:11.800
And I get a^2 - x^2.
00:34:11.800 --> 00:34:15.030
And I have in the
numerator a^2 - x^2 + x^2.
00:34:15.030 --> 00:34:17.590
So the same cancellation occurs.
00:34:17.590 --> 00:34:25.390
But now we get an
a^2 in the numerator.
00:34:25.390 --> 00:34:28.070
So now I can set up the ds.
00:34:28.070 --> 00:34:30.200
And so here's what happens.
00:34:30.200 --> 00:34:35.370
The area of a section of
the sphere, so let's see.
00:34:35.370 --> 00:34:38.920
We're going to start at
some starting place x_1,
00:34:38.920 --> 00:34:40.760
and end at some place x_2.
00:34:40.760 --> 00:34:43.110
So what does that look like?
00:34:43.110 --> 00:34:45.140
Here's the sphere.
00:34:45.140 --> 00:34:47.490
And we're starting
at a place x_1.
00:34:47.490 --> 00:34:49.810
And we're ending at a place x_2.
00:34:49.810 --> 00:34:53.830
And we're taking more or less
the slice here, if you like.
00:34:53.830 --> 00:34:59.480
The section of this sphere.
00:34:59.480 --> 00:35:02.150
So the area's going
to equal this.
00:35:02.150 --> 00:35:06.730
And what is it going to be?
00:35:06.730 --> 00:35:12.820
Well, so I have here 2 pi y.
00:35:12.820 --> 00:35:15.310
I'll write it out, just
leave it as y for now.
00:35:15.310 --> 00:35:19.190
And then I have ds.
00:35:19.190 --> 00:35:22.110
So that's always what the
formula is when you're
00:35:22.110 --> 00:35:25.880
revolving around the x-axis.
00:35:25.880 --> 00:35:29.920
And then I'll plug
in for those things.
00:35:29.920 --> 00:35:35.990
So 2 pi, the formula for y
is square root a^2 - x^2.
00:35:38.700 --> 00:35:42.220
And the formula
for ds, well, it's
00:35:42.220 --> 00:35:44.970
the square root
of this times dx.
00:35:44.970 --> 00:35:48.920
So it's the square root
of a^2 / (a^2 - x^2), dx.
00:35:51.760 --> 00:35:54.880
So this part is ds.
00:35:54.880 --> 00:36:02.170
And this part is y.
00:36:02.170 --> 00:36:07.590
And now, I claim we have a nice
cancellation that takes place.
00:36:07.590 --> 00:36:09.770
Square root of a^2 is a.
00:36:09.770 --> 00:36:12.740
And then there's another
good cancellation.
00:36:12.740 --> 00:36:13.990
As you can see.
00:36:13.990 --> 00:36:17.140
Now, what we get here is the
integral from x_1 to x_2,
00:36:17.140 --> 00:36:21.510
of 2 pi a dx, which is
about the easiest integral
00:36:21.510 --> 00:36:23.340
you can imagine.
00:36:23.340 --> 00:36:24.940
It's just the integral
of a constant.
00:36:24.940 --> 00:36:27.980
So it's 2 pi a (x_2 - x_1).
00:36:36.810 --> 00:36:40.780
Let's check this in
a couple of examples.
00:36:40.780 --> 00:36:48.390
And then see what it's saying
geometrically, a little bit.
00:36:48.390 --> 00:36:53.220
So what this is saying--
So special cases
00:36:53.220 --> 00:36:56.560
that you should
always check, when
00:36:56.560 --> 00:36:59.220
you have a nice formula
like this, at least.
00:36:59.220 --> 00:37:00.730
But really with
anything in order
00:37:00.730 --> 00:37:03.250
to make sure that you've
got the right answer.
00:37:03.250 --> 00:37:05.710
If you take, for
example, the hemisphere.
00:37:05.710 --> 00:37:08.920
So you take 1/2 of this sphere.
00:37:08.920 --> 00:37:11.090
So that would be
starting at 0, sorry.
00:37:11.090 --> 00:37:14.300
And ending at a.
00:37:14.300 --> 00:37:17.270
So that's the
integral from 0 to a.
00:37:17.270 --> 00:37:21.890
So this is the case
x_1 = 0. x_2 = a.
00:37:21.890 --> 00:37:29.190
And what you're going
to get is a hemisphere.
00:37:29.190 --> 00:37:36.270
And the area is 2 pi a times a.
00:37:36.270 --> 00:37:38.150
Or in other words, 2 pi a^2.
00:37:42.410 --> 00:37:48.920
And if you take
the whole sphere,
00:37:48.920 --> 00:37:55.900
that's starting at x_1 = -a,
and x_2 = a, you're getting 2 pi
00:37:55.900 --> 00:37:58.330
a times (a - (-a)).
00:38:02.640 --> 00:38:06.380
Which is 4 pi a^2.
