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PROFESSOR: So we're going
on to the third unit here.
00:00:24.910 --> 00:00:31.030
So we're getting
started with Unit 3.
00:00:31.030 --> 00:00:38.330
And this is our
intro to integration.
00:00:38.330 --> 00:00:44.820
It's basically the
second half of calculus
00:00:44.820 --> 00:00:51.320
after differentiation.
00:00:51.320 --> 00:00:55.600
Today what I'll talk
about is what are
00:00:55.600 --> 00:01:16.450
known as definite integrals.
00:01:16.450 --> 00:01:19.380
Actually, it looks
like, are we missing
00:01:19.380 --> 00:01:21.580
a bunch of overhead lights?
00:01:21.580 --> 00:01:24.260
Is there a reason for that?
00:01:24.260 --> 00:01:27.320
Hmm.
00:01:27.320 --> 00:01:30.460
Let's see.
00:01:30.460 --> 00:01:32.650
Aah.
00:01:32.650 --> 00:01:34.360
All right.
00:01:34.360 --> 00:01:39.200
OK, that's a little
brighter now.
00:01:39.200 --> 00:01:39.700
All right.
00:01:39.700 --> 00:01:47.420
So the idea of
definite integrals
00:01:47.420 --> 00:01:49.740
can be presented in
a number of ways.
00:01:49.740 --> 00:01:53.707
But I will be consistent with
the rest of the presentation
00:01:53.707 --> 00:01:54.290
in the course.
00:01:54.290 --> 00:01:57.060
We're going to start with
the geometric point of view.
00:01:57.060 --> 00:01:59.990
And the geometric
point of view is,
00:01:59.990 --> 00:02:13.680
the problem we want to solve is
to find the area under a curve.
00:02:13.680 --> 00:02:16.460
The other point of
view that one can take,
00:02:16.460 --> 00:02:18.590
and we'll mention that at
the end of this lecture,
00:02:18.590 --> 00:02:26.220
is the idea of a cumulative sum.
00:02:26.220 --> 00:02:30.687
So keep that in mind that
there's a lot going on here.
00:02:30.687 --> 00:02:32.520
And there are many
different interpretations
00:02:32.520 --> 00:02:37.930
of what the integral is.
00:02:37.930 --> 00:02:41.710
Now, so let's draw
a picture here.
00:02:41.710 --> 00:02:44.670
I'll start at a place
a and end at a place b.
00:02:44.670 --> 00:02:46.720
And I have some curve here.
00:02:46.720 --> 00:02:55.100
And what I have in mind
is to find this area here.
00:02:55.100 --> 00:02:56.600
And, of course, in
order to do that,
00:02:56.600 --> 00:02:58.985
I need more information
than just where we start
00:02:58.985 --> 00:02:59.760
and where we end.
00:02:59.760 --> 00:03:01.860
I also need the
bottom and the top.
00:03:01.860 --> 00:03:05.170
By convention, the bottom
is the x axis and the top
00:03:05.170 --> 00:03:11.670
is the curve that we've
specified, which is y = f(x).
00:03:11.670 --> 00:03:15.660
And we have a notation
for this, which
00:03:15.660 --> 00:03:18.810
is the notation using
calculus for this as opposed
00:03:18.810 --> 00:03:20.490
to some geometric notation.
00:03:20.490 --> 00:03:24.180
And that's the
following expression.
00:03:24.180 --> 00:03:26.560
It's called an
integral, but now it's
00:03:26.560 --> 00:03:29.290
going to have what are
known as limits on it.
00:03:29.290 --> 00:03:31.970
It will start at a and end at b.
00:03:31.970 --> 00:03:35.720
And we write in the
function f(x) dx.
00:03:35.720 --> 00:03:40.830
So this is what's known
as a definite integral.
00:03:40.830 --> 00:03:43.620
And it's interpreted
geometrically
00:03:43.620 --> 00:03:46.460
as the area under the curve.
00:03:46.460 --> 00:03:49.260
The only difference between
this collection of symbols
00:03:49.260 --> 00:03:51.690
and what we had before
with indefinite integrals
00:03:51.690 --> 00:03:54.960
is that before we didn't
specify where it started
00:03:54.960 --> 00:04:03.140
and where it ended.
00:04:03.140 --> 00:04:08.670
Now, in order to understand
what to do with this guy,
00:04:08.670 --> 00:04:12.330
I'm going to just describe
very abstractly what we do.
00:04:12.330 --> 00:04:17.050
And then carry out
one example in detail.
00:04:17.050 --> 00:04:24.310
So, to compute this
area, we're going
00:04:24.310 --> 00:04:27.270
to follow initially three steps.
00:04:27.270 --> 00:04:36.870
First of all, we're going
to divide into rectangles.
00:04:36.870 --> 00:04:42.080
And unfortunately, because
it's impossible to divide
00:04:42.080 --> 00:04:45.510
a curvy region into rectangles,
we're going to cheat.
00:04:45.510 --> 00:04:49.310
So they're only
quote-unquote rectangles.
00:04:49.310 --> 00:04:52.720
They're almost rectangles.
00:04:52.720 --> 00:05:01.060
And the second thing we're going
to do is to add up the areas.
00:05:01.060 --> 00:05:04.430
And the third thing
we're going to do
00:05:04.430 --> 00:05:10.970
is to rectify this problem
that we didn't actually
00:05:10.970 --> 00:05:12.870
hit the answer on the nose.
00:05:12.870 --> 00:05:15.310
That we were missing
some pieces or were
00:05:15.310 --> 00:05:17.460
choosing some extra bits.
00:05:17.460 --> 00:05:19.570
And the way we'll
rectify that is
00:05:19.570 --> 00:05:35.070
by taking the limit as
the rectangles get thin.
00:05:35.070 --> 00:05:39.380
Infinitesimally thin, very thin.
00:05:39.380 --> 00:05:43.780
Pictorially, again,
that looks like this.
00:05:43.780 --> 00:05:46.950
We have a and our b, and
we have our guy here,
00:05:46.950 --> 00:05:48.390
this is our curve.
00:05:48.390 --> 00:05:51.850
And I'm going to chop it up.
00:05:51.850 --> 00:05:57.470
First I'm going to chop up the
x axis into little increments.
00:05:57.470 --> 00:06:00.460
And then I'm going to
chop things up here.
00:06:00.460 --> 00:06:03.060
And I'll decide on
some rectangle, maybe
00:06:03.060 --> 00:06:05.450
some staircase pattern here.
00:06:05.450 --> 00:06:12.960
Like this.
00:06:12.960 --> 00:06:16.100
Now, I don't care so much.
00:06:16.100 --> 00:06:19.140
In some cases the rectangles
overshoot; in some cases
00:06:19.140 --> 00:06:20.400
they're underneath.
00:06:20.400 --> 00:06:23.720
So the new area that
I'm adding up is off.
00:06:23.720 --> 00:06:28.260
It's not quite the same as
the area under the curve.
00:06:28.260 --> 00:06:32.720
It's this region here.
