WEBVTT
00:00:00.000 --> 00:00:03.516
The following content is
provided under a Creative
00:00:03.516 --> 00:00:04.312
Commons license.
00:00:04.312 --> 00:00:06.020
Your support will help
MIT OpenCourseWare
00:00:06.020 --> 00:00:09.930
continue to offer high quality
educational resources for free.
00:00:09.930 --> 00:00:15.050
To make a donation, or to
view additional materials
00:00:15.050 --> 00:00:17.370
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:17.370 --> 00:00:21.420
at ocw.mit.edu.
00:00:21.420 --> 00:00:23.730
PROFESSOR: One correction
from last time.
00:00:23.730 --> 00:00:28.240
Sorry to say, I forgot
a very important factor
00:00:28.240 --> 00:00:30.900
when I was telling you
what an average value is.
00:00:30.900 --> 00:00:33.770
If you don't put in
that factor, it's
00:00:33.770 --> 00:00:37.716
only half off on
the exam problem
00:00:37.716 --> 00:00:38.840
that will be given on this.
00:00:38.840 --> 00:00:43.130
So I would have gotten half off
for missing out on this factor,
00:00:43.130 --> 00:00:44.240
too.
00:00:44.240 --> 00:00:46.370
So remember you have
to divide by n here,
00:00:46.370 --> 00:00:49.150
certainly when you're
integrating over 0 to n,
00:00:49.150 --> 00:00:51.460
the Riemann sum is
the numerator here.
00:00:51.460 --> 00:00:52.940
And if I divide
by n on that side,
00:00:52.940 --> 00:00:55.250
I've got to divide by
n on the other side.
00:00:55.250 --> 00:00:57.770
This was meant to illustrate
this idea that we're
00:00:57.770 --> 00:01:00.540
dividing by the total here.
00:01:00.540 --> 00:01:04.170
And we are going to be talking
about average value in more
00:01:04.170 --> 00:01:05.300
detail.
00:01:05.300 --> 00:01:07.690
Not today, though.
00:01:07.690 --> 00:01:16.190
So this has to do
with average value.
00:01:16.190 --> 00:01:20.730
And we'll discuss it
in considerable detail
00:01:20.730 --> 00:01:27.340
in a couple of days, I guess.
00:01:27.340 --> 00:01:33.900
Now, today I want to continue.
00:01:33.900 --> 00:01:36.120
I didn't have time to
finish my discussion
00:01:36.120 --> 00:01:39.751
of the Fundamental
Theorem of Calculus 2.
00:01:39.751 --> 00:01:41.500
And anyway it's very
important to write it
00:01:41.500 --> 00:01:43.350
down on the board
twice, because you
00:01:43.350 --> 00:01:46.410
want to see it at least twice.
00:01:46.410 --> 00:01:48.370
And many more times as well.
00:01:48.370 --> 00:01:51.630
So let's just remind
you, the second version
00:01:51.630 --> 00:01:55.430
of the Fundamental Theorem of
Calculus says the following.
00:01:55.430 --> 00:02:01.000
It says that the
derivative of an integral
00:02:01.000 --> 00:02:03.630
gives you the
function back again.
00:02:03.630 --> 00:02:08.310
So here's the theorem.
00:02:08.310 --> 00:02:12.150
And the way I'd like
to use it today,
00:02:12.150 --> 00:02:14.500
I started this
discussion last time.
00:02:14.500 --> 00:02:16.660
But we didn't get into it.
00:02:16.660 --> 00:02:19.920
And this is something that's
on your problem set along
00:02:19.920 --> 00:02:23.190
with several other examples.
00:02:23.190 --> 00:02:30.770
Is that we can use this to
solve differential equations.
00:02:30.770 --> 00:02:34.620
And in particular,
for example, we
00:02:34.620 --> 00:02:43.540
can solve the equation y'
= 1 / x with this formula.
00:02:43.540 --> 00:02:47.930
Namely, using an integral.
00:02:47.930 --> 00:02:54.770
L(x) is the integral
from 1 to x of dt / t.
00:02:54.770 --> 00:03:00.410
The function f(t) is just 1 / t.
00:03:00.410 --> 00:03:07.070
Now, that formula can be
taken to be the starting place
00:03:07.070 --> 00:03:11.830
for the derivation of all the
properties of the logarithm
00:03:11.830 --> 00:03:12.510
function.
00:03:12.510 --> 00:03:14.090
So what we're going
to do right now
00:03:14.090 --> 00:03:22.750
is we're going to take
this to be the definition
00:03:22.750 --> 00:03:28.190
of the logarithm.
00:03:28.190 --> 00:03:31.870
And if we do that, then I
claim that we can read off
00:03:31.870 --> 00:03:33.910
the properties of the
logarithm just about as
00:03:33.910 --> 00:03:36.150
easily as we could before.
00:03:36.150 --> 00:03:38.730
And so I'll illustrate that now.
00:03:38.730 --> 00:03:41.470
And there are a
few other examples
00:03:41.470 --> 00:03:45.450
of this where somewhat more
unfamiliar functions come up.
00:03:45.450 --> 00:03:50.532
This one is one that in theory
we know something about.
00:03:50.532 --> 00:03:51.990
The first property
of this function
00:03:51.990 --> 00:03:53.950
is the one that's already given.
00:03:53.950 --> 00:03:59.580
Namely, its derivative is 1/x.
00:03:59.580 --> 00:04:02.150
And we get a lot of information
just out of the fact
00:04:02.150 --> 00:04:04.209
that its derivative is 1/x.
00:04:04.209 --> 00:04:05.750
The other thing that
we need in order
00:04:05.750 --> 00:04:08.750
to nail down the function,
besides its derivative,
00:04:08.750 --> 00:04:10.280
is one value of the function.
00:04:10.280 --> 00:04:14.670
Because it's really not
specified by this equation,
00:04:14.670 --> 00:04:17.490
only specified up to a
constant by this equation.
00:04:17.490 --> 00:04:20.020
But we nail down that
constant when we evaluate it
00:04:20.020 --> 00:04:22.640
at this one place, L(1).
00:04:22.640 --> 00:04:24.640
And there we're getting
the integral from 1 to 1
00:04:24.640 --> 00:04:28.430
of dt / t, which is 0.
00:04:28.430 --> 00:04:30.860
And that's the case with all
these definite integrals.
00:04:30.860 --> 00:04:32.900
If you evaluate them at
their starting places,
00:04:32.900 --> 00:04:34.460
the value will be 0.
00:04:34.460 --> 00:04:36.350
And together these
two properties
00:04:36.350 --> 00:04:42.800
specify this function
L(x) uniquely.
00:04:42.800 --> 00:04:46.200
Now, the next step
is to try to think
00:04:46.200 --> 00:04:47.860
about what its properties are.
00:04:47.860 --> 00:04:50.980
And the first approach
to that, and this
00:04:50.980 --> 00:04:52.660
is the approach
that we always take,
00:04:52.660 --> 00:04:55.870
is to maybe graph the function,
to get a feeling for it.
00:04:55.870 --> 00:04:57.970
And so I'm going to take
the second derivative.
00:04:57.970 --> 00:05:00.440
Now, notice that when
you have a function which
00:05:00.440 --> 00:05:02.460
is given as an integral,
its first derivative
00:05:02.460 --> 00:05:04.752
is really easy to compute.
00:05:04.752 --> 00:05:06.460
And then its second
derivative, well, you
00:05:06.460 --> 00:05:08.290
have to differentiate
whatever you get.
00:05:08.290 --> 00:05:09.720
So it may or may not be easy.
00:05:09.720 --> 00:05:12.230
But anyway, it's a
lot harder in the case
00:05:12.230 --> 00:05:15.150
when I start with a function to
get to the second derivative.
00:05:15.150 --> 00:05:19.070
Here it's relatively easy.
00:05:19.070 --> 00:05:21.650
And these are the properties
that I'm going to use.
00:05:21.650 --> 00:05:26.410
I won't really use very much
more about it than that.
00:05:26.410 --> 00:05:29.020
And qualitatively,
the conclusions
00:05:29.020 --> 00:05:31.760
that we can draw from
this are, first of all,
00:05:31.760 --> 00:05:34.100
from this, for example
we see that this thing is
00:05:34.100 --> 00:05:38.970
concave down every place.
