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PROFESSOR: One correction
from last time.

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Sorry to say, I forgot a very
important factor when I was

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telling you what an
average value is.

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If you don't put in that
factor, it's only half off

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on the exam problem that
will be given on this.

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So I would have gotten
half off for missing out

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on this factor, too.
 

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So remember you have to divide
by n here, certainly when

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you're integrating over 0 to
n, the Riemann sum is

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the numerator here.
 

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And if I divide by n on that
side, I've got to divide

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by n on the other side.
 

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This was meant to illustrate
this idea that we're

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dividing by the total here.
 

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And we are going to be
talking about average

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value in more detail.
 

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Not today, though.
 

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So this has to do
with average value.

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And we'll discuss it in
considerable detail in a

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couple of days, I guess.
 

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Now, today I want to continue.
 

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I didn't have time to finish my
discussion of the Fundamental

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Theorem of Calculus 2.
 

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And anyway it's very important
to write it down on the board

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twice, because you want to
see it at least twice.

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And many more times as well.
 

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So let's just remind you, the
second version of the

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Fundamental Theorem of
Calculus says the following.

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It says that the derivative
of an integral gives you

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the function back again.
 

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So here's the theorem.
 

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And the way I'd like to use
it today, I started this

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discussion last time.
 

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But we didn't get into it.
 

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And this is something that's on
your problem set along with

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several other examples.
 

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Is that we can use this to
solve differential equations.

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And in particular, for example,
we can solve the equation y'

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= 1 / x with this formula.
 

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Namely, using an integral.
 

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L ( x ) is the integral
from 1 to x of dt / t.

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The function f ( t
) is just 1 / t.

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Now, that formula can be taken
to be the starting place for

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the derivation of all
the properties of the

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logarithm function.
 

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So what we're going to do right
now is we're going to take this

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to be the definition
of the logarithm.

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And if we do that, then I claim
that we can read off the

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properties of the logarithm
just about as easily

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as we could before.
 

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And so I'll
illustrate that now.

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And there are a few other
examples of this where

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somewhat more unfamiliar
functions come up.

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This one is one that in theory
we know something about.

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The first property of
this function is the one

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that's already given.
 

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Namely, its
derivative is 1 / x.

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And we get a lot of information
just out of the fact that

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its derivative is 1 / x.
 

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The other thing that we need
in order to nail down the

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function, besides its
derivative, is one

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value of the function.
 

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Because it's really not
specified by this equation,

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only specified up to a
constant by this equation.

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But we nail down that constant
when we evaluate it at

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this one place, L ( 1).
 

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And there we're getting
the integral from 1 to 1

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of dt / t, which is 0.
 

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And that's the case with all
these definite integrals.

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If you evaluate them at
their starting places,

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the value will be 0.
 

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And together these two
properties specify this

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function L (x), uniquely.
 

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Now, the next step is to
try to think about what

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its properties are.
 

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And the first approach to that,
and this is the approach that

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we always take, is to maybe
graph the function, to

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get a feeling for it.
 

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And so I'm going to take
the second derivative.

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Now, notice that when you have
a function which is given as an

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integral, its first derivative
is really easy to compute.

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And then its second derivative,
well, you have to differentiate

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whatever you get.
 

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So it may or may not be easy.
 

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But anyway, it's a lot harder
in the case when I start

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with a function to get to
the second derivative.

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Here it's relatively easy.
 

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And these are the properties
that I'm going to use.

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I won't really use very much
more about it than that.

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And qualitatively, the
conclusions that we can draw

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from this are, first of all,
from this, for example we

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see that this thing is
concave down every place.

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And then to get started with
the graph, since I see I have a

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value here, which is L( 1) =
0, I'm going to throw in

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the value of the slope.
 

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So L' ( 1), which I know is 1
of 1, that's reading off from

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this equation here,
so that's 1.

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And now I'm ready to sketch at
least a part of the curve.

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So here's a sketch
of the graph.

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Here's the point (1, 0);
that is, x = 1, y = 0.

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And the tangent line,
I know, has slope 1.

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And the curve is concave down.
 

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So it's going to look
something like this.

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Incidentally, it's
also increasing.

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And that's an important
property, it's

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strictly increasing.
 

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That's because L'
( x) is positive.

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And so, we can get from
this the following

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important definition.
 

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Which, again, is working
backwards from this definition.

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We can get to where we
started with a log in

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our previous discussion.
 

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Namely, if I take the level
here, which is y = 1, then that

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crosses the axis someplace.
 

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And his point is what we're
going to define as e.

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So the definition of e
is that it's the value

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such that L ( e) = 1.
 

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And again, the fact that
there's exactly one such place

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just comes from the fact that
this L' is positive, so

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that L is increasing.
 

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No, there's just one other
feature of this graph that I'm

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going to emphasize to you.
 

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There's one other thing which
I'm not going to check, which

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you would ordinarily
do with graphs.

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Once it's increasing there are
no critical points, so the only

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other interesting
thing is the n's.

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And it turns out that the
limit as you go down to

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0 is minus infinity.
 

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As you go over to the right
here it's plus infinity.

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It does get arbitrarily
high; it doesn't level off.

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But I'm not going to
discuss that here.

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Instead, I'm going to just
remark on one qualitative

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feature of the graph, which is
this remark that the part which

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is to the left of 1 is below 0.
 

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00:08:06 --> 00:08:17
So I just want to remark, why
is L (x) negative for x < 1.

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00:08:17 --> 00:08:19
Maybe I don't have room
for that, so I'll just

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put in here x < 1.
 

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I want to give you two reasons.
 

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Again, we're only working from
very first principles here.

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00:08:27 --> 00:08:33
Just that the property that
L' = 1 / x, and L (1) = 0.

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00:08:33 --> 00:08:39
So our first reason is that,
well, I just said it.

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L ( 1) = 0.
 

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00:08:41 --> 00:08:46
And L is increasing.
 

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00:08:46 --> 00:08:49
And if you read that
backwards, if it gets up

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00:08:49 --> 00:08:54
to 0 here, it must have
been negative before 0.

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00:08:54 --> 00:08:57
So this is one way of seeing
that L ( x) is negative.

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00:08:57 --> 00:09:02
There's a second way of seeing
it, which is equally important.

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And it has to do with just
manipulation of integrals.

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Here I'm going to start out
with L (x), and its definition.

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Which is the integral
from 1 to x dt / t.

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00:09:15 --> 00:09:19
And now I'm going to reverse
the order of integration.

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00:09:19 --> 00:09:21
This is the same, by our
definition of our properties of

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integrals, as the integral from
x to 1 with a minus

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sign dt / t.
 

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00:09:30 --> 00:09:34
Now, I can tell that this
quantity is negative.

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And the reason that I can tell
is that this chunk of it here,

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this piece of it, is
a positive number.

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This part is positive.
 

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00:09:46 --> 00:09:51
And this part is
positive because x < 1.

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00:09:51 --> 00:09:53
So the lower limit is less
than the upper limit, and

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00:09:53 --> 00:09:54
so this is interpreted.
 

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00:09:54 --> 00:09:58
The thing in the green box
is interpreted as an area.