00:06:32.720 --> 00:06:38.110
But it includes these
extra bits here.
00:06:38.110 --> 00:06:42.250
And then it's missing
this little guy here.
00:06:42.250 --> 00:06:47.950
This little bit
there is missing.
00:06:47.950 --> 00:06:51.310
And, as I say, these
little pieces up here,
00:06:51.310 --> 00:06:55.740
this a little bit
up here is extra.
00:06:55.740 --> 00:06:58.406
So that's why we're
not really dividing up
00:06:58.406 --> 00:06:59.530
the region into rectangles.
00:06:59.530 --> 00:07:01.300
We're just taking rectangles.
00:07:01.300 --> 00:07:05.060
And then the idea is that as
these get thinner and thinner,
00:07:05.060 --> 00:07:09.080
the little itty-bitty amounts
that we miss by are going to 0.
00:07:09.080 --> 00:07:10.830
And they're going
to be negligible.
00:07:10.830 --> 00:07:13.840
Already, you can see it's
kind of a thin piece of area,
00:07:13.840 --> 00:07:15.840
so we're not missing by much.
00:07:15.840 --> 00:07:19.150
And as these get thinner and
thinner, the problem goes away
00:07:19.150 --> 00:07:27.040
and we get the answer on
the nose in the limit.
00:07:27.040 --> 00:07:35.050
So here's our first example.
00:07:35.050 --> 00:07:41.270
I'll take the first interesting
curve, which is f(x) = x^2.
00:07:41.270 --> 00:07:45.150
I don't want to do anything more
complicated than one example,
00:07:45.150 --> 00:07:48.030
because this is a
real labor here,
00:07:48.030 --> 00:07:50.910
what we're going to go through.
00:07:50.910 --> 00:07:52.710
And to make things
easier for myself,
00:07:52.710 --> 00:07:55.930
I'm going to start at a = 0.
00:07:55.930 --> 00:07:58.400
But in order to see
what the pattern is,
00:07:58.400 --> 00:08:11.180
I'm going to allow
b to be arbitrary.
00:08:11.180 --> 00:08:15.230
Let's draw the graph and
start breaking things up.
00:08:15.230 --> 00:08:18.015
So here's the
parabola, and there's
00:08:18.015 --> 00:08:19.640
this piece that we
want, which is going
00:08:19.640 --> 00:08:24.170
to stop at this place, b, here.
00:08:24.170 --> 00:08:37.380
And the first step is
to divide into n pieces.
00:08:37.380 --> 00:08:40.840
That means, well, graphically,
I'll just mark the first three.
00:08:40.840 --> 00:08:44.450
And maybe there are
going to be many of them.
00:08:44.450 --> 00:08:47.480
And then I'll draw
some rectangles here,
00:08:47.480 --> 00:08:51.420
and I'm going to choose
to make the rectangles all
00:08:51.420 --> 00:08:53.310
the way from the right.
00:08:53.310 --> 00:08:55.545
That is, I'll make us
this staircase pattern
00:08:55.545 --> 00:08:58.760
here, like this.
00:08:58.760 --> 00:09:00.160
That's my choice.
00:09:00.160 --> 00:09:02.360
I get to choose
whatever level I want,
00:09:02.360 --> 00:09:04.500
and I'm going to
choose the right ends
00:09:04.500 --> 00:09:07.490
as the shape of the staircase.
00:09:07.490 --> 00:09:17.190
So I'm overshooting
with each rectangle.
00:09:17.190 --> 00:09:19.910
And now I have to
write down formulas
00:09:19.910 --> 00:09:23.800
for what these areas are.
00:09:23.800 --> 00:09:26.840
Now, there's one big advantage
that rectangles have.
00:09:26.840 --> 00:09:28.890
And this is the starting place.
00:09:28.890 --> 00:09:33.130
Which is that it's easy
to find their areas.
00:09:33.130 --> 00:09:35.260
All you need to know is
the base and the height,
00:09:35.260 --> 00:09:37.230
and you multiply,
and you get the area.
00:09:37.230 --> 00:09:40.820
That's the reason why we can
get started with rectangles.
00:09:40.820 --> 00:09:43.750
And in this case,
these distances,
00:09:43.750 --> 00:09:46.350
I'm assuming that they're
all equal, equally
00:09:46.350 --> 00:09:48.350
spaced, intervals.
00:09:48.350 --> 00:09:50.260
And I'll always be doing that.
00:09:50.260 --> 00:10:01.360
And so the spacing, the bases,
the base length, is always b/n.
00:10:01.360 --> 00:10:09.590
All equal intervals.
00:10:09.590 --> 00:10:11.260
So that's the base length.
00:10:11.260 --> 00:10:15.370
And next, I need the heights.
00:10:15.370 --> 00:10:17.400
And in order to keep
track of the heights,
00:10:17.400 --> 00:10:21.310
I'm going to draw a little
table here, with x and f(x),
00:10:21.310 --> 00:10:27.460
and plug in a few values just
to see what the pattern is.
00:10:27.460 --> 00:10:34.100
The first place here,
after 0, is b/n.
00:10:34.100 --> 00:10:36.810
So here's b/n,
that's an x-value.
00:10:36.810 --> 00:10:40.590
And the f(x) value
is the height there.
00:10:40.590 --> 00:10:44.980
And that's just, I
evaluate f(x), f(x) is x^2.
00:10:44.980 --> 00:10:49.450
And that's (b/n)^2.
00:10:49.450 --> 00:10:53.350
And similarly, the
next one is 2b/n.
00:10:56.230 --> 00:10:58.560
And the value here is (2b/n)^2.
00:11:01.530 --> 00:11:02.950
That's this.
00:11:02.950 --> 00:11:07.600
This height here is 2b/n.
00:11:07.600 --> 00:11:14.070
That's the second rectangle.
00:11:14.070 --> 00:11:16.960
And I'll write down one more.
00:11:16.960 --> 00:11:18.600
3b/n, that's the third one.
00:11:18.600 --> 00:11:20.090
And the height is (3b/n)^2.
00:11:23.250 --> 00:11:29.940
And so forth.
00:11:29.940 --> 00:11:34.537
Well, my next job is
to add up these areas.
00:11:34.537 --> 00:11:36.620
And I've already prepared
that by finding out what
00:11:36.620 --> 00:11:39.020
the base and the height is.
00:11:39.020 --> 00:11:50.030
So the total area, or the
sum of the areas, let's say,
00:11:50.030 --> 00:12:01.240
of these rectangles, is-- Well,
the first one is (b/n) (b/n)^2.
00:12:01.240 --> 00:12:06.380
The second one is 2b/n --
I'm sorry, is (b/n) (2b/n)^2.
00:12:08.920 --> 00:12:11.070
And it just keeps on going.
00:12:11.070 --> 00:12:13.540
And the last one is
(b/n) (nb / n)^2.
00:12:17.810 --> 00:12:19.870
So it's very important
to figure out
00:12:19.870 --> 00:12:22.720
what the general formula is.