00:05:38.970 --> 00:05:40.920
And then to get
started with the graph,
00:05:40.920 --> 00:05:45.700
since I see I have a value
here, which is L(1) = 0,
00:05:45.700 --> 00:05:48.560
I'm going to throw in
the value of the slope.
00:05:48.560 --> 00:05:50.980
So L'(1), which I
know is 1 over 1,
00:05:50.980 --> 00:05:55.180
that's reading off from this
equation here, so that's 1.
00:05:55.180 --> 00:05:59.200
And now I'm ready to sketch
at least a part of the curve.
00:05:59.200 --> 00:06:07.680
So here's a sketch of the graph.
00:06:07.680 --> 00:06:13.000
Here's the point (1, 0),
that is, x = 1, y = 0.
00:06:13.000 --> 00:06:17.420
And the tangent line,
I know, has slope 1.
00:06:17.420 --> 00:06:20.540
And the curve is concave down.
00:06:20.540 --> 00:06:27.530
So it's going to look
something like this.
00:06:27.530 --> 00:06:31.230
Incidentally, it's
also increasing.
00:06:31.230 --> 00:06:34.359
And that's an
important property,
00:06:34.359 --> 00:06:35.400
it's strictly increasing.
00:06:35.400 --> 00:06:39.570
That's because
L'(x) is positive.
00:06:39.570 --> 00:06:44.220
And so, we can get from this the
following important definition.
00:06:44.220 --> 00:06:46.610
Which, again, is working
backwards from this definition.
00:06:46.610 --> 00:06:48.880
We can get to where
we started with a log
00:06:48.880 --> 00:06:50.620
in our previous discussion.
00:06:50.620 --> 00:06:57.870
Namely, if I take the
level here, which is y = 1,
00:06:57.870 --> 00:06:59.730
then that crosses
the axis someplace.
00:06:59.730 --> 00:07:04.180
And this point is what
we're going to define as e.
00:07:04.180 --> 00:07:11.970
So the definition
of e is that it's
00:07:11.970 --> 00:07:20.220
the value such that L(e) = 1.
00:07:20.220 --> 00:07:22.800
And again, the fact that
there's exactly one such place
00:07:22.800 --> 00:07:25.370
just comes from the fact
that this L' is positive,
00:07:25.370 --> 00:07:29.340
so that L is increasing.
00:07:29.340 --> 00:07:32.910
Now, there's just one
other feature of this graph
00:07:32.910 --> 00:07:35.882
that I'm going to
emphasize to you.
00:07:35.882 --> 00:07:38.090
There's one other thing
which I'm not going to check,
00:07:38.090 --> 00:07:40.060
which you would
ordinarily do with graphs.
00:07:40.060 --> 00:07:42.143
Once it's increasing there
are no critical points,
00:07:42.143 --> 00:07:44.230
so the only other interesting
thing is the ends.
00:07:44.230 --> 00:07:46.400
And it turns out that the
limit as you go down to 0
00:07:46.400 --> 00:07:47.150
is minus infinity.
00:07:47.150 --> 00:07:49.840
As you go over to the right
here it's plus infinity.
00:07:49.840 --> 00:07:53.340
It does get arbitrarily
high; it doesn't level off.
00:07:53.340 --> 00:07:55.820
But I'm not going to
discuss that here.
00:07:55.820 --> 00:07:57.400
Instead, I'm going
to just remark
00:07:57.400 --> 00:08:01.920
on one qualitative feature of
the graph, which is this remark
00:08:01.920 --> 00:08:06.850
that the part which is to
the left of 1 is below 0.
00:08:06.850 --> 00:08:17.292
So I just want to remark, why
is L(x) negative for x < 1.
00:08:17.292 --> 00:08:19.750
Maybe I don't have room for
that, so I'll just put in here:
00:08:19.750 --> 00:08:23.550
x < 1.
00:08:23.550 --> 00:08:25.320
I want to give you two reasons.
00:08:25.320 --> 00:08:27.980
Again, we're only working from
very first principles here.
00:08:27.980 --> 00:08:33.500
Just that-- the property
that L' = 1/x, and L(1) = 0.
00:08:33.500 --> 00:08:39.650
So our first reason is
that, well, I just said it.
00:08:39.650 --> 00:08:41.430
L(1) = 0.
00:08:41.430 --> 00:08:46.910
And L is increasing.
00:08:46.910 --> 00:08:49.920
And if you read that backwards,
if it gets up to 0 here,
00:08:49.920 --> 00:08:54.160
it must have been
negative before 0.
00:08:54.160 --> 00:08:57.960
So this is one way of seeing
that L(x) is negative.
00:08:57.960 --> 00:09:02.410
There's a second way of seeing
it, which is equally important.
00:09:02.410 --> 00:09:07.470
And it has to do with just
manipulation of integrals.
00:09:07.470 --> 00:09:11.430
Here I'm going to start out
with L(x), and its definition.
00:09:11.430 --> 00:09:15.730
Which is the integral
from 1 to x, dt / t.
00:09:15.730 --> 00:09:19.250
And now I'm going to reverse
the order of integration.
00:09:19.250 --> 00:09:21.500
This is the same, by our
definition of our properties
00:09:21.500 --> 00:09:24.010
of integrals, as the
integral from x to 1
00:09:24.010 --> 00:09:30.010
with a minus sign dt / t.
00:09:30.010 --> 00:09:34.140
Now, I can tell that this
quantity is negative.
00:09:34.140 --> 00:09:36.400
And the reason
that I can tell is
00:09:36.400 --> 00:09:41.840
that this chunk of it
here, this piece of it,
00:09:41.840 --> 00:09:44.010
is a positive number.
00:09:44.010 --> 00:09:46.290
This part is positive.
00:09:46.290 --> 00:09:51.290
And this part is
positive because x < 1.
00:09:51.290 --> 00:09:53.494
So the lower limit is
less than the upper limit,
00:09:53.494 --> 00:09:55.785
and so this is interpreted
- the thing in the green box
00:09:55.785 --> 00:09:58.070
is interpreted - as an area.
00:09:58.070 --> 00:09:58.970
It's an area.
00:09:58.970 --> 00:10:02.690
And so negative a positive
quantity is negative,
00:10:02.690 --> 00:10:08.280
minus a positive
quantity's negative.
00:10:08.280 --> 00:10:13.870
So both of these work perfectly
well as interpretations.
00:10:13.870 --> 00:10:16.010
And it's just to
illustrate what we can do.
00:10:16.010 --> 00:10:18.280
Now, there's one
more manipulation
00:10:18.280 --> 00:10:23.380
of integrals that gives us the
fanciest property of the log.
00:10:23.380 --> 00:10:26.410
And that's the last one
that I'm going to do.
00:10:26.410 --> 00:10:29.450
And you have a similar
thing on your homework.
00:10:29.450 --> 00:10:32.090
So I'm going to
prove that-- This
00:10:32.090 --> 00:10:34.970
is, as I say, the fanciest
property of the log.
00:10:34.970 --> 00:10:40.200
On your homework, by the way,
you're going to check that
00:10:40.200 --> 00:10:43.160
L(1/x) = -L(x).
00:10:46.680 --> 00:10:52.070
But we'll do this one.
00:10:52.070 --> 00:10:56.430
The idea is just to plug in the
formula and see what it gives.
00:10:56.430 --> 00:11:02.830
On the left-hand side,
I have 1 to ab, dt / t.
00:11:02.830 --> 00:11:06.040
That's L(ab).
00:11:06.040 --> 00:11:08.900
And then that's certainly
equal to the left-hand side.
00:11:08.900 --> 00:11:12.650
And then I'm going to now
split this into two pieces.
00:11:12.650 --> 00:11:14.600
Again, this is a
property of integrals.
00:11:14.600 --> 00:11:18.790
That if you have an integral
from one place to another,
00:11:18.790 --> 00:11:20.620
you can break it up into pieces.
00:11:20.620 --> 00:11:26.540
So I'm going to start
at 1 but then go to a.
00:11:26.540 --> 00:11:33.950
And then I'm going to
continue from a to ab.
00:11:33.950 --> 00:11:36.130
So this is the
question that we have.
00:11:36.130 --> 00:11:38.010
We haven't proved this.
00:11:38.010 --> 00:11:41.050
Well, this one is actually true.
00:11:41.050 --> 00:11:42.700
If we want this to
be true, we know
00:11:42.700 --> 00:11:44.900
by definition L(ab) is this.
