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PROFESSOR: OK.
 

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Now, today we get to move on
from integral formulas and

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methods of integration
back to some geometry.

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And this is more or less going
to lead into the kinds of

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tools you'll be using in
multivariable calculus.

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The first thing that we're
going to do today is

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discuss arc length.
 

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Like all of the cumulative sums
that we've worked on, this one

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has a storyline and the picture
associated to it, which

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involves dividing things up.
 

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If you have a roadway, if you
like, and you have mileage

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markers along the road, like
this, all the way up to, say,

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sn here, then the length along
the road is described

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by this parameter, s.
 

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Which is arc length.
 

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And if we look at a graph of
this sort of thing, if this is

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the last point b, and this is
the first point a, then you

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can think in terms of having
points above x1, x2, x3, etc.

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The same as we did
with Riemann sums.

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And then the way that we're
going to approximate this is

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by taking the straight lines
between each of these points.

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As things get smaller and
smaller, the straight line

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is going to be fairly
close to the curve.

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And that's the main idea.
 

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So let me just depict one
little chunk of this.

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Which is like this.
 

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One straight line, and here's
the curved surface there.

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And the distance along the
curved surface is what I'm

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calling delta s, the change in
the length between, so this

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would be s2 - s1 if I
depicted that one.

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So this would be delta s
is, say s. si - si - 1,

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some increment there.
 

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And then I can figure out
what the length of the

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orange segment is.
 

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Because the horizontal
distance is delta x.

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And the vertical
distance is delta y.

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And so the formula is that
the hypotenuse is delta

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x ^2 + delta y ^2.
 

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Square root.
 

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And delta s is
approximately that.

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So what we're saying
is that delta s ^2 is

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approximately this.
 

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So this is the hypotenuse.
 

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Squared.
 

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And it's very close to
the length of the curve.

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And the whole idea of calculus
is in the infinitesimal,

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this is exactly correct.
 

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So that's what's going
to happen in the limit.

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And that is the basis for
calculating arc length.

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I'm going to rewrite that
formula on the next board.

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But I'm going to write it in
the more customary fashion.

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We've done this before,
a certain amount.

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But I just want to emphasize it
here because this handwriting

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is a little bit peculiar.
 

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This ds is really
all one thing.

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What I really mean is to put
the parenthesis around it.

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It's one thing.
 

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It's not d * s, it's ds.
 

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It's one thing.
 

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And we square it.
 

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But for whatever reason people
have gotten into the habit

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of omitting the parentheses.
 

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So you're just going to
have to live with that.

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And realize that this is not d
of s ^2 or anything like that.

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And similarly, this is a
single number, and this

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is a single number.
 

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Infinitesimal.
 

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So that's just the way
that this idea here gets

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written in our notation.
 

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And this is the first time
we're dealing with squares

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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

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is take the square root.
 

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If I take the square root,
that's the square root

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of dx ^2 + dy ^2.
 

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And this is the form in which I
always remember this formula.

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Let's put it in some
brightly decorated form.

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But there are about 4, 5,
6 other forms that you'll

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derive from this, which
all mean the same thing.

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So this is, as I say,
the way I remember it.

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But there are other ways
of thinking of it.

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And let's just write a
couple of them down.

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The first one is that you
can factor out the dx.

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So that looks like this.
 

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1 + (dy / dx)^2.
 

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And then I factored out the dx.
 

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So this is a variant.
 

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And this is the one which
actually we'll be using in

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practice right now
on our examples.

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So the conclusion is that the
arc length, which if you like

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is this total sn - s0, if you
like, is going to be equal to

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the integral from a to b
of the square root of

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1 + (dy / dx)^2 dx.
 

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In practice, it's also
very often written

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informally as this.
 

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The integral ds.
 

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So the change in this little
variable s, and this is what

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you'll see notationally
in many textbooks.

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So that's one way of writing
it, and of course the second

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00:06:42 --> 00:06:45
way of writing it which is
practically the same thing is

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00:06:45 --> 00:06:50
square root of 1
+ f ' ( x^2) dx.

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Mixing in a little bit
of Newton's notation.

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And this is with y = f (x).
 

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So this is the formula
for arc length.

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And as I say, I
remember it this way.

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But you're going to have to
derive various variants of it.

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And you'll have to use
some arithmetic to get

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to various formulas.
 

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And there will be more later.
 

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Yeah, question.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: OK, the question
is, is f ' (x)^2 = f '' (x).

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And the answer is no.
 

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And let's just see what it is.
 

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So, for example, if f ( x) = x
^2, which is an example which

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will come up in a few minutes,
then f ' (x) = 2x and f ' ( x

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)^2 = = (2x), which is 4x ^2.
 

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Whereas f '' ( x) is equal to,
if I differentiate this another

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time, it's equal to 2.
 

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So they don't mean
the same thing.

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The same thing over here.
 

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You can see this dy / dx,
this is the rate of change

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of y with respect to x.
 

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The quantity squared.
 

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So in other words, this
thing is supposed to

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mean the same as that.
 

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00:08:13 --> 00:08:13
Yeah.
 

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Another question.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: So the question is,
you got a little nervous

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because I left out
these limits.

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And indeed, I did that on
purpose because I didn't want

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to specify what was going on.
 

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Really, if you wrote it in
terms of ds, you'd have to

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write it as starting at s0 and
ending at sn to be consistent

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00:08:40 --> 00:08:42
with the variable s.
 

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00:08:42 --> 00:08:45
But of course, if you write it
in terms of another variable,

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you put that variable in.
 

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So this is what we do when
we change variables, right?

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We have many different
choices for these limits.

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00:08:51 --> 00:08:54
And this is the clue as to
which variable we use.

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: Correct. s0
and sn are not the

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same thing as a and b.
 

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In fact, this is xn.
 

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And this x0, over here.
 

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That's what a and b are.
 

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But s0 and sn are mileage
markers on the road.

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They're not the same thing
as keeping track of what's

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happening on the x axis.
 

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So when we measure arc length,
remember it's mileage

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along the curved path.
 