00:12:22.720 --> 00:12:25.000
And here we have a base.
00:12:25.000 --> 00:12:27.170
And here we have a
height, and here we
00:12:27.170 --> 00:12:31.930
have the same kind of base,
but we have a new height.
00:12:31.930 --> 00:12:32.880
And so forth.
00:12:32.880 --> 00:12:36.390
And the pattern is that the
coefficient here is 1, then 2,
00:12:36.390 --> 00:12:42.976
then 3, all the way up to n.
00:12:42.976 --> 00:12:44.850
The rectangles are
getting taller and taller,
00:12:44.850 --> 00:12:50.940
and this one, the last
one is the biggest.
00:12:50.940 --> 00:12:55.300
OK, this is a very
complicated gadget.
00:12:55.300 --> 00:12:58.800
and the first thing I
want to do is simplify it
00:12:58.800 --> 00:13:00.600
and then I'm actually
going to evaluate it.
00:13:00.600 --> 00:13:03.129
But actually I'm not going
to evaluate it exactly.
00:13:03.129 --> 00:13:04.670
I'm just going to
evaluate the limit.
00:13:04.670 --> 00:13:07.060
Turns out, limits
are always easier.
00:13:07.060 --> 00:13:10.960
The point about calculus here is
that these rectangles are hard.
00:13:10.960 --> 00:13:13.330
But the limiting value
is an easy value.
00:13:13.330 --> 00:13:16.440
So what we're heading for is
the simple formula, as opposed
00:13:16.440 --> 00:13:19.550
to the complicated one.
00:13:19.550 --> 00:13:22.190
Alright, so the first
thing I'm going to do
00:13:22.190 --> 00:13:25.130
is factor out all
these b/n factors.
00:13:25.130 --> 00:13:27.970
There's a b/n, here, and
there's a (b/n)^2, So all told,
00:13:27.970 --> 00:13:31.390
we have a (b/n)^3.
00:13:31.390 --> 00:13:33.730
As a common factor.
00:13:33.730 --> 00:13:36.830
And then the first term
is 1, and the second term,
00:13:36.830 --> 00:13:39.310
what's left over, is 2^2.
00:13:39.310 --> 00:13:41.100
2^2.
00:13:41.100 --> 00:13:43.670
And then the third
term would be 3^2,
00:13:43.670 --> 00:13:46.970
although I haven't written it.
00:13:46.970 --> 00:13:51.200
In the last term, there's
an extra factor of n^2.
00:13:51.200 --> 00:14:05.050
In the numerator.
00:14:05.050 --> 00:14:09.820
OK, is everybody with me here?
00:14:09.820 --> 00:14:23.930
Now, what I'd like to do is
to eventually take the limit
00:14:23.930 --> 00:14:26.520
as n goes to infinity here.
00:14:26.520 --> 00:14:29.010
And the quantity that's
hard to understand
00:14:29.010 --> 00:14:33.540
is this massive quantity here.
00:14:33.540 --> 00:14:36.690
And there's one change
that I'd like to make,
00:14:36.690 --> 00:14:40.160
but it's a very modest one.
00:14:40.160 --> 00:14:41.410
Extremely minuscule.
00:14:41.410 --> 00:14:43.350
Which is that I'm
going to write 1,
00:14:43.350 --> 00:14:45.450
just to see that there's
a general pattern here.
00:14:45.450 --> 00:14:46.500
Going to write 1 as 1^2.
00:14:52.830 --> 00:14:59.420
And let's put in
the 3 here, why not.
00:14:59.420 --> 00:15:05.400
And now I want to use a trick.
00:15:05.400 --> 00:15:08.560
This trick is not
completely recommended,
00:15:08.560 --> 00:15:12.380
but I will say a
lot more about that
00:15:12.380 --> 00:15:13.780
when we get through to the end.
00:15:13.780 --> 00:15:16.110
I want to understand how
big this quantity is.
00:15:16.110 --> 00:15:18.795
So I'm going to use a
geometric trick to draw
00:15:18.795 --> 00:15:20.880
a picture of this quantity.
00:15:20.880 --> 00:15:23.660
Namely, I'm going
to build a pyramid.
00:15:23.660 --> 00:15:29.510
And the base of the pyramid
is going to be n by n blocks.
00:15:29.510 --> 00:15:32.410
So imagine we're in Egypt
and we're building a pyramid.
00:15:32.410 --> 00:15:39.190
And the next layer is
going to be n-1 by n-1.
00:15:39.190 --> 00:15:43.250
So this next layer
in is n-1 by n-1.
00:15:43.250 --> 00:15:46.590
So the total number of blocks
on the bottom is n squared.
00:15:46.590 --> 00:15:50.020
That's this rightmost term here.
00:15:50.020 --> 00:15:52.460
But the next term, which I
didn't write in but maybe I
00:15:52.460 --> 00:15:57.280
should, the next-to-the-last
term was this one.
00:15:57.280 --> 00:16:00.360
And that's the second
layer that I've put on.
00:16:00.360 --> 00:16:05.500
Now, this is, if you
like, the top view.
00:16:05.500 --> 00:16:08.950
But perhaps we should also
think in terms of a side view.
00:16:08.950 --> 00:16:12.920
So here's the same picture,
we're starting at n
00:16:12.920 --> 00:16:15.920
and we build up this layer here.
00:16:15.920 --> 00:16:18.507
And now we're going to put
a layer on top of it, which
00:16:18.507 --> 00:16:19.340
is a little shorter.
00:16:19.340 --> 00:16:21.600
So the first layer
is of length n.
00:16:21.600 --> 00:16:25.460
And the second layers is of
length n-1, and then on top
00:16:25.460 --> 00:16:28.007
of that we have something
of length n-2, and so forth.
00:16:28.007 --> 00:16:29.340
And we're going to pile them up.
00:16:29.340 --> 00:16:31.800
So we pile them up.
00:16:31.800 --> 00:16:37.630
All the way to the top, which is
just one giant block of stone.
00:16:37.630 --> 00:16:39.490
And that's this last one, 1^2.
00:16:39.490 --> 00:16:43.290
So we're going
backwards in the sum.
00:16:43.290 --> 00:16:46.140
And so I have to build
this whole thing up.
00:16:46.140 --> 00:16:48.350
And I get all the way up
in this staircase pattern
00:16:48.350 --> 00:16:57.720
to this top block, up there.
00:16:57.720 --> 00:17:00.140
So here's the trick
that you can use
00:17:00.140 --> 00:17:02.870
to estimate the size
of this, and it's
00:17:02.870 --> 00:17:06.410
sufficient in the limit
as n goes to infinity.
00:17:06.410 --> 00:17:14.560
The trick is that I can
imagine the solid thing
00:17:14.560 --> 00:17:19.560
underneath the
staircase, like this.
00:17:19.560 --> 00:17:24.390
That's an ordinary pyramid,
not a staircase pyramid.
00:17:24.390 --> 00:17:26.550
Which is inside.