00:11:44.900 --> 00:11:48.460
We know, we can see
it, that L(a) is this.
00:11:48.460 --> 00:11:52.320
So the question that
this boils down to
00:11:52.320 --> 00:11:54.960
is, we want to know that
these two things are equal.
00:11:54.960 --> 00:12:01.320
We want to know that L(b) is
that other integral there.
00:12:01.320 --> 00:12:04.774
So let's check it.
00:12:04.774 --> 00:12:06.190
I'm going to rewrite
the integral.
00:12:06.190 --> 00:12:09.170
It's the integral from--
sorry, from lower limit a
00:12:09.170 --> 00:12:13.940
to upper limit ab of dt / t.
00:12:13.940 --> 00:12:16.640
And now, again, to
illustrate properties
00:12:16.640 --> 00:12:18.170
of integrals, the
key property here
00:12:18.170 --> 00:12:23.740
that we're going to have to
use is change of variables.
00:12:23.740 --> 00:12:26.680
This is a kind of a scaled
integral where everything
00:12:26.680 --> 00:12:28.710
is multiplied by a
factor of a from what
00:12:28.710 --> 00:12:32.010
we want to get to
this L(b) quantity.
00:12:32.010 --> 00:12:38.240
And so this suggests that
we write down t = au.
00:12:38.240 --> 00:12:40.460
That's going to be our trick.
00:12:40.460 --> 00:12:43.440
And if I use that new variable
u, then the change in t,
00:12:43.440 --> 00:12:50.440
dt, is a du.
00:12:50.440 --> 00:12:53.950
And as a result, I can write
this as equal to an integral
00:12:53.950 --> 00:12:58.010
from, let's see, dt = a du.
00:12:58.010 --> 00:13:00.080
And t = au.
00:13:00.080 --> 00:13:05.740
So I've now substituted
in for the integrand.
00:13:05.740 --> 00:13:08.720
But on top of this,
with definite integrals,
00:13:08.720 --> 00:13:13.080
we also have to
check the limits.
00:13:13.080 --> 00:13:17.140
And the limits work
out as follows.
00:13:17.140 --> 00:13:20.180
When t = a, that's
the lower limit.
00:13:20.180 --> 00:13:22.460
Let's just take a look. t = au.
00:13:22.460 --> 00:13:26.600
So that means that
u is equal to, what?
00:13:26.600 --> 00:13:28.710
It's 1.
00:13:28.710 --> 00:13:31.880
Because a * 1 = a.
00:13:31.880 --> 00:13:34.040
So if t = a, this
is if and only if.
00:13:34.040 --> 00:13:36.610
So this lower limit, which
really in disguise was where
00:13:36.610 --> 00:13:45.400
t = a, becomes where u = 1.
00:13:45.400 --> 00:13:53.920
And similarly,
when t = ab, u = b.
00:13:53.920 --> 00:13:56.690
So the upper limit here is b.
00:13:56.690 --> 00:14:00.460
And now, if you
notice, we're just
00:14:00.460 --> 00:14:02.920
going to cancel these
two factors here.
00:14:02.920 --> 00:14:06.810
And now we recognize that
this is just the same
00:14:06.810 --> 00:14:09.940
as the definition of L(b).
00:14:09.940 --> 00:14:13.490
Because L(x) is over
here in the box.
00:14:13.490 --> 00:14:16.270
And the fact that I use the
letter t there is irrelevant;
00:14:16.270 --> 00:14:18.020
it works equally well
with the letter u.
00:14:18.020 --> 00:14:22.430
So this is just L(b).
00:14:22.430 --> 00:14:33.160
Which is what we wanted to show.
00:14:33.160 --> 00:14:34.890
So that's an
example, and you have
00:14:34.890 --> 00:14:45.380
one in your homework,
which is a little similar.
00:14:45.380 --> 00:14:49.300
Now, the last example, that I'm
going to discuss of this type,
00:14:49.300 --> 00:14:51.390
I already mentioned last time.
00:14:51.390 --> 00:14:54.430
Which is the function F(x),
which is the integral from 0
00:14:54.430 --> 00:14:58.150
to x of e^(-t^2) dt.
00:14:58.150 --> 00:15:04.130
This one is even more exotic
because unlike the logarithm
00:15:04.130 --> 00:15:06.030
it's a new function.
00:15:06.030 --> 00:15:09.320
It really is not any
function that you
00:15:09.320 --> 00:15:13.870
can express in terms of the
functions that we know already.
00:15:13.870 --> 00:15:18.340
And the approach, always,
to these new functions
00:15:18.340 --> 00:15:22.000
is to think of what
their properties are.
00:15:22.000 --> 00:15:24.380
And the way we think
of functions in order
00:15:24.380 --> 00:15:26.570
to understand them is
to maybe sketch them.
00:15:26.570 --> 00:15:29.830
And so I'm going to do exactly
the same thing I did over here.
00:15:29.830 --> 00:15:31.630
So, what is it that I
can get out of this?
00:15:31.630 --> 00:15:35.105
Well, immediately I can figure
out what the derivative is.
00:15:35.105 --> 00:15:38.000
I read it off from the
fundamental theorem.
00:15:38.000 --> 00:15:41.610
It's this.
00:15:41.610 --> 00:15:45.470
I also can figure out the
value at the starting place.
00:15:45.470 --> 00:15:48.440
In this case, the
starting place is 0.
00:15:48.440 --> 00:15:53.990
And the value is 0.
00:15:53.990 --> 00:15:57.600
And I should check the second
derivative, which is also not
00:15:57.600 --> 00:15:59.145
so difficult to compute.
00:15:59.145 --> 00:16:03.400
The second derivative
is -2x e^(-x^2).
00:16:06.070 --> 00:16:10.650
And so now I can see that
this function is increasing,
00:16:10.650 --> 00:16:12.890
because this
derivative is positive,
00:16:12.890 --> 00:16:14.670
it's always increasing.
00:16:14.670 --> 00:16:17.830
And it's going to be concave
down when x is positive
00:16:17.830 --> 00:16:20.540
and concave up
when x is negative.
00:16:20.540 --> 00:16:24.900
Because there's a minus sign
here, so the sign is negative.
00:16:24.900 --> 00:16:30.250
This is less than 0 when x is
positive and greater than 0
00:16:30.250 --> 00:16:36.800
when x is negative.
00:16:36.800 --> 00:16:40.160
And maybe to get started
I'll remind you F(0) is 0.
00:16:40.160 --> 00:16:46.750
It's also true that F'(0)-- that
just comes right out of this,
00:16:46.750 --> 00:16:52.450
F'(0) = e^(-0^2), which is 1.
00:16:52.450 --> 00:16:55.420
That means the tangent
line again has slope 1.
00:16:55.420 --> 00:16:57.090
We do this a lot with functions.
00:16:57.090 --> 00:17:00.500
We normalize them so that the
slopes of their tangent lines
00:17:00.500 --> 00:17:03.600
are 1 at convenient spots.
00:17:03.600 --> 00:17:06.390
So here's the tangent
line of slope 1.
00:17:06.390 --> 00:17:10.820
We know this thing is
concave down to the right
00:17:10.820 --> 00:17:14.890
and concave up to the left.
00:17:14.890 --> 00:17:17.570
And so it's going to
look something like this.
00:17:17.570 --> 00:17:20.860
With an inflection point.
00:17:20.860 --> 00:17:26.290
Right?
00:17:26.290 --> 00:17:32.230
Now, I want to say one
more-- make one more remark
00:17:32.230 --> 00:17:33.860
about this function,
or maybe two more
00:17:33.860 --> 00:17:36.050
remarks about this
function, before we go on.
00:17:36.050 --> 00:17:39.230
Really, you want to know this
graph as well as possible.
00:17:39.230 --> 00:17:42.220
And so there are just
a couple more features.
00:17:42.220 --> 00:17:44.640
And one is enormously
helpful because it
00:17:44.640 --> 00:17:47.780
cuts in half all of
the work that you have.
00:17:47.780 --> 00:17:49.750
and that is the
property that turns out
00:17:49.750 --> 00:17:51.800
that this function is odd.
00:17:51.800 --> 00:17:56.860
Namely, - F(-x) = -F(x).
00:17:56.860 --> 00:18:01.050
That's what's known
as an odd function.
