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PROFESSOR: OK.
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00:00:23 --> 00:00:29
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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00:00:38 --> 00:00:43
tools you'll be using in
multivariable calculus.
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00:00:43 --> 00:00:45
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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00:01:09 --> 00:01:12
If you have a roadway, if you
like, and you have mileage
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00:01:12 --> 00:01:19
markers along the road, like
this, all the way up to, say,
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00:01:19 --> 00:01:27
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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00:01:32 --> 00:01:38
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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00:01:50 --> 00:01:53
The same as we did
with Riemann sums.
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00:01:53 --> 00:02:00
And then the way that we're
going to approximate this is
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00:02:00 --> 00:02:09
by taking the straight lines
between each of these points.
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00:02:09 --> 00:02:12
As things get smaller and
smaller, the straight line
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00:02:12 --> 00:02:15
is going to be fairly
close to the curve.
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And that's the main idea.
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00:02:16 --> 00:02:20
So let me just depict one
little chunk of this.
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Which is like this.
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00:02:20 --> 00:02:24
One straight line, and here's
the curved surface there.
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00:02:24 --> 00:02:26
And the distance along the
curved surface is what I'm
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00:02:26 --> 00:02:32
calling delta s, the change in
the length between, so this
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00:02:32 --> 00:02:34
would be s2 - s1 if I
depicted that one.
39
00:02:34 --> 00:02:42
So this would be delta s
is, say s. si - si - 1,
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00:02:42 --> 00:02:45
some increment there.
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00:02:45 --> 00:02:50
And then I can figure out
what the length of the
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00:02:50 --> 00:02:51
orange segment is.
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00:02:51 --> 00:02:54
Because the horizontal
distance is delta x.
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00:02:54 --> 00:02:57
And the vertical
distance is delta y.
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00:02:57 --> 00:03:02
And so the formula is that
the hypotenuse is delta
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00:03:02 --> 00:03:06
x ^2 + delta y ^2.
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00:03:06 --> 00:03:09
Square root.
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00:03:09 --> 00:03:11
And delta s is
approximately that.
49
00:03:11 --> 00:03:13
So what we're saying
is that delta s ^2 is
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approximately this.
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00:03:16 --> 00:03:21
So this is the hypotenuse.
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00:03:21 --> 00:03:23
Squared.
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And it's very close to
the length of the curve.
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00:03:29 --> 00:03:34
And the whole idea of calculus
is in the infinitesimal,
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this is exactly correct.
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00:03:44 --> 00:03:48
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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00:03:52 --> 00:03:56
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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00:04:20 --> 00:04:20
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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00:04:26 --> 00:04:28
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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00:04:31 --> 00:04:33
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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00:04:46 --> 00:04:47
So it's just a
little different.
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00:04:47 --> 00:04:49
But immediately the first
thing we're going to do
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is take the square root.
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00:04:51 --> 00:04:55
If I take the square root,
that's the square root
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00:04:55 --> 00:04:58
of dx ^2 + dy ^2.
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00:04:58 --> 00:05:02
And this is the form in which I
always remember this formula.
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00:05:02 --> 00:05:07
Let's put it in some
brightly decorated form.
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00:05:07 --> 00:05:12
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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00:05:16 --> 00:05:18
So this is, as I say,
the way I remember it.
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00:05:18 --> 00:05:20
But there are other ways
of thinking of it.
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00:05:20 --> 00:05:23
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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00:05:27 --> 00:05:29
So that looks like this.
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00:05:29 --> 00:05:33
1 + (dy / dx)^2.
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00:05:34 --> 00:05:37
And then I factored out the dx.
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00:05:37 --> 00:05:39
So this is a variant.
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00:05:39 --> 00:05:43
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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00:05:47 --> 00:05:57
So the conclusion is that the
arc length, which if you like
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00:05:57 --> 00:06:06
is this total sn - s0, if you
like, is going to be equal to
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00:06:06 --> 00:06:12
the integral from a to b
of the square root of
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00:06:12 --> 00:06:20
1 + (dy / dx)^2 dx.
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00:06:20 --> 00:06:26
In practice, it's also
very often written
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00:06:26 --> 00:06:29
informally as this.
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00:06:29 --> 00:06:30
The integral ds.
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00:06:30 --> 00:06:34
So the change in this little
variable s, and this is what
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00:06:34 --> 00:06:40
you'll see notationally
in many textbooks.