00:17:26.550 --> 00:17:28.940
And this one is inside.
00:17:28.940 --> 00:17:32.150
And so, but it's an
ordinary pyramid as opposed
00:17:32.150 --> 00:17:34.530
to a staircase pyramid.
00:17:34.530 --> 00:17:37.690
And so, we know the formula
for the volume of that.
00:17:37.690 --> 00:17:40.580
Because we know the formula
for volumes of cones.
00:17:40.580 --> 00:17:50.090
And the formula for the volume
of this guy, of the inside,
00:17:50.090 --> 00:17:58.150
is 1/3 base times height.
00:17:58.150 --> 00:18:03.500
And in that case, the
base here-- so that's 1/3,
00:18:03.500 --> 00:18:06.180
and the base is n by n, right?
00:18:06.180 --> 00:18:08.540
So the base is n^2.
00:18:08.540 --> 00:18:10.050
That's the base.
00:18:10.050 --> 00:18:13.340
And the height, it goes all
the way to the top point.
00:18:13.340 --> 00:18:21.290
So the height is n.
00:18:21.290 --> 00:18:27.130
And what we've discovered
here is that this whole sum is
00:18:27.130 --> 00:18:30.910
bigger than 1/3 n^3.
00:18:42.680 --> 00:18:46.050
Now, I claimed that - this
line, by the way has slope 2.
00:18:46.050 --> 00:18:50.140
So you go 1/2 over
each time you go up 1.
00:18:50.140 --> 00:18:52.960
And that's why you
get to the top.
00:18:52.960 --> 00:18:56.950
On the other hand, I can
trap it on the outside,
00:18:56.950 --> 00:19:01.840
too, by drawing a
parallel line out here.
00:19:01.840 --> 00:19:07.100
And this will go down 1/2
more on this side and 1/2 more
00:19:07.100 --> 00:19:08.230
on the other side.
00:19:08.230 --> 00:19:14.870
So the base will be n+1 by
n+1 of this bigger pyramid.
00:19:14.870 --> 00:19:18.120
And it'll go up 1 higher.
00:19:18.120 --> 00:19:22.790
So on the other end, we get that
this is less than 1/3 (n+1)^3.
00:19:25.690 --> 00:19:34.290
Again, (n+1)^2 times n+1, again,
this is a base times a height.
00:19:34.290 --> 00:19:36.920
Of this bigger pyramid.
00:19:36.920 --> 00:19:38.110
Yes, question.
00:19:38.110 --> 00:19:47.860
STUDENT: [INAUDIBLE] and
then equating it to volume.
00:19:47.860 --> 00:19:49.360
PROFESSOR: The
question is, it seems
00:19:49.360 --> 00:19:54.360
as if I'm adding up areas
and equating it to volume.
00:19:54.360 --> 00:19:57.160
But I'm actually
creating volumes
00:19:57.160 --> 00:20:00.060
by making these honest
increments here.
00:20:00.060 --> 00:20:07.750
That is, the base is
n but the height is 1.
00:20:07.750 --> 00:20:09.380
Thank you for pointing that out.
00:20:09.380 --> 00:20:11.220
Each one of these
little staircases
00:20:11.220 --> 00:20:14.150
here has exactly height 1.
00:20:14.150 --> 00:20:15.950
So I'm honestly
sticking blocks there.
00:20:15.950 --> 00:20:18.200
They're sort of square blocks,
and I'm lining them up.
00:20:18.200 --> 00:20:21.450
And I'm thinking of n
by n cubes, if you like.
00:20:21.450 --> 00:20:22.960
Honest cubes, there.
00:20:22.960 --> 00:20:25.447
And the height is 1.
00:20:25.447 --> 00:20:26.280
And the base is n^2.
00:20:33.360 --> 00:20:36.760
Alright, so I claim that I've
trapped this guy in between two
00:20:36.760 --> 00:20:38.520
quantities here.
00:20:38.520 --> 00:20:52.510
And now I'm ready
to take the limit.
00:20:52.510 --> 00:20:55.120
If you look at what
our goal is, we
00:20:55.120 --> 00:20:57.060
want to have an
expression like this.
00:20:57.060 --> 00:21:00.840
And I'm going to-- This was the
massive expression that we had.
00:21:00.840 --> 00:21:03.310
And actually, I'm going
to write it differently.
00:21:03.310 --> 00:21:10.230
I'll write it as b^3 times
1^2 plus 2^2 plus... plus n^2,
00:21:10.230 --> 00:21:12.070
divided by n^3.
00:21:12.070 --> 00:21:15.520
I'm going to combine
all the n's together.
00:21:15.520 --> 00:21:17.420
Alright, so the
right thing to do
00:21:17.420 --> 00:21:20.480
is to divide what
I had up there.
00:21:20.480 --> 00:21:28.280
Divide by n^3 in this set
of inequalities there.
00:21:28.280 --> 00:21:35.930
And what I get here is 1/3 is
less than 1 plus 2^2 plus 3^2
00:21:35.930 --> 00:21:41.740
plus... plus n^2 divided by n^3
is less than 1/3 (n+1)^3 / n^3.
00:21:45.490 --> 00:21:49.230
And that's 1/3 + (1 + 1/n)^3.
00:21:56.230 --> 00:21:59.530
And now, I claim we're done.
00:21:59.530 --> 00:22:03.170
Because this is
1/3, and the limit,
00:22:03.170 --> 00:22:06.450
as n goes to infinity,
of this quantity here,
00:22:06.450 --> 00:22:09.350
is easily seen to be, this
is, as n goes to infinity,
00:22:09.350 --> 00:22:10.470
this goes to 0.
00:22:10.470 --> 00:22:14.790
So this also goes to 1/3.
00:22:14.790 --> 00:22:28.810
And so our total here, so
our total area, under x^2,
00:22:28.810 --> 00:22:33.040
which we sometimes might write
the integral from 0 to b x^2
00:22:33.040 --> 00:22:38.210
dx, is going to be
equal to - well,
00:22:38.210 --> 00:22:40.350
it's this 1/3 that I've got.
00:22:40.350 --> 00:22:43.400
But then there was
also a b^3 there.
00:22:43.400 --> 00:22:45.040
So there's this
extra b cubed here.
00:22:45.040 --> 00:22:49.240
So it's 1/3 b^3.
00:22:49.240 --> 00:22:54.630
That's the result from
this whole computation.
00:22:54.630 --> 00:22:55.620
Yes, question.
00:22:55.620 --> 00:22:57.060
STUDENT: [INAUDIBLE]
00:22:57.060 --> 00:23:05.370
PROFESSOR: So that was
a very good question.
00:23:05.370 --> 00:23:08.960
The question is, why did
we leave the b/n^3 out,
00:23:08.960 --> 00:23:11.190
for this step.
00:23:11.190 --> 00:23:16.290
And a part of the answer
is malice aforethought.
00:23:16.290 --> 00:23:19.290
In other words, we know
what we're heading for.