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So now, I need to give
you some examples.

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And my first example is
going to be really basic.

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But I hope that it helps to
give some perspective here.

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So I'm going to take the
example y = m x, which is

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a linear function,
a straight line.

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And then y ' would = m, and so
ds is going to be the square

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root of 1 + (y ') ^2 dx.
 

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Which is the square
root of 1 + m ^2 dx.

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And now, the length, say, if we
go from, I don't know, let's

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say 0 to 10, let's say.
 

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Of the graph is going to be the
integral from 0 to 10 of the

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square root of 1 + m ^2 dx.
 

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Which of course is just 10
square root of 1 + m ^2.

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Not very surprising.
 

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This is a constant.
 

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It just factors out and
the integral from 0

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to 10 of dx is 10.
 

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00:10:51 --> 00:10:54
Let's just draw a
picture of this.

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This is something which
has slope m here.

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And it's going to 10.
 

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So this horizontal is 10.
 

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And the vertical is 10 m.
 

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Those are the
dimensions of this.

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And the Pythagorean theorem
says that the hypotenuse, not

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surprisingly, let's draw it
in here in orange to remind

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ourselves that it was the same
type of orange that we had over

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there, this length here is the
square root of 10 ^2 + (10m)^2.

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That's the formula
for the hypotenuse.

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And that's exactly
the same as this.

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00:11:38 --> 00:11:40
Maybe you're saying
duh, this is obvious.

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But the point that I'm
trying to make is this.

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If you can figure out these
formulas for linear functions,

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calculus tells you how to
do it for every function.

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The idea of calculus is that
this easy calculation here,

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which you can do without any
calculus at all, all of the

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00:12:01 --> 00:12:05
tools, the notations of
differentials and limits and

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integrals, is going to make you
be able to do it for any curve.

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00:12:10 --> 00:12:12
Because we can break things
up into these little

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infinitesimal bits.
 

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This is the whole idea of all
of the methods that we had

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to set up integrals here.
 

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This is the main point
of these integrals.

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Now, so let's do something
slightly more interesting.

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Our next example is going
to be the circle, so y =

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square root of 1 - x ^2.
 

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00:12:48 --> 00:12:51
If you like, that's the
graph of a semicircle.

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And maybe we'll set
it up here this way.

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00:12:57 --> 00:13:00
So that the semicircle
goes around like this.

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00:13:00 --> 00:13:02
And well start it
here at x = 0.

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00:13:02 --> 00:13:04
And we'll go over to a.
 

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00:13:04 --> 00:13:06
And we'll take this little
piece of the circle.

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00:13:06 --> 00:13:08
So down to here.
 

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00:13:08 --> 00:13:11
If you like.
 

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00:13:11 --> 00:13:16
So here's the portion of the
circle that I'm going to

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00:13:16 --> 00:13:17
measure the length of.
 

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00:13:17 --> 00:13:19
Now, we know that length.
 

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00:13:19 --> 00:13:20
It's called arc length.
 

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00:13:20 --> 00:13:21
And I'm going to give it
a name, I'm going to

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00:13:21 --> 00:13:23
call it alpha here.
 

221
00:13:23 --> 00:13:39
So alpha's the arc length
along the circle.

222
00:13:39 --> 00:13:42
Now, let's figure
out what it is.

223
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First, in order to do this, I
have to figure out what y ' is.

224
00:13:45 --> 00:13:47
Or, if you like, dy / dx.
 

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Now, that's a calculation that
we've done a number of times.

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00:13:50 --> 00:13:52
And I'm going to do
it slightly faster.

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00:13:52 --> 00:13:57
But you remember it gives you a
square root in the denominator.

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00:13:57 --> 00:13:59
And then you have the
derivative of what's

229
00:13:59 --> 00:14:01
inside the square root.
 

230
00:14:01 --> 00:14:02
Which is - 2x.
 

231
00:14:02 --> 00:14:05
But then there's also 1/2,
because in disguise it's

232
00:14:05 --> 00:14:07
really (1 - x ^2)^2 1/2.
 

233
00:14:07 --> 00:14:11
So we've done this calculation
enough times that I'm not going

234
00:14:11 --> 00:14:12
to carry it out completely.
 

235
00:14:12 --> 00:14:14
I want you to think
about what it is.

236
00:14:14 --> 00:14:17
It turns out to - x up
here, because the 1/2

237
00:14:17 --> 00:14:22
and the 2 cancel.
 

238
00:14:22 --> 00:14:27
And now the thing that we have
to integrate is this arc length

239
00:14:27 --> 00:14:30
element, as it's called. ds.
 

240
00:14:30 --> 00:14:38
And that's going to be the
square root of 1 + (y ') ^2 dx.

241
00:14:38 --> 00:14:41
And so I'm going to have to
carry out the calculation,

242
00:14:41 --> 00:14:42
some messy calculation here.
 

243
00:14:42 --> 00:14:49
Which is that this is 1 + ( - x
/ square root of 1 - x ^2) ^2.

244
00:14:49 --> 00:14:52
So I have to figure out what's
under the square root sign over

245
00:14:52 --> 00:14:55
here in order to carry
out this calculation.

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Now let's do that.
 

247
00:14:58 --> 00:15:03
This is 1 + x ^2 / 1 - x ^2.
 

248
00:15:03 --> 00:15:06
That's what this simplifies to.
 

249
00:15:06 --> 00:15:08
And then that's equal
to, over a common

250
00:15:08 --> 00:15:11
denominator, (1 - x ^2).
 

251
00:15:11 --> 00:15:13
1 - x ^2 + x^2.
 

252
00:15:13 --> 00:15:16
And there is a little bit
of simplification now.

253
00:15:16 --> 00:15:19
Because the 2x ^2's cancel.
 

254
00:15:19 --> 00:15:28
And we get 1 / 1 - x^2.
 

255
00:15:28 --> 00:15:35
So now I get to finish
off the calculation by

256
00:15:35 --> 00:15:40
actually figuring out
what the arc length is.

257
00:15:40 --> 00:15:42
And what is it?
 