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00:06:40 --> 00:06:42
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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00:06:50 --> 00:06:52
Mixing in a little bit
of Newton's notation.
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00:06:52 --> 00:06:57
And this is with y = f (x).
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00:06:57 --> 00:07:03
So this is the formula
for arc length.
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00:07:03 --> 00:07:05
And as I say, I
remember it this way.
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00:07:05 --> 00:07:09
But you're going to have to
derive various variants of it.
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00:07:09 --> 00:07:11
And you'll have to use
some arithmetic to get
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00:07:11 --> 00:07:12
to various formulas.
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00:07:12 --> 00:07:15
And there will be more later.
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00:07:15 --> 00:07:16
Yeah, question.
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00:07:16 --> 00:07:20
STUDENT: [INAUDIBLE]
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00:07:20 --> 00:07:27
PROFESSOR: OK, the question
is, is f ' (x)^2 = f '' (x).
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00:07:27 --> 00:07:31
And the answer is no.
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00:07:31 --> 00:07:33
And let's just see what it is.
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00:07:33 --> 00:07:39
So, for example, if f ( x) = x
^2, which is an example which
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00:07:39 --> 00:07:46
will come up in a few minutes,
then f ' (x) = 2x and f ' ( x
126
00:07:46 --> 00:07:52
)^2 = = (2x), which is 4x ^2.
127
00:07:52 --> 00:07:56
Whereas f '' ( x) is equal to,
if I differentiate this another
128
00:07:56 --> 00:07:58
time, it's equal to 2.
129
00:07:58 --> 00:08:03
So they don't mean
the same thing.
130
00:08:03 --> 00:08:04
The same thing over here.
131
00:08:04 --> 00:08:07
You can see this dy / dx,
this is the rate of change
132
00:08:07 --> 00:08:08
of y with respect to x.
133
00:08:08 --> 00:08:09
The quantity squared.
134
00:08:09 --> 00:08:11
So in other words, this
thing is supposed to
135
00:08:11 --> 00:08:13
mean the same as that.
136
00:08:13 --> 00:08:13
Yeah.
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00:08:13 --> 00:08:19
Another question.
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00:08:19 --> 00:08:25
STUDENT: [INAUDIBLE]
139
00:08:25 --> 00:08:28
PROFESSOR: So the question is,
you got a little nervous
140
00:08:28 --> 00:08:30
because I left out
these limits.
141
00:08:30 --> 00:08:32
And indeed, I did that on
purpose because I didn't want
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00:08:32 --> 00:08:34
to specify what was going on.
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00:08:34 --> 00:08:37
Really, if you wrote it in
terms of ds, you'd have to
144
00:08:37 --> 00:08:40
write it as starting at s0 and
ending at sn to be consistent
145
00:08:40 --> 00:08:42
with the variable s.
146
00:08:42 --> 00:08:45
But of course, if you write it
in terms of another variable,
147
00:08:45 --> 00:08:46
you put that variable in.
148
00:08:46 --> 00:08:48
So this is what we do when
we change variables, right?
149
00:08:48 --> 00:08:51
We have many different
choices for these limits.
150
00:08:51 --> 00:08:54
And this is the clue as to
which variable we use.
151
00:08:54 --> 00:08:59
STUDENT: [INAUDIBLE]
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00:08:59 --> 00:09:01
PROFESSOR: Correct. s0
and sn are not the
153
00:09:01 --> 00:09:03
same thing as a and b.
154
00:09:03 --> 00:09:05
In fact, this is xn.
155
00:09:05 --> 00:09:07
And this x0, over here.
156
00:09:07 --> 00:09:08
That's what a and b are.
157
00:09:08 --> 00:09:13
But s0 and sn are mileage
markers on the road.
158
00:09:13 --> 00:09:15
They're not the same thing
as keeping track of what's
159
00:09:15 --> 00:09:16
happening on the x axis.
160
00:09:16 --> 00:09:19
So when we measure arc length,
remember it's mileage
161
00:09:19 --> 00:09:27
along the curved path.
162
00:09:27 --> 00:09:32
So now, I need to give
you some examples.
163
00:09:32 --> 00:09:40
And my first example is
going to be really basic.
164
00:09:40 --> 00:09:45
But I hope that it helps to
give some perspective here.
165
00:09:45 --> 00:09:49
So I'm going to take the
example y = m x, which is
166
00:09:49 --> 00:09:52
a linear function,
a straight line.