00:23:19.290 --> 00:23:21.670
We know, we understand,
this quantity.
00:23:21.670 --> 00:23:23.570
It's all one thing.
00:23:23.570 --> 00:23:26.080
This thing is a sum, which
is growing larger and larger.
00:23:26.080 --> 00:23:28.430
It's not what's
called a closed form.
00:23:28.430 --> 00:23:31.440
So, the thing that's not
known, or not well understood,
00:23:31.440 --> 00:23:33.320
is how big is this
quantity here.
00:23:33.320 --> 00:23:37.050
1^2 + 2^2, the sum
of the squares.
00:23:37.050 --> 00:23:38.620
Whereas, this is
something that's
00:23:38.620 --> 00:23:40.740
quite easy to understand.
00:23:40.740 --> 00:23:42.570
So we factor it out.
00:23:42.570 --> 00:23:47.740
And we analyze carefully the
piece which we don't know yet,
00:23:47.740 --> 00:23:48.920
how big it is.
00:23:48.920 --> 00:23:52.560
And we discovered that it's
very, very similar to n^3.
00:23:52.560 --> 00:23:57.030
But it's more
similar to 1/3 n^3.
00:23:57.030 --> 00:24:00.500
It's almost
identical to 1/3 n^3.
00:24:00.500 --> 00:24:02.080
This extra piece here.
00:24:02.080 --> 00:24:04.120
So that's what's going on.
00:24:04.120 --> 00:24:05.110
And then we match that.
00:24:05.110 --> 00:24:09.230
Since this thing is very
similar to 1/3 n^3 we cancel
00:24:09.230 --> 00:24:24.780
the n^3's and we
have our result.
00:24:24.780 --> 00:24:28.230
Let me just mention that
although this may seem odd,
00:24:28.230 --> 00:24:30.760
in fact this is what
you always do if you
00:24:30.760 --> 00:24:32.630
analyze these kinds of sum.
00:24:32.630 --> 00:24:34.810
You always factor
out whatever you can.
00:24:34.810 --> 00:24:37.250
And then you still are
faced with a sum like this.
00:24:37.250 --> 00:24:40.100
So this will happen
systematically, every time
00:24:40.100 --> 00:24:45.860
you're faced with such a sum.
00:24:45.860 --> 00:24:53.450
OK, now I want to say one
more word about notation.
00:24:53.450 --> 00:25:00.000
Which is that this notation
is an extreme nuisance here.
00:25:00.000 --> 00:25:04.170
And it's really sort of too
large for us to deal with.
00:25:04.170 --> 00:25:08.440
And so, mathematicians
have a shorthand for it.
00:25:08.440 --> 00:25:10.930
Unfortunately, when you
actually do a computation,
00:25:10.930 --> 00:25:15.280
you're going to end up with
this collection of stuff anyway.
00:25:15.280 --> 00:25:19.530
But I want to just show you
this summation notation in order
00:25:19.530 --> 00:25:24.980
to compress it a little bit.
00:25:24.980 --> 00:25:31.600
The idea of summation
notation is the following.
00:25:31.600 --> 00:25:35.380
So this thing tends-- The
ideas are the following.
00:25:35.380 --> 00:25:37.920
I'll illustrate it
with an example first.
00:25:37.920 --> 00:25:45.810
So, the general notation is
the sum of a_i, i = 1 to n,
00:25:45.810 --> 00:25:50.510
is a_1 plus a_2 plus
dot dot dot plus a_n.
00:25:50.510 --> 00:25:53.700
So this is the abbreviation.
00:25:53.700 --> 00:26:03.950
And this is a capital sigma.
00:26:03.950 --> 00:26:06.720
And so, this quantity
here, for instance,
00:26:06.720 --> 00:26:15.380
is 1/n^3 times the
sum i^2, i = 1 to n.
00:26:15.380 --> 00:26:17.350
So that's what this
thing is equal to.
00:26:17.350 --> 00:26:20.590
And what we just showed
is that that tends to 1/3
00:26:20.590 --> 00:26:23.910
as n goes to infinity.
00:26:23.910 --> 00:26:30.630
So this is the way the
summation notation is used.
00:26:30.630 --> 00:26:34.330
There's a formula for each
of these coefficients,
00:26:34.330 --> 00:26:37.360
each of these entries
here, or summands.
00:26:37.360 --> 00:26:39.460
And then this is
just an abbreviation
00:26:39.460 --> 00:26:40.710
for what the sum is.
00:26:40.710 --> 00:26:43.407
And this is the reason
why I stuck in that 1^2
00:26:43.407 --> 00:26:45.740
at the beginning, so that you
could see that the pattern
00:26:45.740 --> 00:26:47.505
worked all the
way down to i = 1.
00:26:47.505 --> 00:26:50.600
It isn't an exception
to the rule.
00:26:50.600 --> 00:26:54.160
It's the same as
all of the others.
00:26:54.160 --> 00:26:59.020
Now, over here, in
this board, we also
00:26:59.020 --> 00:27:02.680
had one of these
extremely long sums.
00:27:02.680 --> 00:27:06.770
And this one can be written
in the following way.
00:27:06.770 --> 00:27:10.830
And I hope you agree, this
is rather hard to scan.
00:27:10.830 --> 00:27:16.385
But one way of writing it is,
it's the sum from i = 1 to n of
00:27:16.385 --> 00:27:19.720
- now I have to write down the
formula for the general term.
00:27:19.720 --> 00:27:23.130
Which is b/n (ib/n)^2.
00:27:29.550 --> 00:27:34.350
So that's a way of abbreviating
this massive formula into one
00:27:34.350 --> 00:27:36.980
which is just a lot shorter.
00:27:36.980 --> 00:27:40.090
And now, the manipulation
that I performed with it,
00:27:40.090 --> 00:27:43.940
which is to factor
out this (b/n)^3,
00:27:43.940 --> 00:27:47.560
is something that I'm perfectly
well allowed to do also over
00:27:47.560 --> 00:27:49.130
here.
00:27:49.130 --> 00:27:51.480
This is the distributive law.
00:27:51.480 --> 00:27:56.506
This, if I factor out b^3 /
n^3, I'm left with the sum,
00:27:56.506 --> 00:28:00.880
i = 1 to n, of i^2, right?
00:28:00.880 --> 00:28:06.040
So these notations make it
a little bit more compact.
00:28:06.040 --> 00:28:10.870
What we're dealing with.
00:28:10.870 --> 00:28:14.940
The conceptual phenomenon
is still the same.
00:28:14.940 --> 00:28:18.160
And the mess is really still
just hiding under the rug.
00:28:18.160 --> 00:28:23.960
But the notation is-- at
least fits with fewer symbols,
00:28:23.960 --> 00:28:32.320
anyway.
00:28:32.320 --> 00:28:39.560
So let's continue here.
00:28:39.560 --> 00:28:41.460
I've given you one calculation.
00:28:41.460 --> 00:28:51.010
And now I want to fit
it into a pattern.
00:28:51.010 --> 00:28:54.650
And here's the thing that
I'd like to calculate.