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00:09:58 --> 00:09:58
It's an area.
 

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00:09:58 --> 00:10:03
And so negative a positive
quantity is negative, minus a

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positive quantity's negative.
 

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00:10:08 --> 00:10:13
So both of these work perfectly
well as interpretations.

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And it's just to illustrate
what we can do.

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Now, there's one more
manipulation of integrals

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00:10:18 --> 00:10:23
that gives us the fanciest
property of the ln.

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00:10:23 --> 00:10:26
And that's the last one
that I'm going to do.

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00:10:26 --> 00:10:29
And you have a similar
thing on your homework.

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00:10:29 --> 00:10:33
So I'm going to prove that this
is, as I say, the fanciest

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00:10:33 --> 00:10:34
property of the log.
 

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00:10:34 --> 00:10:38
On your homework, by the way,
you're going to check that

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L( 1 / x) = - L ( x).
 

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00:10:46 --> 00:10:52
But we'll do this one.
 

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00:10:52 --> 00:10:56
The idea is just to plug in the
formula and see what it gives.

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00:10:56 --> 00:11:02
On the left-hand side, I
have 1 to ab, dt / t.

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00:11:02 --> 00:11:06
That's L ( ab).
 

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00:11:06 --> 00:11:08
And then that's certainly
equal to the left-hand side.

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00:11:08 --> 00:11:12
And then I'm going to now
split this into two pieces.

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00:11:12 --> 00:11:14
Again, this is a
property of integrals.

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00:11:14 --> 00:11:19
That if you have an integral
from one place to another, you

192
00:11:19 --> 00:11:20
can break it up into pieces.
 

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00:11:20 --> 00:11:26
So I'm going to start
at 1 but then go to a.

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00:11:26 --> 00:11:33
And then I'm going to
continue from a to ab.

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00:11:33 --> 00:11:36
So this is the question
that we have.

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We haven't proved this.
 

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00:11:38 --> 00:11:41
Well, this one is
actually true.

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00:11:41 --> 00:11:43
If we want this to be true,
we know by definition

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L ( ab) is this.
 

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We know, we can see it,
that L(a) is this.

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00:11:48 --> 00:11:52
So the question that this
boils down to is, we

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want to know that these
two things are equal.

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00:11:54 --> 00:12:01
We want to know that L ( b) is
that other integral there.

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00:12:01 --> 00:12:04
So let's check it.
 

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00:12:04 --> 00:12:06
I'm going to rewrite
the integral.

206
00:12:06 --> 00:12:09
It's the integral from -
sorry, from lower limit a to

207
00:12:09 --> 00:12:13
upper limit ab of dt / t.
 

208
00:12:13 --> 00:12:17
And now, again, to illustrate
properties of integrals, the

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00:12:17 --> 00:12:20
key property here that we're
going to have to use is

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00:12:20 --> 00:12:23
change of variables.
 

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00:12:23 --> 00:12:26
This is a kind of a scaled
integral where everything is

212
00:12:26 --> 00:12:29
multiplied by a factor of a
from what we want to get

213
00:12:29 --> 00:12:32
to this L ( b) quantity.
 

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00:12:32 --> 00:12:38
And so this suggests that
we write down t = a u.

215
00:12:38 --> 00:12:40
That's going to be our trick.
 

216
00:12:40 --> 00:12:42
And if I use that new
variable u, then the

217
00:12:42 --> 00:12:50
change in t, dt, is adu.
 

218
00:12:50 --> 00:12:53
And as a result, I can write
this as equal to an integral

219
00:12:53 --> 00:12:58
from, let's see, dt = adu.
 

220
00:12:58 --> 00:13:00
And t = au.
 

221
00:13:00 --> 00:13:05
So I've now substituted
in for the integrand.

222
00:13:05 --> 00:13:09
But on top of this, with
definite integrals, we also

223
00:13:09 --> 00:13:13
have to check the limits.
 

224
00:13:13 --> 00:13:17
And the limits work
out as follows.

225
00:13:17 --> 00:13:20
When t = a, that's
the lower limit.

226
00:13:20 --> 00:13:22
Let's just take a
look. t = a u.

227
00:13:22 --> 00:13:26
So that means that u
is equal to, what?

228
00:13:26 --> 00:13:28
It's 1.
 

229
00:13:28 --> 00:13:31
Because a * 1 = a.
 

230
00:13:31 --> 00:13:34
So if t = a, this
is if and only if.

231
00:13:34 --> 00:13:36
So this lower limit, which
really in disguise was where

232
00:13:36 --> 00:13:45
t = a, becomes where u = 1.
 

233
00:13:45 --> 00:13:53
And similarly, when
t = ab, u = b.

234
00:13:53 --> 00:13:56
So the upper limit here is b.
 

235
00:13:56 --> 00:14:01
And now, if you notice,
we're just going to cancel

236
00:14:01 --> 00:14:02
these two factors here.
 

237
00:14:02 --> 00:14:07
And now we recognize that
this is just the same as

238
00:14:07 --> 00:14:09
the definition of L ( b).
 

239
00:14:09 --> 00:14:13
Because L ( x) is over
here in the box.

240
00:14:13 --> 00:14:16
And the fact that I use the
letter t there is irrelevant;

241
00:14:16 --> 00:14:18
it works equally well
with the letter u.

242
00:14:18 --> 00:14:22
So this is just L (b).
 

243
00:14:22 --> 00:14:33
Which is what we
wanted to show.

244
00:14:33 --> 00:14:35
So that's an example, and you
have one in your homework,

245
00:14:35 --> 00:14:45
which is a little similar.
 

246
00:14:45 --> 00:14:49
Now, the last example, that I'm
going to discuss of this type,

247
00:14:49 --> 00:14:51
I already mentioned last time.
 

248
00:14:51 --> 00:14:54
Which is the function F ( x),
which is the integral from

249
00:14:54 --> 00:14:58
0 to x of e ^ -t^2 dt.
 

250
00:14:58 --> 00:15:04
This one is even more exotic
because unlike the logarithm

251
00:15:04 --> 00:15:06
it's a new function.
 

252
00:15:06 --> 00:15:10
It really is not any function
that you can express in terms

253
00:15:10 --> 00:15:13
of the functions that
we know already.

254
00:15:13 --> 00:15:19
And the approach, always, to
these new functions is to think

255
00:15:19 --> 00:15:22
of what their properties are.
 

256
00:15:22 --> 00:15:24
And the way we think of
functions in order to

257
00:15:24 --> 00:15:26
understand them is to
maybe sketch them.

258
00:15:26 --> 00:15:29
And so I'm going to do exactly
the same thing I did over here.

259
00:15:29 --> 00:15:31
So, what is it that I
can get out of this?

260
00:15:31 --> 00:15:35
Well, immediately I can figure
out what the derivative is.

261
00:15:35 --> 00:15:38
I read it off from the
fundamental theorem.

262
00:15:38 --> 00:15:41
It's this.
 

263
00:15:41 --> 00:15:45
I also can figure out the
value at the starting place.

264
00:15:45 --> 00:15:48
In this case, the
starting place is 0.