00:18:01.050 --> 00:18:06.840
Now, the reason why it's odd
is that it's the antiderivative
00:18:06.840 --> 00:18:08.500
of something that's even.
00:18:08.500 --> 00:18:10.430
This function in here is even.
00:18:10.430 --> 00:18:14.860
And we nailed it down
so that it was 0 at 0.
00:18:14.860 --> 00:18:17.690
Another way of interpreting
that, and let me show it to you
00:18:17.690 --> 00:18:20.560
underneath, is the following.
00:18:20.560 --> 00:18:24.440
When we look at its derivative,
its derivative, course,
00:18:24.440 --> 00:18:25.370
is the function e^x.
00:18:28.190 --> 00:18:30.040
Sorry, e^(-x^2).
00:18:30.040 --> 00:18:37.430
So that's this shape here.
00:18:37.430 --> 00:18:41.530
And you can see the slope is
0, but-- fairly close to 0,
00:18:41.530 --> 00:18:42.610
but positive along here.
00:18:42.610 --> 00:18:44.560
It's getting, this is
its steepest point.
00:18:44.560 --> 00:18:45.965
This is the highest point here.
00:18:45.965 --> 00:18:47.340
And then it's
leveling off again.
00:18:47.340 --> 00:18:50.670
The slope is going
down, always positive.
00:18:50.670 --> 00:18:56.860
This is the graph
of F' = e^(-x^2).
00:18:56.860 --> 00:19:01.670
Now, the interpretation of
the function that's up above
00:19:01.670 --> 00:19:08.510
is that the value here
is the area from 0 to x.
00:19:08.510 --> 00:19:12.690
So this is area F(x).
00:19:12.690 --> 00:19:16.570
Maybe I'll color it in,
decorate it a little bit.
00:19:16.570 --> 00:19:25.120
So this area here is F(x).
00:19:25.120 --> 00:19:28.840
Now, I want to show
you this odd property,
00:19:28.840 --> 00:19:30.900
by using this symmetry.
00:19:30.900 --> 00:19:34.590
The graph here is even,
so in other words,
00:19:34.590 --> 00:19:39.290
what's back here is exactly
the same as what's forward.
00:19:39.290 --> 00:19:42.580
But now there's a reversal.
00:19:42.580 --> 00:19:44.650
Because we're keeping
track of the area
00:19:44.650 --> 00:19:46.330
starting from 0 going forward.
00:19:46.330 --> 00:19:47.090
That's positive.
00:19:47.090 --> 00:19:49.940
If we go backwards,
it's counted negatively.
00:19:49.940 --> 00:19:51.810
So if we went
backwards to -x, we'd
00:19:51.810 --> 00:19:54.970
get exactly the same as
that green patch over there.
00:19:54.970 --> 00:19:56.810
We'd get a red patch over here.
00:19:56.810 --> 00:20:01.280
But it would be
counted negatively.
00:20:01.280 --> 00:20:04.210
And that's the
property that it's odd.
00:20:04.210 --> 00:20:07.150
You can also check this
by properties of integrals
00:20:07.150 --> 00:20:08.920
directly.
00:20:08.920 --> 00:20:16.190
That would be just
like this process here.
00:20:16.190 --> 00:20:20.280
So it's completely analogous
to checking this formula
00:20:20.280 --> 00:20:25.410
over there.
00:20:25.410 --> 00:20:29.670
So that's one of the comments
I wanted to make about this.
00:20:29.670 --> 00:20:31.900
And why does this
save us a lot of time,
00:20:31.900 --> 00:20:33.100
if we know this is odd?
00:20:33.100 --> 00:20:35.510
Well, we know that the
shape of this branch
00:20:35.510 --> 00:20:37.840
is exactly the reverse,
or the reflection,
00:20:37.840 --> 00:20:39.940
if you like, of the
shape of this one.
00:20:39.940 --> 00:20:41.950
What we want to do is
flip it under the axis
00:20:41.950 --> 00:20:44.960
and then reflect
it over that way.
00:20:44.960 --> 00:20:53.590
And that's the symmetry
property of the graph of F(x).
00:20:53.590 --> 00:20:56.930
Now, the last property
that I want to mention
00:20:56.930 --> 00:21:00.750
is what's happening
with the ends.
00:21:00.750 --> 00:21:03.570
And at the end
there's an asymptote,
00:21:03.570 --> 00:21:05.820
there's a limit here.
00:21:05.820 --> 00:21:10.532
So this is an asymptote.
00:21:10.532 --> 00:21:11.990
And the same thing
down here, which
00:21:11.990 --> 00:21:13.781
will be exactly because
of the odd feature,
00:21:13.781 --> 00:21:15.790
this'll be exactly negative.
00:21:15.790 --> 00:21:19.570
The opposite value over here.
00:21:19.570 --> 00:21:23.570
And you might ask yourself,
what level is this, exactly.
00:21:23.570 --> 00:21:26.850
Now, that level turns out to
be a very important quantity.
00:21:26.850 --> 00:21:28.910
It's interpreted
down here as the area
00:21:28.910 --> 00:21:31.620
under this whole
infinite stretch.
00:21:31.620 --> 00:21:36.670
It's all the way
out to infinity.
00:21:36.670 --> 00:21:38.960
So, let's see.
00:21:38.960 --> 00:21:48.740
What do you think it is?
00:21:48.740 --> 00:21:49.750
You're all clueless.
00:21:49.750 --> 00:21:52.460
Well, maybe not all of you,
you're just afraid to say.
00:21:52.460 --> 00:21:54.140
So it's obvious.
00:21:54.140 --> 00:21:57.040
It's the square root of pi/2.
00:21:57.040 --> 00:22:00.170
That was right on the tip
of your tongue, wasn't it?
00:22:00.170 --> 00:22:01.610
STUDENT: Ah, yes.
00:22:01.610 --> 00:22:04.100
PROFESSOR: Right, so this
is actually very un-obvious,
00:22:04.100 --> 00:22:06.290
but it's a very
important quantity.
00:22:06.290 --> 00:22:08.750
And it's an amazing
fact that this thing
00:22:08.750 --> 00:22:10.720
approaches this number.
00:22:10.720 --> 00:22:18.060
And it's something that people
worried about for many years
00:22:18.060 --> 00:22:22.860
before actually nailing down.
00:22:22.860 --> 00:22:24.680
And so what I just
claimed here is
00:22:24.680 --> 00:22:28.540
that the limit as x
approaches infinity of F(x)
00:22:28.540 --> 00:22:33.300
is equal to the square
root of pi over 2.
00:22:33.300 --> 00:22:35.340
And similarly, if you
do it to minus infinity,
00:22:35.340 --> 00:22:38.370
you'll get minus square
root of pi over 2.
00:22:38.370 --> 00:22:42.090
And for this reason, people
introduced a new function
00:22:42.090 --> 00:22:44.010
because they like the number 1.
00:22:44.010 --> 00:22:49.740
This function is erf,
short for error function.
00:22:49.740 --> 00:22:54.470
And it's 2 over the square root
of pi times the integral from 0
00:22:54.470 --> 00:22:57.630
to x, e^(-t^2) dt.
00:22:57.630 --> 00:22:59.650
In other words, it's
just our original,
00:22:59.650 --> 00:23:03.110
our previous function
multiplied by 2
00:23:03.110 --> 00:23:08.240
over the square root of pi.
00:23:08.240 --> 00:23:10.880
And that's the function which
gets tabulated quite a lot.
00:23:10.880 --> 00:23:13.260
You'll see it on the
internet everywhere,
00:23:13.260 --> 00:23:15.856
and it's a very
important function.
00:23:15.856 --> 00:23:17.730
There are other
normalizations that are used,
00:23:17.730 --> 00:23:20.640
and the discussions of
the other normalizations
00:23:20.640 --> 00:23:23.500
are in your problems.
00:23:23.500 --> 00:23:27.540
This is one of them, and another
one is in your exercises.
00:23:27.540 --> 00:23:31.130
The standard normal
distribution.
00:23:31.130 --> 00:23:33.200
There are tons of
functions like this,
00:23:33.200 --> 00:23:35.920
which are new functions
that we can get at once we
00:23:35.920 --> 00:23:37.650
have the tool of integrals.
00:23:37.650 --> 00:23:40.460
And I'll write down just
one or two more, just so
00:23:40.460 --> 00:23:42.760
that you'll see the variety.
00:23:42.760 --> 00:23:49.080
Here's one which is
called a Fresnel integral.