00:28:54.650 --> 00:28:59.120
So, first of all let's
try the case-- S I'm
00:28:59.120 --> 00:29:02.697
going to do two more examples.
00:29:02.697 --> 00:29:04.280
I'll do two more
examples, but they're
00:29:04.280 --> 00:29:05.650
going to be much, much easier.
00:29:05.650 --> 00:29:09.160
And then things are going to
get much easier from now on.
00:29:09.160 --> 00:29:19.580
So, the second example is going
to be the function f(x) = x.
00:29:19.580 --> 00:29:23.520
If I draw that, that's
this function here,
00:29:23.520 --> 00:29:26.500
that's the line with slope 1.
00:29:26.500 --> 00:29:29.280
And here's b.
00:29:29.280 --> 00:29:32.300
And so this area
here is the same
00:29:32.300 --> 00:29:36.440
as the area of the triangle
with base b and height b.
00:29:36.440 --> 00:29:44.360
So the area is equal to 1/2
b * b, so this is the base.
00:29:44.360 --> 00:29:45.660
And this is the height.
00:29:45.660 --> 00:29:49.970
We also know how to find
the area of triangles.
00:29:49.970 --> 00:29:52.770
And so, the formula is 1/2 b^2.
00:29:57.270 --> 00:30:02.480
And the third example--
Notice, by the way,
00:30:02.480 --> 00:30:05.920
I didn't have to do this
elaborate summing to do that,
00:30:05.920 --> 00:30:07.770
because we happen
to know this area.
00:30:07.770 --> 00:30:13.370
The third example is going
to be even easier. f(x) = 1.
00:30:13.370 --> 00:30:16.200
By far the most
important example.
00:30:16.200 --> 00:30:20.230
Remarkably, when you get to
18.02, multivariable calculus,
00:30:20.230 --> 00:30:22.420
you will forget
this calculation.
00:30:22.420 --> 00:30:23.160
Somehow.
00:30:23.160 --> 00:30:26.080
And I don't know why, but
it happens to everybody.
00:30:26.080 --> 00:30:30.690
So, the function is just
horizontal, like this.
00:30:30.690 --> 00:30:31.190
Right?
00:30:31.190 --> 00:30:32.610
It's the constant 1.
00:30:32.610 --> 00:30:37.390
And if we stop it at b, then
the area we're interested in
00:30:37.390 --> 00:30:42.190
is just this, from 0 to b.
00:30:42.190 --> 00:30:47.850
And we know that this is
height 1, so this is area
00:30:47.850 --> 00:30:51.600
is base, which is b, times 1.
00:30:51.600 --> 00:31:03.750
So it's b.
00:31:03.750 --> 00:31:13.300
Let's look now at the pattern.
00:31:13.300 --> 00:31:17.730
We're going to look at the
pattern of the function,
00:31:17.730 --> 00:31:21.080
and it's the area
under the curve, which
00:31:21.080 --> 00:31:26.330
is this-- has this
elaborate formula in terms
00:31:26.330 --> 00:31:34.990
of-- so this is just the
area under the curve.
00:31:34.990 --> 00:31:40.500
Between 0 and b.
00:31:40.500 --> 00:31:47.110
And we have x^2, which
turned out to be b^3 / 3.
00:31:47.110 --> 00:31:49.824
And we have x, which
turned out to be-- well,
00:31:49.824 --> 00:31:52.240
let me write them over just a
bit more to give myself some
00:31:52.240 --> 00:31:57.200
room. x, which turns
out to be b^2 / 2.
00:31:57.200 --> 00:32:07.090
And then we have 1,
which turned out to be b.
00:32:07.090 --> 00:32:10.830
So this, I claim, is suggestive.
00:32:10.830 --> 00:32:14.750
If you can figure
out the pattern,
00:32:14.750 --> 00:32:19.750
one way of making it a little
clearer is to see that x is
00:32:19.750 --> 00:32:22.210
x^1.
00:32:22.210 --> 00:32:24.410
And 1 is x^0.
00:32:27.250 --> 00:32:30.070
So this is the case, 0, 1 and 2.
00:32:30.070 --> 00:32:32.660
And b is b^1 / 1.
00:32:40.010 --> 00:32:56.720
So, if you want to guess what
happens when f(x) is x^3,
00:32:56.720 --> 00:33:01.030
well if it's 0, you do b^1 /
1; if it's 1, you do b^2 / 2;
00:33:01.030 --> 00:33:04.270
if it's 2, you do b^3 / 3.
00:33:04.270 --> 00:33:07.850
So it's reasonable to guess
that this should be b^4 / 4.
00:33:11.110 --> 00:33:15.110
That's a reasonable
guess, I would say.
00:33:15.110 --> 00:33:24.740
Now, the strange thing is that
in history, Archimedes figured
00:33:24.740 --> 00:33:27.550
out the area under a parabola.
00:33:27.550 --> 00:33:29.600
So that was a long time ago.
00:33:29.600 --> 00:33:30.910
It was after the pyramids.
00:33:30.910 --> 00:33:34.340
And he used, actually, a
much more complicated method
00:33:34.340 --> 00:33:36.500
than I just described here.
00:33:36.500 --> 00:33:40.700
And his method, which is
just fantastically amazing,
00:33:40.700 --> 00:33:43.850
was so brilliant that it may
have set back mathematics
00:33:43.850 --> 00:33:46.080
by 2,000 years.
00:33:46.080 --> 00:33:48.630
Because people were
so-- it was so difficult
00:33:48.630 --> 00:33:51.070
that people couldn't
see this pattern.
00:33:51.070 --> 00:33:54.312
And couldn't see that, actually,
these kinds of calculations
00:33:54.312 --> 00:33:54.812
are easy.
00:33:54.812 --> 00:33:56.740
So they couldn't
get to the cubic.
00:33:56.740 --> 00:33:58.240
And even when they
got to the cubic,
00:33:58.240 --> 00:33:59.690
they were struggling
with everything else.
00:33:59.690 --> 00:34:01.550
And it wasn't until
calculus fit everything
00:34:01.550 --> 00:34:04.770
together that people were
able to make serious progress
00:34:04.770 --> 00:34:06.640
on calculating these areas.
00:34:06.640 --> 00:34:09.940
Even though he was the expert on
calculating areas and volumes,
00:34:09.940 --> 00:34:12.130
for his time.
00:34:12.130 --> 00:34:15.470
So this is really a great
thing that we now can
00:34:15.470 --> 00:34:16.810
have easy methods of doing it.
00:34:16.810 --> 00:34:21.430
And the main thing that I
want to tell you is that's we
00:34:21.430 --> 00:34:25.740
will not have to labor to
build pyramids to calculate
00:34:25.740 --> 00:34:27.250
all of these quantities.
00:34:27.250 --> 00:34:29.620
We will have a way
faster way of doing it.
00:34:29.620 --> 00:34:32.790
This is the slow, laborious way.