258
00:15:42 --> 00:15:51
Well, this alpha is equal to
the integral from 0 to a of ds.

259
00:15:51 --> 00:15:53
Well, it's going to be
the square root of

260
00:15:53 --> 00:15:55
what I have here.
 

261
00:15:55 --> 00:15:57
This was a square, this is
just what was underneath

262
00:15:57 --> 00:15:58
the square root sign.
 

263
00:15:58 --> 00:16:01
This is 1 + (y ') ^2.
 

264
00:16:01 --> 00:16:03
Have to take the
square root of that.

265
00:16:03 --> 00:16:13
So what I get here is dx /
the square root of 1 - x ^2.

266
00:16:13 --> 00:16:18
And now, we recognize this.
 

267
00:16:18 --> 00:16:21
The antiderivative of this
is something that we know.

268
00:16:21 --> 00:16:23
This is the inverse sine.
 

269
00:16:23 --> 00:16:25
Evaluated at 0 and a.
 

270
00:16:25 --> 00:16:30
Which is just giving us the
inverse sine of a, because

271
00:16:30 --> 00:16:35
the inverse sine of 0 = 0.
 

272
00:16:35 --> 00:16:43
So alpha = the
inverse sine of a.

273
00:16:43 --> 00:16:51
That's a very fancy way of
saying that sine alpha = a.

274
00:16:51 --> 00:16:54
That's the equivalent
statement here.

275
00:16:54 --> 00:16:59
And what's going on here is
something that's just a

276
00:16:59 --> 00:17:01
little deeper than it looks.
 

277
00:17:01 --> 00:17:03
Which is this.
 

278
00:17:03 --> 00:17:08
We've just figured out a
geometric interpretation

279
00:17:08 --> 00:17:09
of what's going on here.
 

280
00:17:09 --> 00:17:13
That is, that we went a
distance alpha along this arc.

281
00:17:13 --> 00:17:28
And now remember that
the radius here is 1.

282
00:17:28 --> 00:17:34
And this horizontal
distance here is a.

283
00:17:34 --> 00:17:37
This distance here is a.
 

284
00:17:37 --> 00:17:43
And so the geometric
interpretation of this is that

285
00:17:43 --> 00:17:51
this angle is alpha radians.
 

286
00:17:51 --> 00:17:54
And sine alpha = a.
 

287
00:17:54 --> 00:17:58
So this is consistent with
our definition previously,

288
00:17:58 --> 00:18:02
our previous geometric
definition of radians.

289
00:18:02 --> 00:18:07
But this is really your first
true definition of radians.

290
00:18:07 --> 00:18:11
We never actually, people told
you that radians were the

291
00:18:11 --> 00:18:12
arc length along this curve.
 

292
00:18:12 --> 00:18:14
This is the first time
you're deriving it.

293
00:18:14 --> 00:18:18
This is the first time you're
seeing it correctly done.

294
00:18:18 --> 00:18:20
And furthermore, this is the
first time you're seeing a

295
00:18:20 --> 00:18:24
correct definition of
the sine function.

296
00:18:24 --> 00:18:27
Remember we had this crazy way,
we we defined the exponential

297
00:18:27 --> 00:18:29
function, then we had another
way of defining the ln

298
00:18:29 --> 00:18:30
function as an integral.
 

299
00:18:30 --> 00:18:32
Then we defined the
exponential in terms of it.

300
00:18:32 --> 00:18:34
Well, this is the
same sort of thing.

301
00:18:34 --> 00:18:37
What's really happening here is
that if you want to know what

302
00:18:37 --> 00:18:40
radians are, you have to
calculate this number.

303
00:18:40 --> 00:18:44
If you've calculated this
number then by definition if

304
00:18:44 --> 00:18:49
sine is the thing whose alpha
radian amount gives you a,

305
00:18:49 --> 00:18:52
then it must be that this
is sine inverse of a.

306
00:18:52 --> 00:18:55
And so the first thing that
gets defined is the arc sine.

307
00:18:55 --> 00:19:00
And the next thing that gets
defined is the sine afterwards.

308
00:19:00 --> 00:19:04
This is the way the
foundational approach actually

309
00:19:04 --> 00:19:06
works when you start
from first principles.

310
00:19:06 --> 00:19:10
This arc length being one
of the first principles.

311
00:19:10 --> 00:19:13
So now we have a solid
foundation for trig functions.

312
00:19:13 --> 00:19:16
And this is giving
that connection.

313
00:19:16 --> 00:19:18
Of course, it's consistent with
everything you already knew, so

314
00:19:18 --> 00:19:22
you don't have to make any
transitional thinking here.

315
00:19:22 --> 00:19:23
It's just that this is the
first time it's being

316
00:19:23 --> 00:19:25
done rigorously.
 

317
00:19:25 --> 00:19:36
Because you only now
have arc length.

318
00:19:36 --> 00:19:39
So these are examples,
as I say, that maybe

319
00:19:39 --> 00:19:41
you already know.
 

320
00:19:41 --> 00:19:44
And maybe we'll do one that
we don't know quite as well.

321
00:19:44 --> 00:19:49
Let's find the length
of a parabola.

322
00:19:49 --> 00:19:59
This is Example 3.
 

323
00:19:59 --> 00:20:00
Now, that was what I
was suggesting we were

324
00:20:00 --> 00:20:03
going to do earlier.
 

325
00:20:03 --> 00:20:09
So this is the function
y = x ^2. y ' = 2x.

326
00:20:09 --> 00:20:20
And so ds = the square
root of 1 + (2x) ^2 dx.

327
00:20:20 --> 00:20:24
And now I can figure out what
a piece of a parabola is.

328
00:20:24 --> 00:20:28
So I'll draw the piece of
parabola up to a, let's

329
00:20:28 --> 00:20:30
say, starting from 0.
 

330
00:20:30 --> 00:20:32
So that's the chunk.
 

331
00:20:32 --> 00:20:45
And then its arc length,
between 0 and a of this curve,

332
00:20:45 --> 00:21:02
is the integral from 0 to a of
square root of 1 + 4x ^2 dx.