00:23:49.080 --> 00:23:51.420
On your problem set
next week, we'll
00:23:51.420 --> 00:23:57.090
do the other Fresnel integral,
we'll look at this one.
00:23:57.090 --> 00:24:01.990
These functions cannot be
expressed in elementary terms.
00:24:01.990 --> 00:24:11.890
The one on your homework
for this week was this one.
00:24:11.890 --> 00:24:14.980
This one comes up
in Fourier analysis.
00:24:14.980 --> 00:24:19.910
And I'm going to just tell you
maybe one more such function.
00:24:19.910 --> 00:24:22.220
There's a function
which is called
00:24:22.220 --> 00:24:30.220
Li(x), logarithmic integral
of x, which is this guy.
00:24:30.220 --> 00:24:33.640
The reciprocal of the
logarithm, the natural log.
00:24:33.640 --> 00:24:36.640
And the significance
of this one is
00:24:36.640 --> 00:24:45.270
that Li(x) is approximately
equal to the number of primes
00:24:45.270 --> 00:24:49.680
less than x.
00:24:49.680 --> 00:24:54.430
And, in fact, if you can make
this as precise as possible,
00:24:54.430 --> 00:24:59.220
you'll be famous for millennia,
because this is known
00:24:59.220 --> 00:25:01.830
as the Riemann hypothesis.
00:25:01.830 --> 00:25:05.370
Exactly how closely this
approximation occurs.
00:25:05.370 --> 00:25:08.470
But it's a hard
problem, and already
00:25:08.470 --> 00:25:10.500
a century ago the
prime number theorem,
00:25:10.500 --> 00:25:14.890
which established this
connection was extremely
00:25:14.890 --> 00:25:18.450
important to progress in math.
00:25:18.450 --> 00:25:19.340
Yeah, question.
00:25:19.340 --> 00:25:21.350
STUDENT: [INAUDIBLE]
00:25:21.350 --> 00:25:23.700
PROFESSOR: Is this stuff
you're supposed to understand.
00:25:23.700 --> 00:25:24.840
That's a good question.
00:25:24.840 --> 00:25:26.300
I love that question.
00:25:26.300 --> 00:25:31.570
The answer is, this is, so we
launched off into something
00:25:31.570 --> 00:25:32.080
here.
00:25:32.080 --> 00:25:34.310
And let me just
explain it to you.
00:25:34.310 --> 00:25:36.560
I'm going to be talking
a fair amount more
00:25:36.560 --> 00:25:41.570
about this particular function,
because it's associated
00:25:41.570 --> 00:25:43.240
to the normal distribution.
00:25:43.240 --> 00:25:45.240
And I'm going to let you
get familiar with it.
00:25:45.240 --> 00:25:47.810
What I'm doing here
is purely cultural.
00:25:47.810 --> 00:25:51.500
Well, after this panel, what
I'm doing is purely cultural.
00:25:51.500 --> 00:25:53.780
Just saying there's
a lot of other beasts
00:25:53.780 --> 00:25:55.810
out there in the world.
00:25:55.810 --> 00:25:57.930
And one of them
is called C of x--
00:25:57.930 --> 00:26:01.460
So we'll have a just a very
passing familiarity with one
00:26:01.460 --> 00:26:02.920
or two of these functions.
00:26:02.920 --> 00:26:05.750
But there are literally
dozens and dozens of them.
00:26:05.750 --> 00:26:08.980
The only thing that you'll
need to do with such functions
00:26:08.980 --> 00:26:12.350
is things like understanding
the derivative,
00:26:12.350 --> 00:26:14.960
the second derivative,
and tracking
00:26:14.960 --> 00:26:16.140
what the function does.
00:26:16.140 --> 00:26:18.950
Sketching the same way you
did with any other tool.
00:26:18.950 --> 00:26:22.870
So we're going to do this type
of thing with these functions.
00:26:22.870 --> 00:26:25.659
And I'll have to
lead you through.
00:26:25.659 --> 00:26:27.450
If I wanted to ask you
a question about one
00:26:27.450 --> 00:26:30.070
of these functions,
I have to tell you
00:26:30.070 --> 00:26:32.470
exactly what I'm aiming for.
00:26:32.470 --> 00:26:35.747
Yeah, another question.
00:26:35.747 --> 00:26:36.580
STUDENT: [INAUDIBLE]
00:26:36.580 --> 00:26:37.538
PROFESSOR: Yeah, I did.
00:26:37.538 --> 00:26:43.520
I called these guys
Fresnel integrals.
00:26:43.520 --> 00:26:47.460
The guy's name is Fresnel.
00:26:47.460 --> 00:26:49.180
It's just named after a person.
00:26:49.180 --> 00:26:51.980
But, and this one, Li
is logarithmic integral,
00:26:51.980 --> 00:26:53.250
it's not named after a person.
00:26:53.250 --> 00:26:56.820
Logarithm is not
somebody's name.
00:26:56.820 --> 00:27:01.510
So look, in fact this
will be mentioned also
00:27:01.510 --> 00:27:03.284
on a problem set,
but I don't expect
00:27:03.284 --> 00:27:04.450
you to remember these names.
00:27:04.450 --> 00:27:05.940
In particular,
that you definitely
00:27:05.940 --> 00:27:07.600
don't want to try to remember.
00:27:07.600 --> 00:27:08.730
Yes, another question.
00:27:08.730 --> 00:27:10.020
STUDENT: [INAUDIBLE]
00:27:10.020 --> 00:27:15.240
PROFESSOR: The question is,
will we prove this limit.
00:27:15.240 --> 00:27:17.470
And the answer is
yes, if we have time.
00:27:17.470 --> 00:27:21.482
It'll be in about a week or so.
00:27:21.482 --> 00:27:22.690
We're not going to do it now.
00:27:22.690 --> 00:27:29.010
It takes us quite a
bit of work to do it.
00:27:29.010 --> 00:27:32.630
OK.
00:27:32.630 --> 00:27:36.260
I'm going to change gears
now, I'm going to shift gears.
00:27:36.260 --> 00:27:41.400
And we're going to go back
to a more standard thing
00:27:41.400 --> 00:27:44.650
which has to do with just
setting up integrals.
00:27:44.650 --> 00:27:47.820
And this has to do with
understanding where integrals
00:27:47.820 --> 00:27:50.590
play a role, and they play
a role in cumulative sums,
00:27:50.590 --> 00:27:52.160
in evaluating things.
00:27:52.160 --> 00:27:54.150
This is much more
closely associated
00:27:54.150 --> 00:27:57.020
with the first
Fundamental Theorem.
00:27:57.020 --> 00:27:59.060
That is, we'll
take, today we were
00:27:59.060 --> 00:28:02.260
talking about how integrals
are formulas for functions.
00:28:02.260 --> 00:28:04.730
Or solutions to
differential equations.
00:28:04.730 --> 00:28:08.400
We're going to go back
and talk about integrals
00:28:08.400 --> 00:28:11.040
as being the answer to
a question as opposed
00:28:11.040 --> 00:28:14.400
to what we've done now.
00:28:14.400 --> 00:28:18.470
So in other words,
and the first example,
00:28:18.470 --> 00:28:20.370
or most of the examples
for now, are going
00:28:20.370 --> 00:28:22.880
to be taken from geometry.
00:28:22.880 --> 00:28:27.440
Later on we'll get
to probability.
00:28:27.440 --> 00:28:44.550
And the first topic is
just areas between curves.
00:28:44.550 --> 00:28:46.890
Here's the idea.
00:28:46.890 --> 00:28:50.310
If you have a couple of curves
that look like this and maybe
00:28:50.310 --> 00:28:54.140
like this, and you want
to start at a place a
00:28:54.140 --> 00:28:59.270
and you want to
end at a place b,
00:28:59.270 --> 00:29:06.720
then you can chop it up the same
way we did with Riemann sums.
00:29:06.720 --> 00:29:10.685
And take a chunk
that looks like this.
00:29:10.685 --> 00:29:12.810
And I'm going to write the
thickness of that chunk.
00:29:12.810 --> 00:29:14.710
Well, let's give
these things names.
00:29:14.710 --> 00:29:20.510
Let's say the top curve is f(x),
and the bottom curve is g(x).
00:29:20.510 --> 00:29:26.980
And then this thickness
is going to be dx.