265
00:15:48 --> 00:15:53
And the value is 0.
 

266
00:15:53 --> 00:15:57
And I should check the second
derivative, which is also not

267
00:15:57 --> 00:15:59
so difficult to compute.
 

268
00:15:59 --> 00:16:03
The second derivative
is - 2x e ^ x^2.

269
00:16:06 --> 00:16:11
And so now I can see that this
function is increasing, because

270
00:16:11 --> 00:16:14
this derivative is positive,
it's always increasing.

271
00:16:14 --> 00:16:17
And it's going to be concave
down when x is positive and

272
00:16:17 --> 00:16:20
concave up when x is negative.
 

273
00:16:20 --> 00:16:24
Because there's a minus sign
here, so the sign is negative.

274
00:16:24 --> 00:16:29
This is less than 0 when x is
positive and greater than

275
00:16:29 --> 00:16:36
0 when x is negative.
 

276
00:16:36 --> 00:16:40
And maybe to get started
I'll remind you F( 0), 0.

277
00:16:40 --> 00:16:45
It's also true that F' ( 0),
that just comes right out of

278
00:16:45 --> 00:16:52
this, F' ( 0) = e ^
- 0^2 which is 1.

279
00:16:52 --> 00:16:55
That means the tangent
line again has slope 1.

280
00:16:55 --> 00:16:57
We do this a lot
with functions.

281
00:16:57 --> 00:17:00
We normalize them so that their
slopes of their tangent lines

282
00:17:00 --> 00:17:03
are 1 at convenient spots.
 

283
00:17:03 --> 00:17:06
So here's the tangent
line of slope 1.

284
00:17:06 --> 00:17:10
We know this thing is concave
down to the right and

285
00:17:10 --> 00:17:14
concave up to the left.
 

286
00:17:14 --> 00:17:17
And so it's going to look
something like this.

287
00:17:17 --> 00:17:20
With an inflection point.
 

288
00:17:20 --> 00:17:26
Right?
 

289
00:17:26 --> 00:17:32
Now, I want to say one more,
make one more remark about

290
00:17:32 --> 00:17:34
this function, or maybe two
more remarks about this

291
00:17:34 --> 00:17:36
function, before we go on.
 

292
00:17:36 --> 00:17:39
Really, you want to know this
graph as well as possible.

293
00:17:39 --> 00:17:42
And so there are just a
couple more features.

294
00:17:42 --> 00:17:45
And one is enormously helpful
because it cuts in half all of

295
00:17:45 --> 00:17:49
the work that you have. and
that is the property that turns

296
00:17:49 --> 00:17:51
out that this function is i.
 

297
00:17:51 --> 00:17:56
Namely, F( - x) = - F(x).
 

298
00:17:56 --> 00:18:01
That's what's known
as an odd function.

299
00:18:01 --> 00:18:06
Now, the reason why it's odd is
that it's the antiderivative

300
00:18:06 --> 00:18:08
of something that's even.
 

301
00:18:08 --> 00:18:10
This function in here is even.
 

302
00:18:10 --> 00:18:14
And we nailed it down
so that it was 0 at 0.

303
00:18:14 --> 00:18:17
Another way of interpreting
that, and let me show it to you

304
00:18:17 --> 00:18:20
underneath, is the following.
 

305
00:18:20 --> 00:18:24
When we look at its derivative,
its derivative, course,

306
00:18:24 --> 00:18:28
is the function e ^ x.
 

307
00:18:28 --> 00:18:30
Sorry, e ^ - x ^2.
 

308
00:18:30 --> 00:18:37
So that's this shape here.
 

309
00:18:37 --> 00:18:41
And you can see the slope is
0, but fairly close to 0,

310
00:18:41 --> 00:18:42
but positive along here.
 

311
00:18:42 --> 00:18:44
It's getting, this is
its steepest point.

312
00:18:44 --> 00:18:46
This is the highest point here.
 

313
00:18:46 --> 00:18:48
And then it's leveling off
again; the slope is going

314
00:18:48 --> 00:18:50
down, always positive.
 

315
00:18:50 --> 00:18:55
This is the graph
of F' = e ^ -x^2.

316
00:18:56 --> 00:19:02
Now, the interpretation of the
function that's up above is

317
00:19:02 --> 00:19:08
that the value here is
the area from 0 to x.

318
00:19:08 --> 00:19:12
So this is area F(x).
 

319
00:19:12 --> 00:19:16
Maybe I'll color it in,
decorate it a little bit.

320
00:19:16 --> 00:19:25
So this area here is F ( x).
 

321
00:19:25 --> 00:19:29
Now, I want to show you
this odd property, by

322
00:19:29 --> 00:19:30
using this symmetry.
 

323
00:19:30 --> 00:19:35
The graph here is even, so in
other words, what's back here

324
00:19:35 --> 00:19:39
is exactly the same
as what's forward.

325
00:19:39 --> 00:19:42
But now there's a reversal.
 

326
00:19:42 --> 00:19:45
Because we're keeping track
of the area starting

327
00:19:45 --> 00:19:46
from 0 going forward.
 

328
00:19:46 --> 00:19:47
That's positive.
 

329
00:19:47 --> 00:19:49
If we go backwards, it's
counted negatively.

330
00:19:49 --> 00:19:52
So if we went backwards to - x,
we'd get exactly the same as

331
00:19:52 --> 00:19:54
that green patch over there.
 

332
00:19:54 --> 00:19:56
We'd get a red patch over here.
 

333
00:19:56 --> 00:20:01
But it would be
counted negatively.

334
00:20:01 --> 00:20:04
And that's the property
that it's odd.

335
00:20:04 --> 00:20:06
You can also check
this by properties of

336
00:20:06 --> 00:20:08
integrals directly.
 

337
00:20:08 --> 00:20:16
That would be just like
this process here.

338
00:20:16 --> 00:20:18
So it's completely
analogous to checking

339
00:20:18 --> 00:20:25
this formula over there.
 

340
00:20:25 --> 00:20:29
So that's one of the comments
I wanted to make about this.

341
00:20:29 --> 00:20:32
And why does this save
us a lot of time, if

342
00:20:32 --> 00:20:33
we know this is odd?
 

343
00:20:33 --> 00:20:36
Well, we know that the shape of
this branch is exactly the

344
00:20:36 --> 00:20:38
reverse, or the reflection,
if you like, of the

345
00:20:38 --> 00:20:39
shape of this one.
 

346
00:20:39 --> 00:20:42
What we want to do is flip
it under the axis and then

347
00:20:42 --> 00:20:44
reflect it over that way.
 

348
00:20:44 --> 00:20:53
And that's the symmetry
property of the graph of F(x).

349
00:20:53 --> 00:20:57
Now, the last property that
I want to mention is what's

350
00:20:57 --> 00:21:00
happening with the ends.
 

351
00:21:00 --> 00:21:03
And at the end there's
an asymptote, there's

352
00:21:03 --> 00:21:05
a limit here.
 

353
00:21:05 --> 00:21:10
So this is an asymptote.
 

354
00:21:10 --> 00:21:12
And the same thing down here,
which will be exactly because

355
00:21:12 --> 00:21:15
of the odd feature, this'll
be exactly negative.