333
00:21:02 --> 00:21:08
OK, now if you like, this is
the answer to the question.

334
00:21:08 --> 00:21:12
But people hate looking at
answers when they're integrals

335
00:21:12 --> 00:21:13
if they can be evaluated.
 

336
00:21:13 --> 00:21:15
So one of the reasons why we
went through all this

337
00:21:15 --> 00:21:19
rigamarole of calculating these
things is to show you that we

338
00:21:19 --> 00:21:22
can actually evaluate a
bunch of these functions

339
00:21:22 --> 00:21:23
here more explicitly.
 

340
00:21:23 --> 00:21:27
It doesn't help a lot,
but there is an explicit

341
00:21:27 --> 00:21:28
calculation of this.
 

342
00:21:28 --> 00:21:30
So remember how you
would do this.

343
00:21:30 --> 00:21:33
So this is just a
little bit of review.

344
00:21:33 --> 00:21:35
What we did in techniques
of integration.

345
00:21:35 --> 00:21:39
The first step is what?
 

346
00:21:39 --> 00:21:40
A substitution.
 

347
00:21:40 --> 00:21:43
It's a trig substitution.
 

348
00:21:43 --> 00:21:45
And what is it?
 

349
00:21:45 --> 00:21:47
STUDENT: [INAUDIBLE]
 

350
00:21:47 --> 00:21:50
PROFESSOR: So x equals
something tan theta.

351
00:21:50 --> 00:21:54
I claim that it's 1/2 tan,
and I'm going to call it u.

352
00:21:54 --> 00:21:56
Because I'm going to use
theta for something else

353
00:21:56 --> 00:21:58
in a couple of days.
 

354
00:21:58 --> 00:21:58
OK?
 

355
00:21:58 --> 00:22:01
So this is the substitution.
 

356
00:22:01 --> 00:22:10
And then of course dx =
1/2 sec ^2 u du , etc.

357
00:22:10 --> 00:22:12
So what happens if you do this?
 

358
00:22:12 --> 00:22:15
I'll write down the
answer, but I'm not

359
00:22:15 --> 00:22:16
going to carry this out.
 

360
00:22:16 --> 00:22:19
Because every one of
these is horrendous.

361
00:22:19 --> 00:22:22
But I think I worked it out.
 

362
00:22:22 --> 00:22:23
Let's see if I'm lucky.
 

363
00:22:23 --> 00:22:24
Oh yeah.
 

364
00:22:24 --> 00:22:26
I think this is what it is.
 

365
00:22:26 --> 00:22:40
It's a 1/4 ln 2x + square
root of 1 + 4x ^2 + 1/2 x (

366
00:22:40 --> 00:22:46
square root of 1 + 4x ^2).
 

367
00:22:46 --> 00:22:52
Evaluated at a and 0.
 

368
00:22:52 --> 00:22:53
So yick.
 

369
00:22:53 --> 00:22:53
I mean, you know.
 

370
00:22:53 --> 00:22:55
STUDENT: [INAUDIBLE]
 

371
00:22:55 --> 00:22:58
PROFESSOR: Why I
did I make it 1/2?

372
00:22:58 --> 00:23:01
Because it turns out that
when you differentiate.

373
00:23:01 --> 00:23:02
So the question is,
why there 1/2 there?

374
00:23:02 --> 00:23:05
If you differentiate it without
the 1/2, you get this term and

375
00:23:05 --> 00:23:07
it looks like it's going
to be just right.

376
00:23:07 --> 00:23:09
But then if you differentiate
this one you get another thing.

377
00:23:09 --> 00:23:12
And it all mixes together.
 

378
00:23:12 --> 00:23:13
And it turns out
that there's more.

379
00:23:13 --> 00:23:15
So it turns out that it's 1/2.
 

380
00:23:15 --> 00:23:18
Differentiate it and check.
 

381
00:23:18 --> 00:23:21
So this just an incredibly
long calculation.

382
00:23:21 --> 00:23:24
It would take fifteen minutes
or something like that.

383
00:23:24 --> 00:23:26
But the point is, you do
know in principle how

384
00:23:26 --> 00:23:27
to do these things.
 

385
00:23:27 --> 00:23:43
STUDENT: [INAUDIBLE]
 

386
00:23:43 --> 00:23:45
PROFESSOR: Oh, he was
talking about this 1/2,

387
00:23:45 --> 00:23:47
not this crazy 1/2 here.
 

388
00:23:47 --> 00:23:47
Sorry.
 

389
00:23:47 --> 00:23:48
STUDENT: [INAUDIBLE]
 

390
00:23:48 --> 00:23:49
PROFESSOR: Yeah, OK.
 

391
00:23:49 --> 00:23:50
So sorry about that.
 

392
00:23:50 --> 00:23:53
Thank you for helping.
 

393
00:23:53 --> 00:23:56
This factor of 1/2 here comes
about because when you square

394
00:23:56 --> 00:23:58
x, you don't get tan ^2.
 

395
00:23:58 --> 00:24:02
When you square 2x, you get
4x ^2 and that matches

396
00:24:02 --> 00:24:03
perfectly with this thing.
 

397
00:24:03 --> 00:24:07
And that's why you need
this factor here.

398
00:24:07 --> 00:24:07
Yeah.
 

399
00:24:07 --> 00:24:08
Another question,
way in the back.

400
00:24:08 --> 00:24:18
STUDENT: [INAUDIBLE]
 

401
00:24:18 --> 00:24:20
PROFESSOR: The question is,
when you do this substitution,

402
00:24:20 --> 00:24:25
doesn't the limit
from 0 to a change.

403
00:24:25 --> 00:24:27
And the answer is,
absolutely yes.

404
00:24:27 --> 00:24:30
The limits in terms of u
are not the same as the

405
00:24:30 --> 00:24:31
limits in terms of a.
 

406
00:24:31 --> 00:24:35
But if I then translate back to
the x variables, which I've

407
00:24:35 --> 00:24:40
done here in this bottom
formula, of x = 0 and x = a,

408
00:24:40 --> 00:24:44
it goes back to those in
the original variables.