00:29:26.980 --> 00:29:30.020
That's the thickness.
00:29:30.020 --> 00:29:32.300
And what is the height?
00:29:32.300 --> 00:29:34.150
Well, the height
is the difference
00:29:34.150 --> 00:29:38.970
between the top value
and the bottom value.
00:29:38.970 --> 00:29:44.120
So here we have
(f(x) - g(x)) dx.
00:29:44.120 --> 00:29:50.120
This is, if you like, base
times-- Whoops, backwards.
00:29:50.120 --> 00:29:54.850
This is height, and this is
the base of the rectangle.
00:29:54.850 --> 00:29:56.850
And these are
approximately correct.
00:29:56.850 --> 00:29:59.690
But of course, only in limit
when this is an infinitesimal,
00:29:59.690 --> 00:30:03.890
is it exactly right.
00:30:03.890 --> 00:30:10.337
In order to get the whole
area, I have add these guys up.
00:30:10.337 --> 00:30:11.920
So I'm going to
integrate from a to b.
00:30:11.920 --> 00:30:14.350
That's summing them,
that's adding them up.
00:30:14.350 --> 00:30:16.470
And that's going to be my area.
00:30:16.470 --> 00:30:27.280
So that's the story here.
00:30:27.280 --> 00:30:30.020
Now, let me just say
two things about this.
00:30:30.020 --> 00:30:33.360
First of all, on a very
abstract level before we get
00:30:33.360 --> 00:30:36.340
started with details of
more complicated problems.
00:30:36.340 --> 00:30:39.390
The first one is
that every problem
00:30:39.390 --> 00:30:41.800
that I'm going to be
talking about from now
00:30:41.800 --> 00:30:45.730
on for several days, involves
the following collection
00:30:45.730 --> 00:30:48.700
of-- the following goals.
00:30:48.700 --> 00:30:52.090
I want to identify
something to integrate.
00:30:52.090 --> 00:30:58.010
That's called an integrand.
00:30:58.010 --> 00:31:06.420
And I want to identify what
are known as the limits.
00:31:06.420 --> 00:31:10.290
The whole game is
simply to figure out
00:31:10.290 --> 00:31:13.240
what a, b, and this
quantity is here.
00:31:13.240 --> 00:31:15.480
Whatever it is.
00:31:15.480 --> 00:31:18.090
And the minute we have that,
we can calculate the integral
00:31:18.090 --> 00:31:19.810
if we like.
00:31:19.810 --> 00:31:21.650
We have numerical
procedures or maybe we
00:31:21.650 --> 00:31:23.490
have analytic
procedures, but anyway we
00:31:23.490 --> 00:31:25.150
can get at the integral.
00:31:25.150 --> 00:31:27.580
The goal here is to set them up.
00:31:27.580 --> 00:31:31.720
And in order to set them up, you
must know these three things.
00:31:31.720 --> 00:31:33.550
The lower limit,
the upper limit,
00:31:33.550 --> 00:31:37.920
and what we're integrating.
00:31:37.920 --> 00:31:42.280
If you leave one of these out,
it's like the following thing.
00:31:42.280 --> 00:31:45.450
I ask you what the
area of this region is.
00:31:45.450 --> 00:31:48.740
If I left out this end,
how could I possibly know?
00:31:48.740 --> 00:31:51.240
I don't even know where it
starts, so how can I figure out
00:31:51.240 --> 00:31:52.990
what this area is if
I haven't identified
00:31:52.990 --> 00:31:55.230
what the left side is.
00:31:55.230 --> 00:31:58.210
I can't leave out the bottom.
00:31:58.210 --> 00:32:00.560
It's sitting here,
in this formula.
00:32:00.560 --> 00:32:03.007
Because I need to
know where it is.
00:32:03.007 --> 00:32:05.340
And I need to know the top
and I need to know this side.
00:32:05.340 --> 00:32:07.600
Those are the four
sides of the figure.
00:32:07.600 --> 00:32:10.190
If I don't incorporate
them into the information,
00:32:10.190 --> 00:32:11.640
I'll never get anything out.
00:32:11.640 --> 00:32:13.620
So I need to know everything.
00:32:13.620 --> 00:32:15.360
And I need to know
exactly those things,
00:32:15.360 --> 00:32:20.740
in order to have a
formula for the area.
00:32:20.740 --> 00:32:23.720
Now, when this gets
carried out in practice,
00:32:23.720 --> 00:32:27.960
as we will do now in
our first example,
00:32:27.960 --> 00:32:29.650
it's more complicated
than it looks.
00:32:29.650 --> 00:32:48.680
So here's our first example:
Find the area between x = y^2
00:32:48.680 --> 00:32:57.650
and y = x - 2.
00:32:57.650 --> 00:33:00.110
This is our first example.
00:33:00.110 --> 00:33:04.950
Let me make sure that I chose
the example that I wanted to.
00:33:04.950 --> 00:33:08.420
Yeah.
00:33:08.420 --> 00:33:20.300
Now, there's a first step in
figuring these things out.
00:33:20.300 --> 00:33:27.514
And this is that you
must draw a picture.
00:33:27.514 --> 00:33:28.930
If you don't draw
a picture you'll
00:33:28.930 --> 00:33:30.827
never figure out
what this area is,
00:33:30.827 --> 00:33:32.410
because you'll never
figure out what's
00:33:32.410 --> 00:33:36.290
what between these curves.
00:33:36.290 --> 00:33:40.540
The first curve, y =
x^2, is a parabola.
00:33:40.540 --> 00:33:42.990
But x is a function of y.
00:33:42.990 --> 00:33:45.090
It's pointing this way.
00:33:45.090 --> 00:33:47.450
So it's this parabola here.
00:33:47.450 --> 00:33:50.690
That's y = x^2.
00:33:50.690 --> 00:33:57.910
Whoops, x = y^2.
00:33:57.910 --> 00:34:06.700
The second curve is a line,
a straight line of slope 1,
00:34:06.700 --> 00:34:09.670
starting at x = 2, y = 0.
00:34:09.670 --> 00:34:13.160
It goes through this place
here, which is 2 over
00:34:13.160 --> 00:34:20.060
and has slope 1,
so it does this.
00:34:20.060 --> 00:34:22.590
And this shape in
here is what we mean
00:34:22.590 --> 00:34:24.100
by the area between the curves.
00:34:24.100 --> 00:34:27.050
Now that we see what it
is, we have a better idea
00:34:27.050 --> 00:34:28.130
of what our goal is.
00:34:28.130 --> 00:34:39.310
If you haven't drawn
it, you have no hope.
00:34:39.310 --> 00:34:45.310
Now, I'm going to describe two
ways of getting at this area
00:34:45.310 --> 00:34:50.660
here.
00:34:50.660 --> 00:34:59.970
And the first one is
motivated by the shape
00:34:59.970 --> 00:35:02.970
that I just
described right here.
00:35:02.970 --> 00:35:07.210
Namely, I'm going to use it
in a straightforward way.
00:35:07.210 --> 00:35:12.560
I'm going to chop things up
into these vertical pieces
00:35:12.560 --> 00:35:17.160
just as I did right there.
00:35:17.160 --> 00:35:19.415
Now, here's the
difficulty with that.
00:35:19.415 --> 00:35:27.000
The difficulty is that the
upper curve here has one formula
00:35:27.000 --> 00:35:28.990
but the lower curve
shifts from being
00:35:28.990 --> 00:35:33.120
a part of the parabola to being
a part of the straight line.
00:35:33.120 --> 00:35:35.160
That means that there are
two different formulas
00:35:35.160 --> 00:35:36.930
for the lower function.
00:35:36.930 --> 00:35:39.200
And the only way
to accommodate that
00:35:39.200 --> 00:35:42.780
is to separate this
up into two halves.
00:35:42.780 --> 00:35:44.980
Separate it out into two halves.
00:35:44.980 --> 00:35:50.320
I'm going to have to
divide it right here.
00:35:50.320 --> 00:35:52.870
So we must break
it into two pieces
00:35:52.870 --> 00:35:57.280
and find the integral of
one half and the other half.
00:35:57.280 --> 00:35:57.780
Question?
00:35:57.780 --> 00:36:06.269
STUDENT: [INAUDIBLE]
00:36:06.269 --> 00:36:08.060
PROFESSOR: So, you're
one step ahead of me.