356
00:21:15 --> 00:21:19
The opposite value over here.
 

357
00:21:19 --> 00:21:23
And you might ask yourself,
what level is this, exactly.

358
00:21:23 --> 00:21:26
Now, that level turns out to
be a very important quantity.

359
00:21:26 --> 00:21:29
It's interpreted down here
as the area under this

360
00:21:29 --> 00:21:31
whole infinite stretch.
 

361
00:21:31 --> 00:21:36
It's all the way
out to infinity.

362
00:21:36 --> 00:21:38
So, let's see.
 

363
00:21:38 --> 00:21:48
What do you think it is?
 

364
00:21:48 --> 00:21:49
You're all clueless.
 

365
00:21:49 --> 00:21:52
Well, maybe not all of you,
you're just afraid to say.

366
00:21:52 --> 00:21:54
So it's obvious.
 

367
00:21:54 --> 00:21:57
It's the square root of pi/2.
 

368
00:21:57 --> 00:22:00
That was right on the tip
of your tongue, wasn't it?

369
00:22:00 --> 00:22:01
STUDENT: Ah, yes.
 

370
00:22:01 --> 00:22:04
PROFESSOR: Right, so this is
actually very un-obvious, but

371
00:22:04 --> 00:22:06
it's a very important quantity.
 

372
00:22:06 --> 00:22:08
And it's an amazing
fact that this thing

373
00:22:08 --> 00:22:10
approaches this number.
 

374
00:22:10 --> 00:22:18
And it's something that people
worried about for many years

375
00:22:18 --> 00:22:22
before actually nailing down.
 

376
00:22:22 --> 00:22:26
And so what I just claimed
here is that the limit as x

377
00:22:26 --> 00:22:33
approaches infinity of F ( x)
= the square root of pi / 2.

378
00:22:33 --> 00:22:35
And similarly, if you do it to
minus infinity, you'll get

379
00:22:35 --> 00:22:38
minus square root of pi/2.
 

380
00:22:38 --> 00:22:42
And for this reason, people
introduce a new function

381
00:22:42 --> 00:22:44
because they like the number 1.
 

382
00:22:44 --> 00:22:49
This function is erf,
short for error function.

383
00:22:49 --> 00:22:53
And it's 2 / the square root
of pi times the integral

384
00:22:53 --> 00:22:57
from 0 to x, e ^ - t ^2 dt.
 

385
00:22:57 --> 00:23:00
In other words, it's just our
original, our previous function

386
00:23:00 --> 00:23:08
multiplied by 2 / the
square root of pi.

387
00:23:08 --> 00:23:10
And that's the function which
gets tabulated quite a lot.

388
00:23:10 --> 00:23:14
You'll see it on the internet
everywhere, and it's a

389
00:23:14 --> 00:23:15
very important function.
 

390
00:23:15 --> 00:23:18
There are other normalizations
that are used, and the

391
00:23:18 --> 00:23:20
discussions of the other
normalizations are

392
00:23:20 --> 00:23:23
in your problems.
 

393
00:23:23 --> 00:23:25
This is one of them,
and another one is

394
00:23:25 --> 00:23:27
in your exercises.
 

395
00:23:27 --> 00:23:31
The standard normal
distribution.

396
00:23:31 --> 00:23:33
There are tons of functions
like this, which are new

397
00:23:33 --> 00:23:36
functions that we can get
at once we have the

398
00:23:36 --> 00:23:37
tool of integrals.
 

399
00:23:37 --> 00:23:40
And I'll write down just one
or two more, just so that

400
00:23:40 --> 00:23:42
you'll see the variety.
 

401
00:23:42 --> 00:23:49
Here's one which is called
a Fresnel integral.

402
00:23:49 --> 00:23:51
On your problem set next
week, we'll do the other

403
00:23:51 --> 00:23:57
Fresnel integral, we'll
look at this one.

404
00:23:57 --> 00:24:01
These functions cannot be
expressed in elementary terms.

405
00:24:01 --> 00:24:11
The one on your homework for
this week was this one.

406
00:24:11 --> 00:24:14
This one comes up in
Fourier analysis.

407
00:24:14 --> 00:24:19
And I'm going to just tell you
maybe one more such function.

408
00:24:19 --> 00:24:23
There's a function which is
called Li ( x), logarithmic

409
00:24:23 --> 00:24:30
integral of x,
which is this guy.

410
00:24:30 --> 00:24:33
The reciprocal of the
logarithm, the natural log.

411
00:24:33 --> 00:24:38
And the significance of this
one is that Li ( x) is

412
00:24:38 --> 00:24:49
approximately equal to the
number of primes < x.

413
00:24:49 --> 00:24:54
And, in fact, if you can make
this as precise as possible,

414
00:24:54 --> 00:24:59
you'll be famous for millennia,
because this is known as

415
00:24:59 --> 00:25:01
the Riemann hypothesis.
 

416
00:25:01 --> 00:25:05
Exactly how closely this
approximation occurs.

417
00:25:05 --> 00:25:09
But it's a hard problem, and
already a century ago the

418
00:25:09 --> 00:25:12
prime number theorem, which
established this connection

419
00:25:12 --> 00:25:18
was extremely important
to progress in math.

420
00:25:18 --> 00:25:19
Yeah, question.
 

421
00:25:19 --> 00:25:21
STUDENT: [INAUDIBLE]
 

422
00:25:21 --> 00:25:23
PROFESSOR: Is this stuff you're
supposed to understand.

423
00:25:23 --> 00:25:24
That's a good question.
 

424
00:25:24 --> 00:25:26
I love that question.
 

425
00:25:26 --> 00:25:31
The answer is, this is,
so we launched off

426
00:25:31 --> 00:25:32
into something here.
 

427
00:25:32 --> 00:25:34
And let me just
explain it to you.

428
00:25:34 --> 00:25:37
I'm going to be talking a
fair amount more about this

429
00:25:37 --> 00:25:41
particular function, because
it's associated to the

430
00:25:41 --> 00:25:43
normal distribution.
 

431
00:25:43 --> 00:25:45
And I'm going to let you
get familiar with it.

432
00:25:45 --> 00:25:47
What I'm doing here
is purely cultural.

433
00:25:47 --> 00:25:51
Well, after this panel, what
I'm doing is purely cultural.

434
00:25:51 --> 00:25:54
Just saying there's a
lot of other beasts out

435
00:25:54 --> 00:25:55
there in the world.
 

436
00:25:55 --> 00:25:57
And one of them is called
C of [INAUDIBLE].

437
00:25:57 --> 00:26:01
So we'll have a just a very
passing familiarity with one

438
00:26:01 --> 00:26:02
or two of these functions.
 

439
00:26:02 --> 00:26:05
But there are literally
dozens and dozens of them.

440
00:26:05 --> 00:26:09
The only thing that you'll need
to do with such functions is

441
00:26:09 --> 00:26:13
things like understanding the
derivative, the second

442
00:26:13 --> 00:26:16
derivative, and tracking
what the function does.

443
00:26:16 --> 00:26:18
Sketching the same way you
did with any other tool.