409
00:24:44 --> 00:24:46
So if I write things in
the original variables, I

410
00:24:46 --> 00:24:48
have the original limits.
 

411
00:24:48 --> 00:24:52
If I use the u variables, I
would have to change limits.

412
00:24:52 --> 00:24:53
But I'm not carrying out
the integration, because

413
00:24:53 --> 00:24:55
I don't want to.
 

414
00:24:55 --> 00:25:00
So I brought it back
to the x formula.

415
00:25:00 --> 00:25:07
Other questions.
 

416
00:25:07 --> 00:25:11
OK, so now we're ready to
launch into three-space

417
00:25:11 --> 00:25:14
a little bit here.
 

418
00:25:14 --> 00:25:41
We're going to talk
about surface area.

419
00:25:41 --> 00:25:46
You're going to be doing a
lot with surface area in

420
00:25:46 --> 00:25:48
multivariable calculus.
 

421
00:25:48 --> 00:25:50
It's one of the
really fun things.

422
00:25:50 --> 00:25:56
And just remember, when it gets
complicated, that the simplest

423
00:25:56 --> 00:25:58
things are the most important.
 

424
00:25:58 --> 00:26:00
And the simple things are, if
you can handle things for

425
00:26:00 --> 00:26:02
linear functions, you
know all the rest.

426
00:26:02 --> 00:26:04
So there's going to be some
complicated stuff but it'll

427
00:26:04 --> 00:26:09
really only involve what's
happening on planes.

428
00:26:09 --> 00:26:11
So let's start with
surface area.

429
00:26:11 --> 00:26:18
And the example that I'd like
to give, this is the only type

430
00:26:18 --> 00:26:28
of example that we'll have,
is the surface of rotation.

431
00:26:28 --> 00:26:31
And as long as we have
our parabola there,

432
00:26:31 --> 00:26:33
we'll use that one.
 

433
00:26:33 --> 00:26:51
So we have y = x ^2,
rotated around the x axis.

434
00:26:51 --> 00:26:54
So let's take a look at
what this looks like.

435
00:26:54 --> 00:26:57
It's the parabola, which
is going like that.

436
00:26:57 --> 00:27:01
And then it's being spun
around the x axis.

437
00:27:01 --> 00:27:08
So some kind of shape like
this with little circles.

438
00:27:08 --> 00:27:17
It's some kind of
trumpet shape, right?

439
00:27:17 --> 00:27:19
And that's the
shape that we're.

440
00:27:19 --> 00:27:20
Now, again, it's the surface.
 

441
00:27:20 --> 00:27:27
It's just the metal of the
trumpet, not the insides.

442
00:27:27 --> 00:27:33
Now, the principle for figuring
out what the formula for area

443
00:27:33 --> 00:27:37
is, is not that different from
what we did for surfaces

444
00:27:37 --> 00:27:38
of revolution.
 

445
00:27:38 --> 00:27:42
But it just requires a little
bit of thought and imagination.

446
00:27:42 --> 00:27:50
We have a little chunk of
arc length along here.

447
00:27:50 --> 00:27:55
And we're going to spin
that around this axis.

448
00:27:55 --> 00:28:02
Now, if this were a horizontal
piece of arc length, then

449
00:28:02 --> 00:28:04
it would spin around
just like a shell.

450
00:28:04 --> 00:28:07
It would just be a surface.
 

451
00:28:07 --> 00:28:12
But if it's tilted, if it's
tilted, then there's more

452
00:28:12 --> 00:28:17
surface area proportional to
the amount that it's tilted.

453
00:28:17 --> 00:28:19
So it's proportional to
the length of the segment

454
00:28:19 --> 00:28:22
that you spin around.
 

455
00:28:22 --> 00:28:29
So the total is going to be ds,
that's one of the factors here.

456
00:28:29 --> 00:28:32
Maybe I'll write that second.
 

457
00:28:32 --> 00:28:33
That's one of the dimensions.
 

458
00:28:33 --> 00:28:36
And then the other dimension
is the circumference.

459
00:28:36 --> 00:28:43
Which is 2 pi, in this case y.
 

460
00:28:43 --> 00:28:46
So that's the end of
the calculation.

461
00:28:46 --> 00:28:55
This is the area element
of surface area.

462
00:28:55 --> 00:29:00
Now, when you get to 18.02, and
maybe even before that, you'll

463
00:29:00 --> 00:29:03
also see some people referring
to this area element when it's

464
00:29:03 --> 00:29:09
a curvy surface like this
with a notation d S.

465
00:29:09 --> 00:29:10
That's a little confusing
because we have a

466
00:29:10 --> 00:29:12
lower case s here.
 

467
00:29:12 --> 00:29:15
We're not going to
use it right now.

468
00:29:15 --> 00:29:17
But the lower case s is
usually arc length.

469
00:29:17 --> 00:29:23
The upper case s is
usually surface area.

470
00:29:23 --> 00:29:25
So.
 

471
00:29:25 --> 00:29:32
Also used for dA.
 

472
00:29:32 --> 00:29:33
The area element.
 

473
00:29:33 --> 00:29:39
Because this is a
curved area element.

474
00:29:39 --> 00:29:47
So let's figure
out this example.

475
00:29:47 --> 00:29:54
So in the example, is equal to
x ^2 then the situation is, we

476
00:29:54 --> 00:30:01
have the surface area is equal
to the integral from, I don't

477
00:30:01 --> 00:30:03
know, 0 to a if those are
the limits that we

478
00:30:03 --> 00:30:05
wanted to choose.
 

479
00:30:05 --> 00:30:10
Of 2 pi x ^2, right?
 

480
00:30:10 --> 00:30:17
Because y = x ^2 ( the square
root of 1 + 4x ^2) dx.

481
00:30:17 --> 00:30:20
Remember we had this from
our previous example.

482
00:30:20 --> 00:30:32
This was ds from previous.
 

483
00:30:32 --> 00:30:41
And this, of course, is 2 pi y.
 