00:36:08.060 --> 00:36:09.570
We'll also have to be
sure to distinguish
00:36:09.570 --> 00:36:12.130
between the top branch and the
bottom branch of the parabola,
00:36:12.130 --> 00:36:14.430
which we're about to do.
00:36:14.430 --> 00:36:17.870
Now, in order to
distinguish what's going on
00:36:17.870 --> 00:36:22.000
I actually have to
use multi colors here.
00:36:22.000 --> 00:36:24.680
And so we will do that.
00:36:24.680 --> 00:36:29.720
First there's the top
part, which is orange.
00:36:29.720 --> 00:36:32.200
That's the top part.
00:36:32.200 --> 00:36:33.990
I'll call it top.
00:36:33.990 --> 00:36:41.600
And then there's the bottom
part, which has two halves.
00:36:41.600 --> 00:36:53.280
They are pink, and I
guess this is blue.
00:36:53.280 --> 00:37:01.240
All right, so now let's
see what's happening.
00:37:01.240 --> 00:37:07.110
The most important two points
that I have to figure out
00:37:07.110 --> 00:37:08.580
in order to get started here.
00:37:08.580 --> 00:37:10.480
Well, really I'm going to have
to figure out three points,
00:37:10.480 --> 00:37:11.010
I claim.
00:37:11.010 --> 00:37:13.460
I'm going to have to figure
out where this point is.
00:37:13.460 --> 00:37:17.650
Where this point is,
and where that point is.
00:37:17.650 --> 00:37:20.220
If I know where these
three points are,
00:37:20.220 --> 00:37:23.990
then I have a chance of knowing
where to start, where to end,
00:37:23.990 --> 00:37:25.390
and so forth.
00:37:25.390 --> 00:37:26.170
Another question.
00:37:26.170 --> 00:37:27.490
STUDENT: [INAUDIBLE]
00:37:27.490 --> 00:37:31.110
PROFESSOR: Could you speak up?
00:37:31.110 --> 00:37:37.539
STUDENT: [INAUDIBLE]
00:37:37.539 --> 00:37:39.080
PROFESSOR: The
question is, why do we
00:37:39.080 --> 00:37:41.450
need to split up the area.
00:37:41.450 --> 00:37:44.170
And I think in order to
answer that question further,
00:37:44.170 --> 00:37:46.930
I'm going to have to go into
the details of the method,
00:37:46.930 --> 00:37:51.480
and then you'll see
where it's necessary.
00:37:51.480 --> 00:37:54.440
So the first step is that
I'm going to figure out
00:37:54.440 --> 00:37:57.160
what these three points are.
00:37:57.160 --> 00:38:02.460
This one is kind of easy;
it's the point (0, 0).
00:38:02.460 --> 00:38:04.880
This point down here
and this point up here
00:38:04.880 --> 00:38:08.010
are intersections
of the two curves.
00:38:08.010 --> 00:38:11.150
I can identify them by
the following equation.
00:38:11.150 --> 00:38:21.330
I need to see where
these curves intersect.
00:38:21.330 --> 00:38:26.690
At what, well, if I plug in
x = y^2, I get y = y^2 - 2.
00:38:26.690 --> 00:38:29.010
And then I can solve
this quadratic equation.
00:38:29.010 --> 00:38:33.840
y^2 - y - 2 = 0.
00:38:33.840 --> 00:38:36.620
So (y - 2) (y+1) = 0.
00:38:36.620 --> 00:38:39.360
= 0.
00:38:39.360 --> 00:38:47.960
And this is telling me
that y = 2 or y = -1.
00:38:52.630 --> 00:38:54.250
So I've found y = -1.
00:38:54.250 --> 00:39:00.430
That means this point down
here has second entry -1.
00:39:00.430 --> 00:39:04.770
Its first entry, its x-value,
I can get from this formula
00:39:04.770 --> 00:39:06.890
here or the other formula.
00:39:06.890 --> 00:39:10.800
I have to square,
this, -1^2 = 1.
00:39:10.800 --> 00:39:15.020
So that's the formula
for this point.
00:39:15.020 --> 00:39:20.180
And the other point
has second entry 2.
00:39:20.180 --> 00:39:22.590
And, again, with
his formula y = x^2,
00:39:22.590 --> 00:39:31.790
I have to square y to
get x, so this is 4.
00:39:31.790 --> 00:39:37.340
Now, I claim I have enough
data to get started.
00:39:37.340 --> 00:39:41.160
But maybe I'll identify
one more thing.
00:39:41.160 --> 00:39:48.890
I need the top, the bottom
left, and the bottom right.
00:39:48.890 --> 00:39:54.770
The top is the formula for
this branch of x = y^2,
00:39:54.770 --> 00:39:57.880
which is in the
positive y region.
00:39:57.880 --> 00:40:05.130
And that is y is equal
to square root of x.
00:40:05.130 --> 00:40:09.390
The bottom curve,
part of the parabola,
00:40:09.390 --> 00:40:21.980
so this is the bottom left, is
y equals minus square root x.
00:40:21.980 --> 00:40:24.210
That's the other branch
of the square root.
00:40:24.210 --> 00:40:26.561
And this is exactly what
you were asking before.
00:40:26.561 --> 00:40:28.810
And this is, we have to
distinguish between these two.
00:40:28.810 --> 00:40:31.510
And the point is, these
formulas really are different.
00:40:31.510 --> 00:40:34.400
They're not the same.
00:40:34.400 --> 00:40:37.860
Now, the last bit is the
bottom right chunk here,
00:40:37.860 --> 00:40:39.910
which is this pink part.
00:40:39.910 --> 00:40:44.330
Bottom right.
00:40:44.330 --> 00:40:49.060
And that one is the
formula for the line.
00:40:49.060 --> 00:40:55.990
And that's y = x - 2.
00:40:55.990 --> 00:41:03.450
Now I'm ready to find the area.
00:41:03.450 --> 00:41:06.070
It's going to be in two chunks.
00:41:06.070 --> 00:41:15.060
This is the left part,
plus the right part.
00:41:15.060 --> 00:41:18.210
And the left part, and I want
to set it up as an integral,
00:41:18.210 --> 00:41:21.350
I want there to be a dx and here
I want to set up an integral
00:41:21.350 --> 00:41:23.740
and I want it to be dx.
00:41:23.740 --> 00:41:26.011
I need to figure out
what the range of x is.
00:41:26.011 --> 00:41:27.510
So, first I'm going
to-- well, let's
00:41:27.510 --> 00:41:37.640
leave ourselves a little
more room than that.
00:41:37.640 --> 00:41:39.310
Just to be safe.
00:41:39.310 --> 00:41:45.740
OK, here's the right.
00:41:45.740 --> 00:41:48.490
So here we have our dx.
00:41:48.490 --> 00:41:53.830
Now, I need to figure out the
starting place and the ending
00:41:53.830 --> 00:41:54.960
place.
00:41:54.960 --> 00:41:57.425
So the starting place
is the leftmost place.
00:41:57.425 --> 00:42:00.000
The leftmost place is over here.
00:42:00.000 --> 00:42:02.890
And x = 0 there.
00:42:02.890 --> 00:42:05.900
So we're going to travel
from this vertical line
00:42:05.900 --> 00:42:08.420
to the green line.
00:42:08.420 --> 00:42:09.070
Over here.
00:42:09.070 --> 00:42:13.100
And that's from 0 to 1.
00:42:13.100 --> 00:42:18.400
And the difference between the
orange curve and the blue curve
00:42:18.400 --> 00:42:22.330
is what I call top and
bottom left, over there.
00:42:22.330 --> 00:42:32.280
So that is square root of x
minus minus square root of x.
00:42:32.280 --> 00:42:41.170
Again, this is what I call
top, and this was bottom.
00:42:41.170 --> 00:42:48.850
But only the left.
00:42:48.850 --> 00:42:51.960
I claim that's giving me
the left half of this,
00:42:51.960 --> 00:42:55.650
the left section
of this diagram.
00:42:55.650 --> 00:42:59.780
Now I'm going to do the
right section of the diagram.
00:42:59.780 --> 00:43:02.040
I start at 1.
00:43:02.040 --> 00:43:04.160
The lower limit is 1.
00:43:04.160 --> 00:43:08.490
And I go all the way
to this point here.
00:43:08.490 --> 00:43:11.160
Which is the last bit.
00:43:11.160 --> 00:43:13.070
And that's going to be x = 4.