444
00:26:18 --> 00:26:22
So we're going to do this type
of thing with these functions.

445
00:26:22 --> 00:26:25
And I'll have to
lead you through.

446
00:26:25 --> 00:26:27
If I wanted to ask you a
question about one of these

447
00:26:27 --> 00:26:32
functions, I have to tell you
exactly what I'm aiming for.

448
00:26:32 --> 00:26:35
Yeah, another question.
 

449
00:26:35 --> 00:26:36
STUDENT: [INAUDIBLE]
 

450
00:26:36 --> 00:26:36
PROFESSOR: Yeah, I did.
 

451
00:26:36 --> 00:26:43
I called these guys
Fresnel integrals.

452
00:26:43 --> 00:26:47
The guy's name is Fresnel.
 

453
00:26:47 --> 00:26:49
It's just named after a person.
 

454
00:26:49 --> 00:26:52
But, and this one, Li's
logarithmic integral, it's

455
00:26:52 --> 00:26:53
not named after a person.
 

456
00:26:53 --> 00:26:56
Logarithm is not
somebody's name.

457
00:26:56 --> 00:27:01
So look, in fact this will be
mentioned also on a problem

458
00:27:01 --> 00:27:04
set, but I don't expect you
to remember these names.

459
00:27:04 --> 00:27:06
In particular, that you
definitely don't want

460
00:27:06 --> 00:27:07
to try to remember.
 

461
00:27:07 --> 00:27:08
Yes, another question.
 

462
00:27:08 --> 00:27:10
STUDENT: [INAUDIBLE]
 

463
00:27:10 --> 00:27:15
PROFESSOR: The question is,
will we prove this limit.

464
00:27:15 --> 00:27:17
And the answer is yes,
if we have time.

465
00:27:17 --> 00:27:21
It'll be in about a week or so.
 

466
00:27:21 --> 00:27:22
We're not going to do it now.
 

467
00:27:22 --> 00:27:29
It takes us quite a
bit of work to do it.

468
00:27:29 --> 00:27:32
OK.
 

469
00:27:32 --> 00:27:36
I'm going to change years now,
I'm going to shift gears.

470
00:27:36 --> 00:27:41
And we're going to go back
to a more standard thing

471
00:27:41 --> 00:27:44
which has to do with just
setting up integrals.

472
00:27:44 --> 00:27:47
And this has to do with
understanding where integrals

473
00:27:47 --> 00:27:50
play a role, and they play a
role in cumulative sums,

474
00:27:50 --> 00:27:52
in evaluating things.
 

475
00:27:52 --> 00:27:55
This is much more closely
associated with the first

476
00:27:55 --> 00:27:57
Fundamental Theorem.
 

477
00:27:57 --> 00:27:59
That is, we'll take, today
we were talking about how

478
00:27:59 --> 00:28:02
integrals are formulas
for functions.

479
00:28:02 --> 00:28:04
Or solutions to
differential equations.

480
00:28:04 --> 00:28:09
We're going to go back and talk
about integrals as being the

481
00:28:09 --> 00:28:14
answer to a question as opposed
to. what we've done now.

482
00:28:14 --> 00:28:19
So in other words, and the
first example, or most of the

483
00:28:19 --> 00:28:22
examples for now, are going
to be taken from geometry.

484
00:28:22 --> 00:28:27
Later on we'll get
to probability.

485
00:28:27 --> 00:28:44
And the first topic is just
areas between curves.

486
00:28:44 --> 00:28:46
Here's the idea.
 

487
00:28:46 --> 00:28:50
If you have a couple of curves
that look like this and maybe

488
00:28:50 --> 00:28:54
like this, and you want to
start at a place a and you want

489
00:28:54 --> 00:29:03
to end at a place b, then you
can chop it up the same way

490
00:29:03 --> 00:29:06
we did with Riemann sums.
 

491
00:29:06 --> 00:29:10
And take a chunk that
looks like this.

492
00:29:10 --> 00:29:12
And I'm going to write the
thickness of that chunk.

493
00:29:12 --> 00:29:14
Well, let's give
these things names.

494
00:29:14 --> 00:29:17
Let's say the top curve
is f(x), and the bottom

495
00:29:17 --> 00:29:20
curve is g ( x).
 

496
00:29:20 --> 00:29:26
And then this thickness
is going to be dx.

497
00:29:26 --> 00:29:30
That's the thickness.
 

498
00:29:30 --> 00:29:32
And what is the height?
 

499
00:29:32 --> 00:29:34
Well, the height is the
difference between the top

500
00:29:34 --> 00:29:38
value and the bottom value.
 

501
00:29:38 --> 00:29:44
So here we have (f
( x) - g ( x)) dx.

502
00:29:44 --> 00:29:50
This is, if you like, base
times - whoops, backwards.

503
00:29:50 --> 00:29:54
This is height, and this is
the base of the rectangle.

504
00:29:54 --> 00:29:56
And these are
approximately correct.

505
00:29:56 --> 00:29:59
But of course, only in limit
when this is an infinitesimal,

506
00:29:59 --> 00:30:03
is it exactly right.
 

507
00:30:03 --> 00:30:10
In order to get the whole area,
I have add these guys up.

508
00:30:10 --> 00:30:11
So I'm going to
integrate from a to b.

509
00:30:11 --> 00:30:14
That's summing them,
that's adding them up.

510
00:30:14 --> 00:30:16
And that's going to be my area.
 

511
00:30:16 --> 00:30:27
So that's the story here.
 

512
00:30:27 --> 00:30:30
Now, let me just say
two things about this.

513
00:30:30 --> 00:30:33
First of all, on a very
abstract level before we get

514
00:30:33 --> 00:30:36
started with details of
more complicated problems.

515
00:30:36 --> 00:30:40
The first one is that every
problem that I'm going to be

516
00:30:40 --> 00:30:44
talking about from now on for,
several days, involves the

517
00:30:44 --> 00:30:48
following collection of,
the following goals.

518
00:30:48 --> 00:30:52
I want to identify
something to integrate.

519
00:30:52 --> 00:30:58
That's called an integrand.
 

520
00:30:58 --> 00:31:06
And I want to identify what
are known as the limits.

521
00:31:06 --> 00:31:11
The whole game is simply to
figure out what a, b, and

522
00:31:11 --> 00:31:13
this quantity is here.
 

523
00:31:13 --> 00:31:15
Whatever it is.
 

524
00:31:15 --> 00:31:17
And the minute we have
that, we can calculate

525
00:31:17 --> 00:31:19
the integral if we like.
 

526
00:31:19 --> 00:31:22
We have numerical procedures
or maybe we have analytic

527
00:31:22 --> 00:31:25
procedures, but anyway we
can get at the integral.

528
00:31:25 --> 00:31:27
The goal here is
to set them up.

529
00:31:27 --> 00:31:30
And in order to set
them up, you must know

530
00:31:30 --> 00:31:31
these three things.
 

531
00:31:31 --> 00:31:33
The lower limit, the
upper limit, and what

532
00:31:33 --> 00:31:37
we're integrating.
 