00:34:32.790 --> 00:34:37.080
And we will be able to do it
so easily that it will happen
00:34:37.080 --> 00:34:39.590
as fast as you differentiate.
00:34:39.590 --> 00:34:42.360
So that's coming up tomorrow.
00:34:42.360 --> 00:34:45.920
But I want you to know that
it's going to be-- However,
00:34:45.920 --> 00:34:52.550
we're going to go through just
a little pain before we do it.
00:34:52.550 --> 00:34:59.400
And I'll just tell you one
more piece of notation here.
00:34:59.400 --> 00:35:01.410
So you need to have a
little practice just
00:35:01.410 --> 00:35:04.850
to recognize how much
savings we're going to make.
00:35:04.850 --> 00:35:07.340
But never again will
you have to face
00:35:07.340 --> 00:35:16.190
elaborate geometric
arguments like this.
00:35:16.190 --> 00:35:21.110
So let me just add a
little bit of notation
00:35:21.110 --> 00:35:27.910
for definite integrals.
00:35:27.910 --> 00:35:35.810
And this goes under the
name of Riemann sums.
00:35:35.810 --> 00:35:44.140
Named after a mathematician
from the 1800s.
00:35:44.140 --> 00:36:01.150
So this is the general procedure
for definite integrals.
00:36:01.150 --> 00:36:04.890
We divide it up into pieces.
00:36:04.890 --> 00:36:07.430
And how do we do that?
00:36:07.430 --> 00:36:16.120
Well, so here's our
a and here's our b.
00:36:16.120 --> 00:36:19.600
And what we're going to do is
break it up into little pieces.
00:36:19.600 --> 00:36:22.620
And we're going to give
a name to the increment.
00:36:22.620 --> 00:36:28.510
And we're going to
call that delta x.
00:36:28.510 --> 00:36:30.380
So we divide up into these.
00:36:30.380 --> 00:36:32.110
So how many pieces are there?
00:36:32.110 --> 00:36:37.570
If there are n pieces,
then the general formula
00:36:37.570 --> 00:36:43.390
is always the delta x is
1/n times the total length.
00:36:43.390 --> 00:36:44.880
So it has to be (b-a) / n.
00:36:48.170 --> 00:36:50.550
We will always use
these equal increments,
00:36:50.550 --> 00:36:53.020
although you don't
absolutely have to do it.
00:36:53.020 --> 00:37:01.080
We will, for these Riemann sums.
00:37:01.080 --> 00:37:07.610
And now there's only
one bit of flexibility
00:37:07.610 --> 00:37:10.560
that we will allow ourselves.
00:37:10.560 --> 00:37:13.020
Which is this.
00:37:13.020 --> 00:37:29.720
We're going to pick any height
of f between-- in the interval,
00:37:29.720 --> 00:37:34.610
in each interval.
00:37:34.610 --> 00:37:39.020
So what that means is,
let me just show it
00:37:39.020 --> 00:37:43.870
to you on the picture here.
00:37:43.870 --> 00:37:47.200
Is, I just pick any
value in between,
00:37:47.200 --> 00:37:49.770
I'll call it c_i,
which is in there.
00:37:49.770 --> 00:37:51.420
And then I go up here.
00:37:51.420 --> 00:37:55.180
And I have the level,
which is f(c_i).
00:37:55.180 --> 00:37:58.730
And that's the
rectangle that I choose.
00:37:58.730 --> 00:38:01.530
In the case that
we did, we always
00:38:01.530 --> 00:38:03.730
chose the right-hand,
which turned out
00:38:03.730 --> 00:38:04.930
to be the largest one.
00:38:04.930 --> 00:38:07.800
But I could've chosen
some level in between.
00:38:07.800 --> 00:38:09.140
Or even the left-hand end.
00:38:09.140 --> 00:38:11.223
Which would have meant
that the staircase would've
00:38:11.223 --> 00:38:13.580
been quite a bit lower.
00:38:13.580 --> 00:38:17.950
So any of these staircases
will work perfectly well.
00:38:17.950 --> 00:38:25.650
So that means were picking
f(c_i), and that's a height.
00:38:25.650 --> 00:38:33.210
And now we're just going
to add them all up.
00:38:33.210 --> 00:38:35.680
And this is the sum of the
areas of the rectangles,
00:38:35.680 --> 00:38:37.350
because this is the height.
00:38:37.350 --> 00:38:43.700
And this is the base.
00:38:43.700 --> 00:38:46.160
This notation is
supposed to be, now,
00:38:46.160 --> 00:38:54.640
very suggestive of the
notation that Leibniz used.
00:38:54.640 --> 00:38:58.250
Which is that in the limit,
this becomes an integral from a
00:38:58.250 --> 00:39:01.010
to b of f(x) dx.
00:39:01.010 --> 00:39:05.230
And notice that the delta
x gets replaced by a dx.
00:39:05.230 --> 00:39:07.960
So this is what
happens in the limit.
00:39:07.960 --> 00:39:10.600
As the rectangles get thin.
00:39:10.600 --> 00:39:17.170
So that's as delta x goes to 0.
00:39:17.170 --> 00:39:21.780
And these gadgets are
called Riemann sums.
00:39:21.780 --> 00:39:29.740
This is called a Riemann sum.
00:39:29.740 --> 00:39:31.580
And we already worked
out an example.
00:39:31.580 --> 00:39:40.680
This very complicated guy was
an example of a Riemann sum.
00:39:40.680 --> 00:39:42.060
So that's a notation.
00:39:42.060 --> 00:39:44.182
And we'll give you
a chance to get
00:39:44.182 --> 00:39:45.640
used to it a little
more when we do
00:39:45.640 --> 00:39:51.680
some numerical work at the end.
00:39:51.680 --> 00:39:55.130
Now, the last
thing for today is,
00:39:55.130 --> 00:40:05.240
I promised you an example
which was not an area example.
00:40:05.240 --> 00:40:10.010
I want to be able to show
you that integrals can be
00:40:10.010 --> 00:40:21.630
interpreted as cumulative sums.
00:40:21.630 --> 00:40:36.480
Integrals as cumulative sums.
00:40:36.480 --> 00:40:39.020
So this is just an example.
00:40:39.020 --> 00:40:48.650
And, so here's the way it goes.
00:40:48.650 --> 00:40:51.440
So we're going to
consider a function f,
00:40:51.440 --> 00:40:55.460
we're going to consider a
variable t, which is time.
00:40:55.460 --> 00:40:59.340
In years.
00:40:59.340 --> 00:41:02.170
And we'll consider
a function f(t),
00:41:02.170 --> 00:41:06.560
which is in dollars per year.
00:41:06.560 --> 00:41:09.470
Right, this is a
financial example here.
00:41:09.470 --> 00:41:13.250
That's the unit here,
dollars per year.
00:41:13.250 --> 00:41:21.500
And this is going to
be a borrowing rate.
00:41:21.500 --> 00:41:24.000
Now, the reason why I
want to put units in here
00:41:24.000 --> 00:41:27.320
is to show you that
there's a good reason
00:41:27.320 --> 00:41:33.920
for this strange dx, which
we append on these integrals.