00:38:06.380 --> 00:38:09.230
That's the whole thing.
00:38:09.230 --> 00:38:10.140
Yeah, question.
00:38:10.140 --> 00:38:21.780
STUDENT: [INAUDIBLE]
00:38:21.780 --> 00:38:23.340
PROFESSOR: The
question is, would it
00:38:23.340 --> 00:38:27.730
be possible to rotate
around the y-axis?
00:38:27.730 --> 00:38:30.330
And the answer is yes.
00:38:30.330 --> 00:38:34.550
It's legal to rotate
around the y-axis.
00:38:34.550 --> 00:38:43.170
And there is-- If you use
vertical slices as we did here,
00:38:43.170 --> 00:38:45.490
that is, well they're
sort of tips of slices,
00:38:45.490 --> 00:38:46.790
it's a different idea.
00:38:46.790 --> 00:38:49.780
But anyway, it's using
dx as the integral
00:38:49.780 --> 00:38:52.670
of the variable of integration.
00:38:52.670 --> 00:38:55.310
So we're checking
each little piece,
00:38:55.310 --> 00:38:58.750
each little strip of that type.
00:38:58.750 --> 00:39:00.540
If we use dx here, we get this.
00:39:00.540 --> 00:39:02.710
If you did the same thing
rotated the other way,
00:39:02.710 --> 00:39:06.140
and use dy as the variable, you
get exactly the same answer.
00:39:06.140 --> 00:39:08.120
And it would be the
same calculation.
00:39:08.120 --> 00:39:14.080
Because they're parallel.
00:39:14.080 --> 00:39:14.870
So you're, yep.
00:39:14.870 --> 00:39:17.280
STUDENT: [INAUDIBLE]
00:39:17.280 --> 00:39:19.770
PROFESSOR: Can you do
surface area with shells?
00:39:19.770 --> 00:39:26.540
Well, the shell shape-- The
short answer is not quite.
00:39:26.540 --> 00:39:30.851
The shell shape is
a vertical shell
00:39:30.851 --> 00:39:32.600
which is itself already
three-dimensional,
00:39:32.600 --> 00:39:34.850
and it has a thickness.
00:39:34.850 --> 00:39:37.270
So this is just a matter
of terminology, though.
00:39:37.270 --> 00:39:41.210
This thickness is this dx,
when we do this rotation here.
00:39:41.210 --> 00:39:43.820
And then there are
two other dimensions.
00:39:43.820 --> 00:39:46.190
If we have a curved
surface, there's
00:39:46.190 --> 00:39:56.460
no other dimension
left to form a shell.
00:39:56.460 --> 00:39:59.421
But basically, you can chop
things up into any bits
00:39:59.421 --> 00:40:00.670
that you can actually measure.
00:40:00.670 --> 00:40:04.020
That you can figure
out what the area is.
00:40:04.020 --> 00:40:08.270
That's the main point.
00:40:08.270 --> 00:40:10.160
Now, I said we were
going to, we've
00:40:10.160 --> 00:40:12.910
just launched into
three-dimensional space.
00:40:12.910 --> 00:40:21.650
And I want to now move on to
other space-like phenomena.
00:40:21.650 --> 00:40:26.780
But we're going to do this.
00:40:26.780 --> 00:40:31.510
So this is also a
preparation for 18.02,
00:40:31.510 --> 00:40:34.840
where you'll be doing
this a tremendous amount.
00:40:34.840 --> 00:40:49.530
We're going to talk now
about parametric equations.
00:40:49.530 --> 00:40:57.102
Really just parametric curves.
00:40:57.102 --> 00:40:58.810
So you're going to
see this now and we're
00:40:58.810 --> 00:41:00.539
going to interpret
it a couple of times,
00:41:00.539 --> 00:41:02.580
and we're going to think
about polar coordinates.
00:41:02.580 --> 00:41:05.920
These are all preparation for
thinking in more variables,
00:41:05.920 --> 00:41:08.570
and thinking in a
different way than you've
00:41:08.570 --> 00:41:09.720
been thinking before.
00:41:09.720 --> 00:41:11.970
So I want you to
prepare your brain
00:41:11.970 --> 00:41:13.975
to make a transition here.
00:41:13.975 --> 00:41:15.600
This is the beginning
of the transition
00:41:15.600 --> 00:41:21.580
to multivariable thinking.
00:41:21.580 --> 00:41:26.350
We're going to consider
curves like this.
00:41:26.350 --> 00:41:29.850
Which are described with x
being a function of t and y
00:41:29.850 --> 00:41:31.410
being a function of t.
00:41:31.410 --> 00:41:35.630
And this letter t is
called the parameter.
00:41:35.630 --> 00:41:38.080
In this case you should
think of it-- the easiest way
00:41:38.080 --> 00:41:39.700
to think of it is as time.