533
00:31:37 --> 00:31:42
If you leave one of these out,
it's like the following thing.

534
00:31:42 --> 00:31:45
I ask you what the area
of this region is.

535
00:31:45 --> 00:31:48
If I left out this end, how
could I possibly know?

536
00:31:48 --> 00:31:51
I don't even know where it
starts, so how can I figure

537
00:31:51 --> 00:31:52
out what this area is
if I haven't identified

538
00:31:52 --> 00:31:55
what the left side is.
 

539
00:31:55 --> 00:31:58
I can't leave out the bottom.
 

540
00:31:58 --> 00:32:00
It's sitting here,
in this formula.

541
00:32:00 --> 00:32:03
Because I need to
know where it is.

542
00:32:03 --> 00:32:05
And I need to know the top and
I need to know this side.

543
00:32:05 --> 00:32:07
Those are the four
sides of the figure.

544
00:32:07 --> 00:32:10
If I don't incorporate them
into the information, I'll

545
00:32:10 --> 00:32:11
never get anything out.
 

546
00:32:11 --> 00:32:13
So I need to know everything.
 

547
00:32:13 --> 00:32:15
And I need to know exactly
those things, in order to

548
00:32:15 --> 00:32:20
have a formula for the area.
 

549
00:32:20 --> 00:32:24
Now, when this gets carried out
in practice, as we will do now

550
00:32:24 --> 00:32:29
in our first example, it's more
complicated than it looks.

551
00:32:29 --> 00:32:44
So here's our first example:
Find the area between x

552
00:32:44 --> 00:32:57
= y ^2 and y = x - 2.
 

553
00:32:57 --> 00:33:00
This is our first example.
 

554
00:33:00 --> 00:33:04
Let me make sure that I chose
the example that I wanted to.

555
00:33:04 --> 00:33:08
Yeah.
 

556
00:33:08 --> 00:33:20
Now, there's a first step in
figuring these things out.

557
00:33:20 --> 00:33:27
And this is that you
must draw a picture.

558
00:33:27 --> 00:33:30
If you don't draw a picture
you'll never figure out what

559
00:33:30 --> 00:33:32
this area is, because you'll
never figure out what's

560
00:33:32 --> 00:33:36
what between these curves.
 

561
00:33:36 --> 00:33:40
The first curve, y =
x ^2, is a parabola.

562
00:33:40 --> 00:33:42
But x is a function of y.
 

563
00:33:42 --> 00:33:45
It's pointing this way.
 

564
00:33:45 --> 00:33:47
So it's this parabola here.
 

565
00:33:47 --> 00:33:50
That's y = x^2.
 

566
00:33:50 --> 00:33:57
Whoops, x = y ^2.
 

567
00:33:57 --> 00:34:06
The second curve is a line, a
straight line of slope 1,

568
00:34:06 --> 00:34:09
starting at (x = 2, y = 0).
 

569
00:34:09 --> 00:34:14
It goes through this place
here, which is 2 over and has

570
00:34:14 --> 00:34:20
slope 1, so it does this.
 

571
00:34:20 --> 00:34:23
And this shape in here is
what we mean by the area

572
00:34:23 --> 00:34:24
between the curves.
 

573
00:34:24 --> 00:34:27
Now that we see what it
is, we have a better idea

574
00:34:27 --> 00:34:28
of what our goal is.
 

575
00:34:28 --> 00:34:39
If you haven't drawn
it, you have no hope.

576
00:34:39 --> 00:34:43
Now, I'm going to describe
two ways of getting

577
00:34:43 --> 00:34:50
at this area here.
 

578
00:34:50 --> 00:35:00
And the first one is motivated
by the shape that I just

579
00:35:00 --> 00:35:02
described right here.
 

580
00:35:02 --> 00:35:07
Namely, I'm going to use it
in a straightforward way.

581
00:35:07 --> 00:35:12
I'm going to chop things up
into these vertical pieces

582
00:35:12 --> 00:35:17
just as I did right there.
 

583
00:35:17 --> 00:35:19
Now, here's the
difficulty with that.

584
00:35:19 --> 00:35:26
The difficulty is that the
upper curve here has one

585
00:35:26 --> 00:35:29
formula but the lower curve
shifts from being a part of

586
00:35:29 --> 00:35:33
the parabola to being a
part of the straight line.

587
00:35:33 --> 00:35:35
That means that there are
two different formulas

588
00:35:35 --> 00:35:36
for the lower function.
 

589
00:35:36 --> 00:35:40
And the only way to accommodate
that is to separate this

590
00:35:40 --> 00:35:42
up into two halves.
 

591
00:35:42 --> 00:35:44
Separate it out
into two halves.

592
00:35:44 --> 00:35:50
I'm going to have to
divide it right here.

593
00:35:50 --> 00:35:54
So we must break it into two
pieces and find the integral of

594
00:35:54 --> 00:35:57
one half and the other half.
 

595
00:35:57 --> 00:35:57
Question?
 

596
00:35:57 --> 00:36:07
STUDENT: [INAUDIBLE]
 

597
00:36:07 --> 00:36:08
PROFESSOR: So, you're
one step ahead of me.

598
00:36:08 --> 00:36:10
We'll also have to be sure to
distinguish between the top

599
00:36:10 --> 00:36:12
branch and the bottom branch
of the parabola, which

600
00:36:12 --> 00:36:14
we're about to do.
 

601
00:36:14 --> 00:36:18
Now, in order to distinguish
what's going on I actually have

602
00:36:18 --> 00:36:22
to use multi colors here.
 

603
00:36:22 --> 00:36:24
And so we will do that.
 

604
00:36:24 --> 00:36:29
First there's the top
part, which is orange.

605
00:36:29 --> 00:36:32
That's the top part.
 

606
00:36:32 --> 00:36:33
I'll call it top.
 

607
00:36:33 --> 00:36:41
And then there's the bottom
part, which has two halves.

608
00:36:41 --> 00:36:53
They are pink, and I
guess this is blue.

609
00:36:53 --> 00:37:01
Alright, so now let's
see what's happening.

610
00:37:01 --> 00:37:07
The most important two points
that I have to figure out in

611
00:37:07 --> 00:37:08
order to get started here.
 

612
00:37:08 --> 00:37:09
Well, really I'm going
to have to figure out

613
00:37:09 --> 00:37:11
three points, I claim.
 

614
00:37:11 --> 00:37:13
I'm going to have to figure
out where this point is.

615
00:37:13 --> 00:37:17
Where this point is, and
where that point is.

616
00:37:17 --> 00:37:20
If I know where these three
points are, then I have a

617
00:37:20 --> 00:37:23
chance of knowing where
to start, where to

618
00:37:23 --> 00:37:25
end, and so forth.
 

619
00:37:25 --> 00:37:26
Another question.
 

620
00:37:26 --> 00:37:27
STUDENT: [INAUDIBLE]
 

621
00:37:27 --> 00:37:31
PROFESSOR: Could you speak up?
 

622
00:37:31 --> 00:37:37
STUDENT: [INAUDIBLE]
 

623
00:37:37 --> 00:37:39
PROFESSOR: The question
is, why do we need to

624
00:37:39 --> 00:37:41
split up the area.
 