484
00:30:41 --> 00:30:49
Now again, the calculation of
this integral is kind of long.

485
00:30:49 --> 00:30:52
And I'm going to omit it.
 

486
00:30:52 --> 00:30:54
But let me just point out
that it follows from

487
00:30:54 --> 00:30:56
the same substitution.
 

488
00:30:56 --> 00:31:01
Namely, x = 1/2 tan u.
 

489
00:31:01 --> 00:31:05
Is going to work
for this integral.

490
00:31:05 --> 00:31:06
It's kind of a mess.
 

491
00:31:06 --> 00:31:08
There's a tan ^2 here
and the sec ^2.

492
00:31:08 --> 00:31:10
There's another
secant and so on.

493
00:31:10 --> 00:31:13
So it's one of these trig
integrals that then

494
00:31:13 --> 00:31:19
takes a while to do.
 

495
00:31:19 --> 00:31:22
So that just is going to
trail off into nothing.

496
00:31:22 --> 00:31:25
And the reason is that
what's important here

497
00:31:25 --> 00:31:27
is more the method.
 

498
00:31:27 --> 00:31:29
And the setup of the integrals.
 

499
00:31:29 --> 00:31:33
The actual computation, in
fact, you could go to a program

500
00:31:33 --> 00:31:35
and you could plug in something
like this and you would spit

501
00:31:35 --> 00:31:37
out an answer immediately.
 

502
00:31:37 --> 00:31:41
So really what we just want is
for you to have enough control

503
00:31:41 --> 00:31:43
to see that it's an integral
that's a manageable one.

504
00:31:43 --> 00:31:45
And also to know that if
you plugged it in, you

505
00:31:45 --> 00:31:50
would get an answer.
 

506
00:31:50 --> 00:31:53
When I actually do carry out a
calculation, though, what I

507
00:31:53 --> 00:31:58
want to do is to do something
that has an answer that

508
00:31:58 --> 00:32:00
you can remember.
 

509
00:32:00 --> 00:32:02
And that's a nice answer.
 

510
00:32:02 --> 00:32:05
So that turns out to be the
example of the surface

511
00:32:05 --> 00:32:07
area of a sphere.
 

512
00:32:07 --> 00:32:10
So it's analogous
to this 2 here.

513
00:32:10 --> 00:32:15
And maybe I should remember
this result here.

514
00:32:15 --> 00:32:24
Which was that the arc length
element was given by this.

515
00:32:24 --> 00:32:38
So we'll save that
for a second.

516
00:32:38 --> 00:32:41
So we're going to do
this surface area now.

517
00:32:41 --> 00:32:43
So if you like, this
is another example.

518
00:32:43 --> 00:32:54
The surface area of a sphere.
 

519
00:32:54 --> 00:33:00
This is a good example, and
one, as I say, that has

520
00:33:00 --> 00:33:01
a really nice answer.
 

521
00:33:01 --> 00:33:07
So it's worth doing.
 

522
00:33:07 --> 00:33:09
So first of all, I'm not
going to set it up quite

523
00:33:09 --> 00:33:11
the way I did in Example 2.
 

524
00:33:11 --> 00:33:13
Instead, I'm going to take
the general sphere, because

525
00:33:13 --> 00:33:18
I'd like to watch the
dependence on the radius.

526
00:33:18 --> 00:33:22
So here this is going
to be the radius.

527
00:33:22 --> 00:33:27
It's going to be radius a.
 

528
00:33:27 --> 00:33:31
And now, if I carry out the
same calculations as before,

529
00:33:31 --> 00:33:34
if you think about it for a
second, you're going

530
00:33:34 --> 00:33:39
to get this result.
 

531
00:33:39 --> 00:33:43
And then, the rest of the
arithmetic, which is sitting

532
00:33:43 --> 00:33:53
up there in the case, a = 1,
will give us that ds = what?

533
00:33:53 --> 00:33:56
Well, maybe I'll
just carry it out.

534
00:33:56 --> 00:33:58
Because that's always nice.
 

535
00:33:58 --> 00:34:03
So we have 1 + x
^2 / a ^2 - x ^2.

536
00:34:03 --> 00:34:07
That's 1 + (y ') ^2.
 

537
00:34:07 --> 00:34:09
And now I put this over
a common denominator.

538
00:34:09 --> 00:34:11
And I get a ^2 - x ^2.
 

539
00:34:11 --> 00:34:15
And I have in the numerator
a ^2 - x ^2 + x^2.

540
00:34:15 --> 00:34:17
So the same
cancellation occurs.

541
00:34:17 --> 00:34:25
But now we get an a
^2 in the numerator.

542
00:34:25 --> 00:34:28
So now I can set up the ds.
 

543
00:34:28 --> 00:34:30
And so here's what happens.
 

544
00:34:30 --> 00:34:35
The area of a section of
the sphere, so let's see.

545
00:34:35 --> 00:34:39
We're going to start at some
starting place x1, and

546
00:34:39 --> 00:34:40
end at some place x2.
 

547
00:34:40 --> 00:34:43
So what does that look like?
 

548
00:34:43 --> 00:34:45
Here's the sphere.
 

549
00:34:45 --> 00:34:47
And we're starting
at a place x1.

550
00:34:47 --> 00:34:49
And we're ending at a place x2.
 

551
00:34:49 --> 00:34:53
And we're taking more or less
the slice here, if you like.

552
00:34:53 --> 00:34:59
The section of this sphere.
 

553
00:34:59 --> 00:35:02
So the area's going
to equal this.

554
00:35:02 --> 00:35:06
And what is it going to be?
 

555
00:35:06 --> 00:35:12
Well, so I have here 2 pi y.
 

556
00:35:12 --> 00:35:15
I'll write it out, just
leave it as y for now.

557
00:35:15 --> 00:35:19
And then I have ds.
 

558
00:35:19 --> 00:35:22
So that's always what the
formula is when you're

559
00:35:22 --> 00:35:25
revolving around the x axis.
 

560
00:35:25 --> 00:35:29
And then I'll plug in
for those things.