167
00:09:52 --> 00:09:58
And then y ' would = m, and so
ds is going to be the square
168
00:09:58 --> 00:10:02
root of 1 + (y ') ^2 dx.
169
00:10:02 --> 00:10:10
Which is the square
root of 1 + m ^2 dx.
170
00:10:10 --> 00:10:17
And now, the length, say, if we
go from, I don't know, let's
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00:10:17 --> 00:10:24
say 0 to 10, let's say.
172
00:10:24 --> 00:10:29
Of the graph is going to be the
integral from 0 to 10 of the
173
00:10:29 --> 00:10:33
square root of 1 + m ^2 dx.
174
00:10:33 --> 00:10:39
Which of course is just 10
square root of 1 + m ^2.
175
00:10:39 --> 00:10:41
Not very surprising.
176
00:10:41 --> 00:10:43
This is a constant.
177
00:10:43 --> 00:10:45
It just factors out and
the integral from 0
178
00:10:45 --> 00:10:51
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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00:10:54 --> 00:10:57
This is something which
has slope m here.
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00:10:57 --> 00:10:59
And it's going to 10.
182
00:10:59 --> 00:11:02
So this horizontal is 10.
183
00:11:02 --> 00:11:05
And the vertical is 10 m.
184
00:11:05 --> 00:11:08
Those are the
dimensions of this.
185
00:11:08 --> 00:11:12
And the Pythagorean theorem
says that the hypotenuse, not
186
00:11:12 --> 00:11:15
surprisingly, let's draw it
in here in orange to remind
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00:11:15 --> 00:11:18
ourselves that it was the same
type of orange that we had over
188
00:11:18 --> 00:11:25
there, this length here is the
square root of 10 ^2 + (10m)^2.
189
00:11:27 --> 00:11:31
That's the formula
for the hypotenuse.
190
00:11:31 --> 00:11:38
And that's exactly
the same as this.
191
00:11:38 --> 00:11:40
Maybe you're saying
duh, this is obvious.
192
00:11:40 --> 00:11:44
But the point that I'm
trying to make is this.
193
00:11:44 --> 00:11:48
If you can figure out these
formulas for linear functions,
194
00:11:48 --> 00:11:52
calculus tells you how to
do it for every function.
195
00:11:52 --> 00:11:56
The idea of calculus is that
this easy calculation here,
196
00:11:56 --> 00:12:01
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
198
00:12:05 --> 00:12:10
integrals, is going to make you
be able to do it for any curve.
199
00:12:10 --> 00:12:12
Because we can break things
up into these little
200
00:12:12 --> 00:12:13
infinitesimal bits.
201
00:12:13 --> 00:12:16
This is the whole idea of all
of the methods that we had
202
00:12:16 --> 00:12:18
to set up integrals here.
203
00:12:18 --> 00:12:25
This is the main point
of these integrals.
204
00:12:25 --> 00:12:32
Now, so let's do something
slightly more interesting.
205
00:12:32 --> 00:12:40
Our next example is going
to be the circle, so y =
206
00:12:40 --> 00:12:48
square root of 1 - x ^2.
207
00:12:48 --> 00:12:51
If you like, that's the
graph of a semicircle.
208
00:12:51 --> 00:12:57
And maybe we'll set
it up here this way.
209
00:12:57 --> 00:13:00
So that the semicircle
goes around like this.
210
00:13:00 --> 00:13:02
And well start it
here at x = 0.
211
00:13:02 --> 00:13:04
And we'll go over to a.
212
00:13:04 --> 00:13:06
And we'll take this little
piece of the circle.
213
00:13:06 --> 00:13:08
So down to here.
214
00:13:08 --> 00:13:11
If you like.
215
00:13:11 --> 00:13:16
So here's the portion of the
circle that I'm going to
216
00:13:16 --> 00:13:17
measure the length of.
217
00:13:17 --> 00:13:19
Now, we know that length.
218
00:13:19 --> 00:13:20
It's called arc length.
219
00:13:20 --> 00:13:21
And I'm going to give it
a name, I'm going to
220
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
00:13:42 --> 00:13:45
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.
225
00:13:47 --> 00:13:50
Now, that's a calculation that
we've done a number of times.
226
00:13:50 --> 00:13:52
And I'm going to do
it slightly faster.
227
00:13:52 --> 00:13:57
But you remember it gives you a
square root in the denominator.
228
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.
246
00:14:55 --> 00:14:58
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