00:43:13.070 --> 00:43:19.130
The upper limit here is 4.
00:43:19.130 --> 00:43:21.410
And now I have to take the
difference between the top
00:43:21.410 --> 00:43:22.570
and the bottom again.
00:43:22.570 --> 00:43:24.800
The top is square root
of x all over again.
00:43:24.800 --> 00:43:26.570
But the bottom has changed.
00:43:26.570 --> 00:43:28.660
The bottom is now
the quantity x - 2.
00:43:31.430 --> 00:43:32.990
Please don't forget
your parenthesis.
00:43:32.990 --> 00:43:44.000
There's going to be minus
signs and cancellations.
00:43:44.000 --> 00:43:46.420
Now, this is almost
the end of the problem.
00:43:46.420 --> 00:43:48.270
The rest of it is routine.
00:43:48.270 --> 00:43:52.440
We would just have to
evaluate these integrals.
00:43:52.440 --> 00:43:57.500
And, fortunately, I'm
going to spare you that.
00:43:57.500 --> 00:43:59.330
We're not going to
bother to do it.
00:43:59.330 --> 00:44:01.050
That's the easy part.
00:44:01.050 --> 00:44:03.466
We're not going to do it.
00:44:03.466 --> 00:44:05.840
But I'm going to show you that
there's a much quicker way
00:44:05.840 --> 00:44:07.320
with this integral.
00:44:07.320 --> 00:44:09.980
And with this area calculation.
00:44:09.980 --> 00:44:10.910
Right now.
00:44:10.910 --> 00:44:18.030
The quicker way is what you see
when you see how long this is.
00:44:18.030 --> 00:44:20.310
And you see that
there's another device
00:44:20.310 --> 00:44:24.180
that you can use that looks
similar in principle to this,
00:44:24.180 --> 00:44:28.250
but reverses the
roles of x and y.
00:44:28.250 --> 00:44:35.530
And the other device, which I'll
draw over here, schematically.
00:44:35.530 --> 00:44:47.130
No, maybe I'll draw it
on this blackboard here.
00:44:47.130 --> 00:44:55.380
So, Method 2, if you
like, this was Method 1,
00:44:55.380 --> 00:45:04.180
and we should call
it the hard way.
00:45:04.180 --> 00:45:10.970
Method 2, which is
better in this case,
00:45:10.970 --> 00:45:24.710
is to use horizontal slices.
00:45:24.710 --> 00:45:33.100
Let me draw the picture,
at least schematically.
00:45:33.100 --> 00:45:35.230
Here's our picture
that we had before.
00:45:35.230 --> 00:45:38.460
And now instead of
slicing it vertically,
00:45:38.460 --> 00:45:40.530
I'm going to slice
it horizontally.
00:45:40.530 --> 00:45:44.910
Like this.
00:45:44.910 --> 00:45:49.740
Now, the dimensions
have different names.
00:45:49.740 --> 00:45:51.990
But the principle is similar.
00:45:51.990 --> 00:45:55.170
The width, we now call dy.
00:45:55.170 --> 00:45:58.600
Because it's the change in y.
00:45:58.600 --> 00:46:05.700
And this distance here, from
the left end to the right end,
00:46:05.700 --> 00:46:09.950
we have to figure out what the
formulas for those things are.
00:46:09.950 --> 00:46:16.500
So on the left, maybe I'll
draw them color coded again.
00:46:16.500 --> 00:46:19.630
So here's a left.
00:46:19.630 --> 00:46:23.400
And, whoops, orange
is right, I guess.
00:46:23.400 --> 00:46:24.580
So here we go.
00:46:24.580 --> 00:46:30.470
So we have the left -
which is this green -
00:46:30.470 --> 00:46:46.090
is x = y^2 And the right,
which is orange, is y = x - 2.
00:46:46.090 --> 00:46:49.820
And now in order
to use this, it's
00:46:49.820 --> 00:46:51.865
going to turn out that
we want to write x
00:46:51.865 --> 00:46:53.170
as-- we want to reverse roles.
00:46:53.170 --> 00:46:57.820
So we want to write this
as x is a function of y.
00:46:57.820 --> 00:47:04.560
So we'll use it in this form.
00:47:04.560 --> 00:47:11.950
And now I want to set
up the integral for you.
00:47:11.950 --> 00:47:21.200
This time, the area is equal to
an integral in the dy variable.
00:47:21.200 --> 00:47:26.350
And its starting
place is down here.
00:47:26.350 --> 00:47:28.960
And its ending
place is up there.
00:47:28.960 --> 00:47:31.900
This is the lowest value of y,
and this is the top value of y.
00:47:31.900 --> 00:47:34.940
And we've already
computed those things.
00:47:34.940 --> 00:47:39.830
The lowest level of y is -1.
00:47:39.830 --> 00:47:42.200
So this is y = -1.
00:47:42.200 --> 00:47:45.660
And this top value is y = 2.
00:47:45.660 --> 00:47:51.490
So this goes from -1 to 2.
00:47:51.490 --> 00:47:56.690
And now the difference
is this distance here,
00:47:56.690 --> 00:47:58.870
the distance between
the rightmost point
00:47:58.870 --> 00:47:59.880
and the leftmost point.
00:47:59.880 --> 00:48:02.020
Those are the two dimensions.
00:48:02.020 --> 00:48:04.590
So again, it's a rectangle
but its horizontal is long
00:48:04.590 --> 00:48:07.570
and its vertical is very short.
00:48:07.570 --> 00:48:08.320
And what are they?
00:48:08.320 --> 00:48:11.530
It's the difference between
the right and the left.
00:48:11.530 --> 00:48:21.760
The right-hand is y + 2,
and the right-hand is y^2.
00:48:21.760 --> 00:48:26.990
So this is the formula.
00:48:26.990 --> 00:48:35.750
STUDENT: [INAUDIBLE]
00:48:35.750 --> 00:48:39.090
PROFESSOR: What
was the question?
00:48:39.090 --> 00:48:41.630
Why is it right minus left?
00:48:41.630 --> 00:48:42.770
That's very important.
00:48:42.770 --> 00:48:44.540
Why is it right minus left?
00:48:44.540 --> 00:48:47.230
And that's actually the point
that I was about to make.
00:48:47.230 --> 00:48:48.400
Which is this.
00:48:48.400 --> 00:48:53.860
That y + 2, which is the
right, is bigger than y^2,
00:48:53.860 --> 00:48:55.480
which is the left.
00:48:55.480 --> 00:49:00.320
So that means that y
+ 2 - y^2 is positive.
00:49:00.320 --> 00:49:02.830
If you do it backwards, you'll
always get a negative number
00:49:02.830 --> 00:49:07.300
and you'll always
get the wrong answer.
00:49:07.300 --> 00:49:10.860
So this is the right-hand
end minus the left-hand end
00:49:10.860 --> 00:49:14.300
gives you a positive number.
00:49:14.300 --> 00:49:16.760
And it's not obvious,
actually, where you are.
00:49:16.760 --> 00:49:19.030
There's another
double-check, by the way.
00:49:19.030 --> 00:49:22.630
When you look at this quantity,
you see that the ends pinch.
00:49:22.630 --> 00:49:25.320
And that's exactly
the crossover points.
00:49:25.320 --> 00:49:30.950
That is, when y =
-1, y + 2 - y^2 = 0.
00:49:30.950 --> 00:49:36.940
And when y = 2, y + 2 - y^2 = 0.
00:49:36.940 --> 00:49:38.740
And that's not an
accident, that's
00:49:38.740 --> 00:49:44.600
exactly the geometry of the
shape that we picked out there.
00:49:44.600 --> 00:49:46.560
So this is the technique.
00:49:46.560 --> 00:49:51.200
Now, this is a much
more routine integral.
00:49:51.200 --> 00:49:54.080
I'm not going to carry it out,
I'll just do one last step.
00:49:54.080 --> 00:50:01.400
Which is that this is
y^2 / 2 + 2y - y^3 / 3,
00:50:01.400 --> 00:50:03.600
evaluated at -1 and 2.
00:50:03.600 --> 00:50:08.310
Which, if you work
it out, is 9/2.
00:50:08.310 --> 00:50:10.230
So we're done for today.
00:50:10.230 --> 00:50:12.960
And tomorrow we'll do
more volumes, more things,
00:50:12.960 --> 00:50:15.070
including three dimensions.