561
00:35:29 --> 00:35:38
So 2 pi, the formula for y
is square root a ^2 - x ^2.

562
00:35:38 --> 00:35:42
And the formula for ds,
well, it's the square

563
00:35:42 --> 00:35:44
root of this times dx.
 

564
00:35:44 --> 00:35:51
So it's the square root of
a ^2 / a ^2 - x ^2 dx.

565
00:35:51 --> 00:35:54
So this part is ds.
 

566
00:35:54 --> 00:36:02
And this part is y.
 

567
00:36:02 --> 00:36:07
And now, I claim we have a nice
cancellation that takes place.

568
00:36:07 --> 00:36:09
Square root of a ^2 = a.
 

569
00:36:09 --> 00:36:12
And then there's another
good cancellation.

570
00:36:12 --> 00:36:13
As you can see.
 

571
00:36:13 --> 00:36:17
Now, what we get here is the
integral from x1 to x2, of 2

572
00:36:17 --> 00:36:21
pi a dx, which is about the
easiest integral

573
00:36:21 --> 00:36:23
you can imagine.
 

574
00:36:23 --> 00:36:24
It's just the integral
of a constant.

575
00:36:24 --> 00:36:36
So it's 2 pi a ( x2 - x1).
 

576
00:36:36 --> 00:36:40
Let's check this in a
couple of examples.

577
00:36:40 --> 00:36:48
And then see what it's saying
geometrically, a little bit.

578
00:36:48 --> 00:36:54
So what this is saying, so
special cases that you should

579
00:36:54 --> 00:36:57
always check when you have a
nice formula like

580
00:36:57 --> 00:36:59
this, at least.
 

581
00:36:59 --> 00:37:01
But really with anything in
order to make sure that

582
00:37:01 --> 00:37:03
you've got the right answer.
 

583
00:37:03 --> 00:37:05
If you take, for example,
the hemisphere.

584
00:37:05 --> 00:37:08
So you take 1/2 of this sphere.
 

585
00:37:08 --> 00:37:11
So that would be
starting at 0, sorry.

586
00:37:11 --> 00:37:14
And ending at a.
 

587
00:37:14 --> 00:37:17
So that's the integral
from 0 to a.

588
00:37:17 --> 00:37:21
So this is the case
x1 = 0. x2 = a.

589
00:37:21 --> 00:37:29
And what you're going to
get is a hemisphere.

590
00:37:29 --> 00:37:36
And the area is (2 pi a ) a.
 

591
00:37:36 --> 00:37:42
Or in other words, 2 pi a ^2.
 

592
00:37:42 --> 00:37:51
And if you take the whole
sphere, that's starting at x1 =

593
00:37:51 --> 00:38:02
- a, and x2 = a, you're getting
(2 pi a) ( a - (- a)).

594
00:38:02 --> 00:38:06
Which is 4 pi a ^2.
 

595
00:38:06 --> 00:38:09
That's the whole thing.
 

596
00:38:09 --> 00:38:10
Yeah, question.
 

597
00:38:10 --> 00:38:21
STUDENT: [INAUDIBLE]
 

598
00:38:21 --> 00:38:24
PROFESSOR: The question is,
would it be possible to

599
00:38:24 --> 00:38:27
rotate around the y axis?
 

600
00:38:27 --> 00:38:30
And the answer is yes.
 

601
00:38:30 --> 00:38:34
It's legal to rotate
around the y axis.

602
00:38:34 --> 00:38:43
And there is, if you use
vertical slices as we did here,

603
00:38:43 --> 00:38:45
that is, well they're sort of
tips of slices, it's

604
00:38:45 --> 00:38:46
a different idea.
 

605
00:38:46 --> 00:38:50
But anyway, it's using dx
as the integral of the

606
00:38:50 --> 00:38:52
variable of integration.
 

607
00:38:52 --> 00:38:55
So we're checking each
little piece, each little

608
00:38:55 --> 00:38:58
strip of that type.
 

609
00:38:58 --> 00:39:00
If we use dx here, we get this.
 

610
00:39:00 --> 00:39:03
If you did the same thing
rotated the other way, and use

611
00:39:03 --> 00:39:06
dy as the variable, you get
exactly the same answer.

612
00:39:06 --> 00:39:08
And it would be the
same calculation.

613
00:39:08 --> 00:39:14
Because they're parallel.
 

614
00:39:14 --> 00:39:14
So you're, yep.
 

615
00:39:14 --> 00:39:17
STUDENT: [INAUDIBLE]
 

616
00:39:17 --> 00:39:19
PROFESSOR: Can you do
service area with shells?

617
00:39:19 --> 00:39:23
Well, ah shell shape.
 

618
00:39:23 --> 00:39:26
The short answer is not quite.
 

619
00:39:26 --> 00:39:32
The shell shape is a vertical
shell which is itself already

620
00:39:32 --> 00:39:34
three-dimensional, and
it has a thickness.

621
00:39:34 --> 00:39:37
So this is just a matter
of terminology, though.

622
00:39:37 --> 00:39:41
This thickness is this dx, when
we do this rotation here.

623
00:39:41 --> 00:39:43
And then there are two
other dimensions.

624
00:39:43 --> 00:39:47
If we have a curved surface,
there's no other dimension

625
00:39:47 --> 00:39:56
left to form a shell.
 

626
00:39:56 --> 00:39:59
But basically, you can chop
things up into any bits that

627
00:39:59 --> 00:40:00
you can actually measure.
 

628
00:40:00 --> 00:40:04
That you can figure
out what the area is.

629
00:40:04 --> 00:40:08
That's the main point.
 

630
00:40:08 --> 00:40:10
Now, I said we were going to,
we've just launched into

631
00:40:10 --> 00:40:12
three-dimensional space.
 

632
00:40:12 --> 00:40:21
And I want to now move on to
other space-like phenomena.

633
00:40:21 --> 00:40:26
But we're going to do this.
 

634
00:40:26 --> 00:40:31
So this is also a preparation
for 18.02, where you'll be

635
00:40:31 --> 00:40:34
doing this a tremendous amount.
 