625
00:37:41 --> 00:37:44
And I think in order to answer
that question further, I'm

626
00:37:44 --> 00:37:47
going to have to go into the
details of the method, and then

627
00:37:47 --> 00:37:51
you'll see where
it's necessary.

628
00:37:51 --> 00:37:54
So the first step is that I'm
going to figure out what

629
00:37:54 --> 00:37:57
these three points are.
 

630
00:37:57 --> 00:38:02
This one is kind of easy;
it's the point (0, 0).

631
00:38:02 --> 00:38:06
This point down here and this
point up here are intersections

632
00:38:06 --> 00:38:08
of the two curves.
 

633
00:38:08 --> 00:38:11
I can identify them by
the following equation.

634
00:38:11 --> 00:38:21
I need to see where
these curves intersect.

635
00:38:21 --> 00:38:26
At what, well, if I plug in
x = y^2, I get y = y^2 - 2.

636
00:38:26 --> 00:38:28
And then I can solve this
quadratic equation.

637
00:38:28 --> 00:38:33
y ^2 - y - 2 = 0.
 

638
00:38:33 --> 00:38:39
So (y - 2)( y + 1) = 0.
 

639
00:38:39 --> 00:38:52
And this is telling me
that y = 2 or y = - 1.

640
00:38:52 --> 00:38:54
So I've found y = - 1.
 

641
00:38:54 --> 00:39:00
That means this point down
here has second entry - 1.

642
00:39:00 --> 00:39:04
Its first entry, its x value,
I can get from this formula

643
00:39:04 --> 00:39:06
here or the other formula.
 

644
00:39:06 --> 00:39:10
I have to square,
this - 1 ^2 = 1.

645
00:39:10 --> 00:39:15
So that's the formula
for this point.

646
00:39:15 --> 00:39:20
And the other point
has second entry 2.

647
00:39:20 --> 00:39:23
And, again, with his formula
y = x ^2, I have to square

648
00:39:23 --> 00:39:31
y to get x, so this is 4.
 

649
00:39:31 --> 00:39:37
Now, I claim I have enough
data to get started.

650
00:39:37 --> 00:39:41
But maybe I'll identify
one more thing.

651
00:39:41 --> 00:39:48
I need the top, the bottom
left, and the bottom right.

652
00:39:48 --> 00:39:55
The top is the formula for this
branch of x = y ^2, which is

653
00:39:55 --> 00:39:57
in the positive y region.
 

654
00:39:57 --> 00:40:05
And that is y = the
square root of x.

655
00:40:05 --> 00:40:09
The bottom curve, part of
the parabola, so this is

656
00:40:09 --> 00:40:21
the bottom left, is y
= minus square root x.

657
00:40:21 --> 00:40:24
That's the other branch
of the square root.

658
00:40:24 --> 00:40:26
And this is exactly what
you were asking before.

659
00:40:26 --> 00:40:28
And this is, we have to
distinguish between these two.

660
00:40:28 --> 00:40:31
And the point is, these
formulas really are different.

661
00:40:31 --> 00:40:34
They're not the same.
 

662
00:40:34 --> 00:40:38
Now, the last bit is the bottom
right chunk here, which

663
00:40:38 --> 00:40:39
is this pink part.
 

664
00:40:39 --> 00:40:44
Bottom right.
 

665
00:40:44 --> 00:40:49
And that one is the
formula for the line.

666
00:40:49 --> 00:40:55
And that's y = x - 2.
 

667
00:40:55 --> 00:41:03
Now I'm ready to find the area.
 

668
00:41:03 --> 00:41:06
It's going to be in two chunks.
 

669
00:41:06 --> 00:41:15
This is the left part,
plus the right part.

670
00:41:15 --> 00:41:18
And the left part, and I want
to set it up as an integral, I

671
00:41:18 --> 00:41:21
want there to be a dx and here
I want to set up an integral

672
00:41:21 --> 00:41:23
and I want it to be dx.
 

673
00:41:23 --> 00:41:26
I need to figure out
what the range of x is.

674
00:41:26 --> 00:41:28
So, first I'm going to - well,
let's leave ourselves a

675
00:41:28 --> 00:41:37
little more room than that.
 

676
00:41:37 --> 00:41:39
Just to be safe.
 

677
00:41:39 --> 00:41:45
OK, here's the right.
 

678
00:41:45 --> 00:41:48
So here we have our dx.
 

679
00:41:48 --> 00:41:53
Now, I need to figure
out the starting place

680
00:41:53 --> 00:41:54
and the ending place.
 

681
00:41:54 --> 00:41:57
So the starting place
is the leftmost place.

682
00:41:57 --> 00:42:00
The leftmost place
is over here.

683
00:42:00 --> 00:42:02
And x = 0 there.
 

684
00:42:02 --> 00:42:05
So we're going to travel
from this vertical line

685
00:42:05 --> 00:42:08
to the green line.
 

686
00:42:08 --> 00:42:09
Over here.
 

687
00:42:09 --> 00:42:13
And that's from 0 to 1.
 

688
00:42:13 --> 00:42:17
And the difference between the
orange curve and the blue

689
00:42:17 --> 00:42:22
curve is what I call top and
bottom left, over there.

690
00:42:22 --> 00:42:32
So that is square root of x
minus minus square root of x.

691
00:42:32 --> 00:42:41
Again, this is what I call
top, and this was bottom.

692
00:42:41 --> 00:42:48
But only the left.
 

693
00:42:48 --> 00:42:52
I claim that's giving me the
left half of this, the left

694
00:42:52 --> 00:42:55
section of this diagram.
 

695
00:42:55 --> 00:42:59
Now I'm going to do the right
section of the diagram.

696
00:42:59 --> 00:43:02
I start at 1.
 

697
00:43:02 --> 00:43:04
The lower limit is 1.
 

698
00:43:04 --> 00:43:08
And I go all the way
to this point here.

699
00:43:08 --> 00:43:11
Which is the last bit.
 

700
00:43:11 --> 00:43:13
And that's going to be x = 4.
 

701
00:43:13 --> 00:43:19
The upper limit here is 4.
 

702
00:43:19 --> 00:43:21
And now I have to take the
difference between the top

703
00:43:21 --> 00:43:22
and the bottom again.
 

704
00:43:22 --> 00:43:24
The top is square root
of x all over again.

705
00:43:24 --> 00:43:26
But the bottom has changed.
 

706
00:43:26 --> 00:43:31
The bottom is now the
quantity (x - 2).

707
00:43:31 --> 00:43:32
Please don't forget
your parenthesis.

708
00:43:32 --> 00:43:44
There's going to be minus
signs and cancellations.

709
00:43:44 --> 00:43:46
Now, this is almost the
end of the problem.

710
00:43:46 --> 00:43:48
The rest of it is routine.
 

711
00:43:48 --> 00:43:52
We would just have to
evaluate these integrals.

712
00:43:52 --> 00:43:57
And, fortunately, I'm
going to spare you that.

713
00:43:57 --> 00:43:59
We're not going to
bother to do it.