00:41:39.700 --> 00:41:43.840
And what you have is
what's called a trajectory.
00:41:43.840 --> 00:41:48.090
So this is also
called a trajectory.
00:41:48.090 --> 00:41:54.590
And its location, let's say, at
time 0, is this location here.
00:41:54.590 --> 00:41:58.310
(x(0), y(0)), that's
a point in the plane.
00:41:58.310 --> 00:42:01.380
And then over here, for
instance, maybe it's
00:42:01.380 --> 00:42:04.410
(x(1), y(1)).
00:42:04.410 --> 00:42:06.540
And I drew arrows
along here to indicate
00:42:06.540 --> 00:42:10.110
that we're going from this
place over to that place.
00:42:10.110 --> 00:42:16.190
These are later times. t = 1
is a later time than t = 0.
00:42:16.190 --> 00:42:20.170
So that's just a
very casual, it's
00:42:20.170 --> 00:42:22.680
just the way we use
these notations.
00:42:22.680 --> 00:42:31.670
Now let me give you the first
example, which is x = a cos
00:42:31.670 --> 00:42:40.340
t, y = a sin t.
00:42:40.340 --> 00:42:43.630
And the first thing to figure
out is what kind of curve
00:42:43.630 --> 00:42:45.490
this is.
00:42:45.490 --> 00:42:47.880
And to do that, we
want to figure out
00:42:47.880 --> 00:42:52.000
what equation it satisfies
in rectangular coordinates.
00:42:52.000 --> 00:42:53.650
So to figure out
what curve this is,
00:42:53.650 --> 00:42:59.730
we recognize that if we square
and add-- So we add x^2 to y^2,
00:42:59.730 --> 00:43:02.670
we're going to get something
very nice and clean.
00:43:02.670 --> 00:43:08.700
We're going to get a^2
cos^2 t + a^2 sin^2 t.
00:43:11.990 --> 00:43:13.500
Yeah that's right, OK.
00:43:13.500 --> 00:43:19.230
Which is just a ^2 a^2 (cos^2 +
sin^2), or in other words a^2.
00:43:19.230 --> 00:43:23.450
So lo and behold, what
we have is a circle.
00:43:23.450 --> 00:43:27.840
And then we know what
shape this is now.
00:43:27.840 --> 00:43:33.240
And the other thing I'd
like to keep track of
00:43:33.240 --> 00:43:35.990
is which direction we're
going on the circle.
00:43:35.990 --> 00:43:40.020
Because there's more to this
parameter then just the shape.
00:43:40.020 --> 00:43:42.570
There's also where
we are at what time.
00:43:42.570 --> 00:43:46.610
This would be, think of it like
the trajectory of a planet.
00:43:46.610 --> 00:43:51.600
So here, I have to do this
by plotting the picture
00:43:51.600 --> 00:43:53.250
and figuring out what happens.
00:43:53.250 --> 00:43:59.980
So at t = 0, we have
(x, y) is equal to,
00:43:59.980 --> 00:44:06.190
plug in here (a cos 0, a sin 0).
00:44:06.190 --> 00:44:10.570
Which is just a * 1
+ a * 0, so (a, 0).
00:44:10.570 --> 00:44:11.830
And that's here.
00:44:11.830 --> 00:44:14.660
That's the point (a, 0).
00:44:14.660 --> 00:44:18.474
We know that it's the
circle of radius a.
00:44:18.474 --> 00:44:19.890
So we know that
the curve is going
00:44:19.890 --> 00:44:22.140
to go around like this somehow.
00:44:22.140 --> 00:44:26.390
So let's see what
happens at t = pi / 2.
00:44:26.390 --> 00:44:30.950
So at that point, we have (x,
y) = (a cos(pi/2), a sin(pi/2)).
00:44:37.310 --> 00:44:41.580
Which is (0, a), because
sine of pi / 2 is 1.
00:44:41.580 --> 00:44:43.190
So that's up here.
00:44:43.190 --> 00:44:46.100
So this is what
happens at t = 0.
00:44:46.100 --> 00:44:49.220
This is what happens
at t = pi / 2.
00:44:49.220 --> 00:44:51.390
And the trajectory
clearly goes this way.
00:44:51.390 --> 00:44:54.800
In fact, this turns
out to be t = pi, etc.
00:44:54.800 --> 00:44:58.330
And it repeats at t = 2pi.
00:44:58.330 --> 00:45:00.610
So the other feature
that we have,
00:45:00.610 --> 00:45:11.560
which is qualitative feature,
is that it's counterclockwise.
00:45:11.560 --> 00:45:17.020
Now the last little bit is
going to be the arc length.
00:45:17.020 --> 00:45:19.030
Keeping track of the arc length.
00:45:19.030 --> 00:45:23.390
And we'll do that next time.