636
00:40:34 --> 00:40:49
We're going to talk now
about parametric equations.

637
00:40:49 --> 00:40:57
Really just parametric curves.
 

638
00:40:57 --> 00:40:59
So you're going to see this now
and we're going to interpret it

639
00:40:59 --> 00:41:01
a couple of times, and we're
going to think about

640
00:41:01 --> 00:41:02
polar coordinates.
 

641
00:41:02 --> 00:41:06
These are all preparation for
thinking in more variables, and

642
00:41:06 --> 00:41:08
thinking in a different way
than you've been

643
00:41:08 --> 00:41:09
thinking before.
 

644
00:41:09 --> 00:41:12
So I want you to prepare
your brain to make

645
00:41:12 --> 00:41:14
a transition here.
 

646
00:41:14 --> 00:41:16
This is the beginning
of the transition to

647
00:41:16 --> 00:41:21
multivariable thinking.
 

648
00:41:21 --> 00:41:26
We're going to consider
curves like this.

649
00:41:26 --> 00:41:29
Which are described with x
being a function of t and

650
00:41:29 --> 00:41:31
y being a function of t.
 

651
00:41:31 --> 00:41:35
And this letter t is
called the parameter.

652
00:41:35 --> 00:41:38
In this case you should think
of it, the easiest way to

653
00:41:38 --> 00:41:39
think of it is as time.
 

654
00:41:39 --> 00:41:43
And what you have is what's
called a trajectory.

655
00:41:43 --> 00:41:48
So this is also
called a trajectory.

656
00:41:48 --> 00:41:54
And its location, let's say, at
time 0, is this location here.

657
00:41:54 --> 00:41:58
Of (0, y ( 0)), that's
a point in the plane.

658
00:41:58 --> 00:42:01
And then over here, for
instance, maybe it's

659
00:42:01 --> 00:42:04
(x ( 1), y (1)).
 

660
00:42:04 --> 00:42:08
And I drew arrows along here to
indicate that we're going from

661
00:42:08 --> 00:42:10
this place over to that place.
 

662
00:42:10 --> 00:42:16
These are later times. t = 1
is a later time than t = 0.

663
00:42:16 --> 00:42:20
So that's just a very casual,
it's just the way we

664
00:42:20 --> 00:42:22
use these notations.
 

665
00:42:22 --> 00:42:28
Now let me give you the
first example, which is x

666
00:42:28 --> 00:42:40
= a cos t, y = a sin t.
 

667
00:42:40 --> 00:42:42
And the first thing to
figure out is what

668
00:42:42 --> 00:42:45
kind of curve this is.
 

669
00:42:45 --> 00:42:48
And to do that, we want to
figure out what equation it

670
00:42:48 --> 00:42:52
satisfies in rectangular
coordinates.

671
00:42:52 --> 00:42:54
So to figure out what curve
this is, we recognize that

672
00:42:54 --> 00:42:57
if we square and add.
 

673
00:42:57 --> 00:43:00
So we add x ^2 to y^2, we're
going to get something

674
00:43:00 --> 00:43:02
very nice and clean.
 

675
00:43:02 --> 00:43:11
We're going to get a^2 cos
^2 t + a ^2 sin ^2 t.

676
00:43:11 --> 00:43:13
Yeah that's right, OK.
 

677
00:43:13 --> 00:43:19
Which is just a ^2 (cos^2 + sin
^2), or in other words a^2.

678
00:43:19 --> 00:43:23
So lo and behold, what
we have is a circle.

679
00:43:23 --> 00:43:27
And then we know what
shape this is now.

680
00:43:27 --> 00:43:33
And the other thing I'd like
to keep track of is which

681
00:43:33 --> 00:43:35
direction we're going
on the circle.

682
00:43:35 --> 00:43:40
Because there's more to this
parameter then just the shape.

683
00:43:40 --> 00:43:42
There's also where we
are at what time.

684
00:43:42 --> 00:43:46
This would be, think of it like
the trajectory of a planet.

685
00:43:46 --> 00:43:52
So here, I have to do this by
plotting the picture and

686
00:43:52 --> 00:43:53
figuring out what happens.
 

687
00:43:53 --> 00:44:00
So at t = 0, we have (x,
y) is equal to, plug in

688
00:44:00 --> 00:44:06
here (a cos 0, a sin 0).
 

689
00:44:06 --> 00:44:10
Which is just a *
1 + a * 0, so a0.

690
00:44:10 --> 00:44:11
And that's here.
 

691
00:44:11 --> 00:44:14
That's the point (a, 0).
 

692
00:44:14 --> 00:44:18
We know that it's the
circle of radius a.

693
00:44:18 --> 00:44:20
So we know that the curve
is going to go around

694
00:44:20 --> 00:44:22
like this somehow.
 

695
00:44:22 --> 00:44:26
So let's see what
happens at t = pi / 2.

696
00:44:26 --> 00:44:31
So at that point, we
have (x,y) = ( a cos

697
00:44:31 --> 00:44:37
pi / 2, a sin pi / 2).
 

698
00:44:37 --> 00:44:41
Which is (0, a), because
sine of pi / 2 = 1.

699
00:44:41 --> 00:44:43
So that's up here.
 

700
00:44:43 --> 00:44:46
So this is what
happens at t = 0.

701
00:44:46 --> 00:44:49
This is what happens
at t = pi / 2.

702
00:44:49 --> 00:44:51
And the trajectory
clearly goes this way.

703
00:44:51 --> 00:44:54
In fact, this turns out
to be t = pi, etc.

704
00:44:54 --> 00:44:58
And it repeats at t = 2 pi.
 

705
00:44:58 --> 00:45:01
So the other feature that we
have, which is qualitative

706
00:45:01 --> 00:45:11
feature, is that it's
counterclockwise.

707
00:45:11 --> 00:45:17
No the last little bit is
going to be the arc length.

708
00:45:17 --> 00:45:19
Keeping track of
the arc length.

709
00:45:19 --> 00:45:23
And we'll do that next time.
 

710
00:45:23 --> 00:45:24