00:41:33.920 --> 00:41:34.890
This notation.
00:41:34.890 --> 00:41:36.520
It allows us to
change variables,
00:41:36.520 --> 00:41:39.020
it allows this to be
consistent with units.
00:41:39.020 --> 00:41:42.360
And allows us to develop
meaningful formulas, which are
00:41:42.360 --> 00:41:44.130
consistent across the board.
00:41:44.130 --> 00:41:46.020
And so I want to
emphasize the units
00:41:46.020 --> 00:41:51.620
in this when I set up this
modeling problem here.
00:41:51.620 --> 00:41:59.660
Now, you're borrowing
money, let's say, every day.
00:41:59.660 --> 00:42:06.160
So that means delta t = 1/365.
00:42:06.160 --> 00:42:08.450
That's almost 1 /
infinity, from the point
00:42:08.450 --> 00:42:11.700
of view of various purposes.
00:42:11.700 --> 00:42:15.180
So this is how much
you're borrowing.
00:42:15.180 --> 00:42:17.820
In each time increment
you're borrowing.
00:42:17.820 --> 00:42:23.990
And let's say that you borrow--
your rate varies over the year.
00:42:23.990 --> 00:42:27.140
I mean, sometimes you need more
money sometimes you need less.
00:42:27.140 --> 00:42:29.650
Certainly any business
would be that way.
00:42:29.650 --> 00:42:32.440
And so here you are,
you've got your money.
00:42:32.440 --> 00:42:35.070
And you're borrowing
but the rate is varying.
00:42:35.070 --> 00:42:36.960
And so how much did you borrow?
00:42:36.960 --> 00:42:51.230
Well, in day 45, which
corresponds to t is 45/365,
00:42:51.230 --> 00:42:55.210
you borrowed the
following amount.
00:42:55.210 --> 00:43:00.770
Here was your borrowing
rate times this quantity.
00:43:00.770 --> 00:43:02.900
So, dollars per year.
00:43:02.900 --> 00:43:05.500
And so this is, if
you like-- I want
00:43:05.500 --> 00:43:11.170
to emphasize the scaling
that comes about here.
00:43:11.170 --> 00:43:14.910
You have dollars per year.
00:43:14.910 --> 00:43:21.060
And this is this
number of years.
00:43:21.060 --> 00:43:23.180
So that comes out
to be in dollars.
00:43:23.180 --> 00:43:24.050
This final amount.
00:43:24.050 --> 00:43:25.883
This is the amount that
you actually borrow.
00:43:25.883 --> 00:43:30.250
So you borrow this amount.
00:43:30.250 --> 00:43:37.880
And now, if I want to
add up how much you get--
00:43:37.880 --> 00:43:39.920
you've borrowed in
the entire year.
00:43:39.920 --> 00:43:50.380
That's this sum. i = 1 to 365
of f of, well, it's (i / 365)
00:43:50.380 --> 00:43:50.880
delta t.
00:43:50.880 --> 00:43:53.220
Which I'll just leave
as delta t here.
00:43:53.220 --> 00:44:01.620
This is total amount borrowed.
00:44:01.620 --> 00:44:02.830
This is kind of a messy sum.
00:44:02.830 --> 00:44:05.570
In fact, your bank probably
will keep track of it
00:44:05.570 --> 00:44:06.830
and they know how to do that.
00:44:06.830 --> 00:44:09.679
But when we're modeling things
with strategies, you know,
00:44:09.679 --> 00:44:11.220
trading strategies,
of course, you're
00:44:11.220 --> 00:44:13.910
really some kind of
financial engineer
00:44:13.910 --> 00:44:17.000
and you want to cleverly
optimize how much you borrow.
00:44:17.000 --> 00:44:19.610
And how much you spend,
and how much you invest.
00:44:19.610 --> 00:44:23.900
This is going to be very,
very similar to the integral
00:44:23.900 --> 00:44:29.460
from 0 to 1 of f(t) dt.
00:44:29.460 --> 00:44:36.340
At the scale of 1/35,
it's probably-- 365,
00:44:36.340 --> 00:44:39.800
it's probably enough
for many purposes.
00:44:39.800 --> 00:44:44.942
Now, however,
there's another thing
00:44:44.942 --> 00:44:46.150
that you would want to model.
00:44:46.150 --> 00:44:47.670
Which is equally important.
00:44:47.670 --> 00:44:49.810
This is how much you
borrowed, but there's also
00:44:49.810 --> 00:44:53.380
how much you owe the back
at the end of the year.
00:44:53.380 --> 00:44:56.680
And the amount that you owe the
bank at the end of the year,
00:44:56.680 --> 00:44:58.680
I'm going to do
it in a fancy way.
00:44:58.680 --> 00:45:04.950
It's, the interest, we'll say,
is compounded continuously.
00:45:04.950 --> 00:45:07.780
So the interest rate,
if you start out with P
00:45:07.780 --> 00:45:20.120
as your principal, then after
time t you owe-- So borrow P,
00:45:20.120 --> 00:45:30.000
after time t, you owe P e^(rt),
where r is your interest rate.
00:45:30.000 --> 00:45:36.070
Say .05 per year.
00:45:36.070 --> 00:45:40.320
That would be an example
of an interest rate.
00:45:40.320 --> 00:45:45.740
And so, if you want to
understand how much money
00:45:45.740 --> 00:45:52.330
you actually owe at
the end of the year.
00:45:52.330 --> 00:45:54.380
At the end of the
year what you owe is,
00:45:54.380 --> 00:46:02.690
well, you borrowed
these amounts here.
00:46:02.690 --> 00:46:04.680
But now you owe more
at the end of the year.
00:46:04.680 --> 00:46:10.050
You owe e^r times the amount
of time left in the year.
00:46:10.050 --> 00:46:15.270
So the amount of time left
in the year is 1 - i / 365.
00:46:15.270 --> 00:46:18.900
Or 365 - i days left.
00:46:18.900 --> 00:46:26.600
So this is 1 - i / 365.
00:46:26.600 --> 00:46:34.770
And this is what you have to
add up, to see how much you owe.
00:46:34.770 --> 00:46:39.540
And that is essentially
the integral from 0 to 1.
00:46:39.540 --> 00:46:41.310
The delta t comes out.
00:46:41.310 --> 00:46:49.940
And you have here e^(r(1-t)),
so the t is replacing this i /
00:46:49.940 --> 00:46:54.880
365, f(t) dt.
00:46:54.880 --> 00:46:58.630
And so when you start computing
and thinking about what's
00:46:58.630 --> 00:47:04.170
the right strategy, you're faced
with integrals of this type.
00:47:04.170 --> 00:47:06.140
So that's just an example.
00:47:06.140 --> 00:47:08.930
And see you next time.
00:47:08.930 --> 00:47:10.640
Remember to think
about questions
00:47:10.640 --> 00:47:12.667
that you'll ask next time.