714
00:43:59 --> 00:44:01
That's the easy part.
 

715
00:44:01 --> 00:44:03
We're not going to do it.
 

716
00:44:03 --> 00:44:05
But I'm going to show you
that there's a much quicker

717
00:44:05 --> 00:44:07
way with this integral.
 

718
00:44:07 --> 00:44:09
And with this area calculation.
 

719
00:44:09 --> 00:44:10
Right now.
 

720
00:44:10 --> 00:44:18
The quicker way is what you see
when you see how long this is.

721
00:44:18 --> 00:44:21
And you see that there's
another device that you can use

722
00:44:21 --> 00:44:25
that looks similar in principle
to this, but reverses

723
00:44:25 --> 00:44:28
the roles of x and y.
 

724
00:44:28 --> 00:44:33
And the other device, which
I'll draw over here,

725
00:44:33 --> 00:44:35
schematically.
 

726
00:44:35 --> 00:44:47
No, maybe I'll draw it on
this blackboard here.

727
00:44:47 --> 00:44:55
So, Method 2, if you like, this
was Method 1, and we should

728
00:44:55 --> 00:45:04
call it the hard way.
 

729
00:45:04 --> 00:45:11
Method 2, which is better
in this case, is to

730
00:45:11 --> 00:45:24
use horizontal slices.
 

731
00:45:24 --> 00:45:33
Let me draw the picture,
at least schematically.

732
00:45:33 --> 00:45:35
Here's our picture
that we had before.

733
00:45:35 --> 00:45:38
And now instead of slicing
it vertically, I'm going

734
00:45:38 --> 00:45:40
to slice it horizontally.
 

735
00:45:40 --> 00:45:44
Like this.
 

736
00:45:44 --> 00:45:49
Now, the dimensions
have different names.

737
00:45:49 --> 00:45:51
But the principle is similar.
 

738
00:45:51 --> 00:45:55
The width, we now call dy.
 

739
00:45:55 --> 00:45:58
Because it's the change in y.
 

740
00:45:58 --> 00:46:05
And this distance here, from
the left end to the right end,

741
00:46:05 --> 00:46:09
we have to figure out what the
formulas for those things are.

742
00:46:09 --> 00:46:16
So on the left, maybe I'll
draw them color coded again.

743
00:46:16 --> 00:46:19
So here's a left.
 

744
00:46:19 --> 00:46:23
And, whoops, orange
is right, I guess.

745
00:46:23 --> 00:46:24
So here we go.
 

746
00:46:24 --> 00:46:30
So we have the left,
which is this green.

747
00:46:30 --> 00:46:35
Is x = y ^2.
 

748
00:46:35 --> 00:46:46
And the right, which is
orange, is y = x - 2.

749
00:46:46 --> 00:46:50
And now in order to use this,
it's going to turn out that we

750
00:46:50 --> 00:46:53
want to write x as, we
want to reverse roles.

751
00:46:53 --> 00:46:57
So we want to write this
as x is a function of y.

752
00:46:57 --> 00:47:04
So we'll use it in this form.
 

753
00:47:04 --> 00:47:11
And now I want to set up
the integral for you.

754
00:47:11 --> 00:47:21
This time, the area is equal to
an integral in the dy variable.

755
00:47:21 --> 00:47:26
And its starting
place is down here.

756
00:47:26 --> 00:47:28
And it's ending
place is up there.

757
00:47:28 --> 00:47:31
This is the lowest value of y,
and this is the top value of y.

758
00:47:31 --> 00:47:34
And we've already
computed those things.

759
00:47:34 --> 00:47:39
The lowest level of y is - 1.
 

760
00:47:39 --> 00:47:42
So this is y = - 1.
 

761
00:47:42 --> 00:47:45
And this top value is y = 2.
 

762
00:47:45 --> 00:47:51
So this goes from - 1 to 2.
 

763
00:47:51 --> 00:47:57
And now the difference is this
distance here, the distance

764
00:47:57 --> 00:47:59
between the rightmost point
and the leftmost point.

765
00:47:59 --> 00:48:02
Those are the two dimensions.
 

766
00:48:02 --> 00:48:04
So again, it's a rectangle but
its horizontal is long and

767
00:48:04 --> 00:48:07
its vertical is very short.
 

768
00:48:07 --> 00:48:08
And what are they?
 

769
00:48:08 --> 00:48:11
It's the difference between
the right and the left.

770
00:48:11 --> 00:48:21
The right-hand is (y + 2),
and the right-hand is y ^2.

771
00:48:21 --> 00:48:26
So this is the formula.
 

772
00:48:26 --> 00:48:35
STUDENT: [INAUDIBLE]
 

773
00:48:35 --> 00:48:39
PROFESSOR: What
was the question?

774
00:48:39 --> 00:48:41
Why is it right minus left?
 

775
00:48:41 --> 00:48:42
That's very important.
 

776
00:48:42 --> 00:48:44
Why is it right minus left?
 

777
00:48:44 --> 00:48:47
And that's actually the point
that I was about to make.

778
00:48:47 --> 00:48:48
Which is this.
 

779
00:48:48 --> 00:48:53
That y + 2, which is the
right, is bigger than y

780
00:48:53 --> 00:48:55
^2, which is the left.
 

781
00:48:55 --> 00:49:00
So that means that y +
2 - y ^2 is positive.

782
00:49:00 --> 00:49:02
If you do it backwards, you'll
always get a negative number

783
00:49:02 --> 00:49:07
and you'll always get
the wrong answer.

784
00:49:07 --> 00:49:11
So this is the right-hand end
minus the left-hand end gives

785
00:49:11 --> 00:49:14
you a positive number.
 

786
00:49:14 --> 00:49:16
And it's not obvious,
actually, where you are.

787
00:49:16 --> 00:49:19
There's another
double-check, by the way.

788
00:49:19 --> 00:49:22
When you look at this quantity,
you see that the ends pinch.

789
00:49:22 --> 00:49:25
And that's exactly the
crossover points.

790
00:49:25 --> 00:49:30
That is, when y = -
1, y + 2 - y ^2 = 0.

791
00:49:30 --> 00:49:36
And when y = 2, y
+ 2 - y ^2 = 0.

792
00:49:36 --> 00:49:40
And that's not an accident,
that's exactly the geometry

793
00:49:40 --> 00:49:44
of the shape that we
picked out there.

794
00:49:44 --> 00:49:46
So this is the technique.
 

795
00:49:46 --> 00:49:51
Now, this is a much
more routine integral.

796
00:49:51 --> 00:49:54
I'm not going to carry it out,
I'll just do one last step.

797
00:49:54 --> 00:50:01
Which is that this is (y ^2
/ 2) + 2y - (y ^3 / 3),

798
00:50:01 --> 00:50:03
evaluated at - 1 and 2.
 

799
00:50:03 --> 00:50:08
Which, if you work
it out, is 9/2.

800
00:50:08 --> 00:50:10
So we're done for today.
 

801
00:50:10 --> 00:50:12
And tomorrow we'll do more
volumes, more things

802
00:50:12 --> 00:50:15
including three dimensions.
 

803
00:50:15 --> 00:50:16