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PROFESSOR: So we're going
on to the third unit here.
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00:00:24 --> 00:00:31
So we're getting
started with Unit 3.
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00:00:31 --> 00:00:38
And this is our intro
to integration.
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It's basically the second
half of calculus after
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differentiation.
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00:00:51 --> 00:00:55
Today what I'll talk about
is what are known as
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definite integrals.
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Actually, it looks like,
are we missing a bunch
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of overhead lights?
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Is there a reason for that?
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Hmm.
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Let's see.
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00:01:30 --> 00:01:32
Ahh.
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Alright.
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00:01:34 --> 00:01:39
OK, that's a little
brighter now.
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00:01:39 --> 00:01:39
Alright.
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00:01:39 --> 00:01:48
So the idea of definite
integrals can be presented
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in a number of ways.
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But I will be consistent
with the rest of the
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presentation in the course.
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00:01:54 --> 00:01:57
We're going to start with the
geometric point of view.
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00:01:57 --> 00:02:00
And the geometric point of view
is, the problem we want to
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solve us to find the
area under a curve.
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00:02:13 --> 00:02:17
The other point of view that
one can take, and we'll mention
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00:02:17 --> 00:02:19
that at the end of this
lecture, is the idea
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00:02:19 --> 00:02:26
of a cumulative sum.
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00:02:26 --> 00:02:30
So keep that in mind that
there's a lot going on here.
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And there are many different
interpretations of
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00:02:32 --> 00:02:37
what the integral is.
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00:02:37 --> 00:02:41
Now, so let's draw
a picture here.
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I'll start at a place a
and end at a place b.
40
00:02:44 --> 00:02:46
And I have some curve here.
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00:02:46 --> 00:02:55
And what I have in mind is
to find this area here.
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00:02:55 --> 00:02:57
And, of course, in order to do
that, I need more information
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than just where we start
and where we end.
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00:02:59 --> 00:03:01
I also need the
bottom and the top.
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00:03:01 --> 00:03:05
By convention, the bottom is
the x axis and the top is the
46
00:03:05 --> 00:03:11
curve that we've specified,
which is y = f(x).
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00:03:11 --> 00:03:17
And we have a notation for
this, which is the notation
48
00:03:17 --> 00:03:19
using calculus for this
as opposed to some
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geometric notation.
50
00:03:20 --> 00:03:24
And that's the
following expression.
51
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It's called an integral, but
now it's going to have what
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are known as limits on it.
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It will start at
a and end at b.
54
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And we write in the
function f(x) dx.
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So this is what's known
as a definite integral.
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And it's interpreted
geometrically as the
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area under the curve.
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00:03:46 --> 00:03:49
The only difference between
this collection of symbols
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and what we had before with
indefinite integrals is that
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00:03:52 --> 00:04:03
before we didn't specify where
it started and where it ended.
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Now, in order to understand
what to do with this guy, I'm
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going to just describe very
abstractly what we do.
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And then carry out one
example in detail.
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So, to compute this area,
we're going to follow
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initially three steps.
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First of all, we're going
to divide into rectangles.
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And unfortunately, because it's
impossible to divide a curvy
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region into rectangles,
we're going to cheat.
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So they're only
quote-unquote rectangles.
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They're almost rectangles.
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And the second thing
we're going to do is
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to add up the areas.
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And the third thing we're going
to do is to rectify this
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problem that we didn't actually
hit the answer on the nose.
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That we were missing some
pieces or were choosing
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some extra bits.
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And the way we'll rectify that
is by taking the limit as
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the rectangles get thin.
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Infinitesimally
thin, very thin.
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Pictorially, again,
that looks like this.
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We have a and our b, and
we have our guy here,
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this is our curve.
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And I'm going to chop it up.
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00:05:51 --> 00:05:57
First I'm going to chop up the
x axis into little increments.
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00:05:57 --> 00:06:00
And then I'm going to
chop things up here.
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00:06:00 --> 00:06:03
And I'll decide on some
rectangle, maybe some
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staircase pattern here.
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Like this.
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Now, I don't care so much.
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In some cases the rectangles
overshoot; in some cases
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they're underneath.
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00:06:20 --> 00:06:23
So the new area that
I'm adding up is off.
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It's not quite the same as
the area under the curve.
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It's this region here.
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But it includes these
extra bits here.
96
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And then it's missing
this little guy here.
97
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This little bit
there is missing.
98
00:06:47 --> 00:06:51
And, as I say, these little
pieces up here, this a little
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bit up here is extra.
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00:06:55 --> 00:06:58
So that's why we're not really
dividing up the region
101
00:06:58 --> 00:06:59
into rectangles.
102
00:06:59 --> 00:07:01
We're just taking rectangles.
103
00:07:01 --> 00:07:05
And then the idea is that as
these get thinner and thinner,
104
00:07:05 --> 00:07:09
the little itty bitty amounts
that we miss by are going to 0.
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00:07:09 --> 00:07:10
And they're going
to be negligible.
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Already, you can see it's kind
of a thin piece of area, so
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we're not missing by much.
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And as these get thinner and
thinner, the problem goes away
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00:07:19 --> 00:07:27
and we get the answer on
the nose in the limit.
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00:07:27 --> 00:07:35
So here's our first example.
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I'll take the first interesting
curve, which is f ( x) = x^2.
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00:07:41 --> 00:07:44
I don't want to do anything
more complicated than one
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example, because this is
a real labor here, what
114
00:07:48 --> 00:07:50
we're going to go through.
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And to make things easier
for myself, I'm going
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00:07:52 --> 00:07:55
to start at a = 0.
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But in order to see what the
pattern is, I'm going to
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allow b to be arbitrary.
119
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Let's draw the graph and
start breaking things up.
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00:08:15 --> 00:08:18
So here's the parabola, and
there's this piece that we
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00:08:18 --> 00:08:24
want, which is going to stop
at this place, b, here.
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00:08:24 --> 00:08:37
And the first step is to
divide into n pieces.
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00:08:37 --> 00:08:40
That means, well, graphically,
I'll just mark the first three.
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00:08:40 --> 00:08:44
And maybe there are going
to be many of them.
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And then I'll draw some
rectangles here, and I'm
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00:08:48 --> 00:08:51
going to choose to make
the rectangles all the
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00:08:51 --> 00:08:53
way from the right.
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That is, I'll make us
this staircase pattern
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00:08:55 --> 00:08:58
here, like this.
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00:08:58 --> 00:09:00
That's my choice.
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I get to choose whatever level
I want, and I'm going to
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choose the right ends as
the shape of the staircase.
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So I'm overshooting
with each rectangle.
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And now I have to write
down formulas for
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00:09:20 --> 00:09:23
what these areas are.
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Now, there's one big advantage
that rectangles have.
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00:09:26 --> 00:09:28
And this is the starting place.
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00:09:28 --> 00:09:33
Which is that it's easy
to find their areas.
139
00:09:33 --> 00:09:35
All you need to know is the
base and the height, and you
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00:09:35 --> 00:09:37
multiply, and you get the area.
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00:09:37 --> 00:09:40
That's the reason why we can
get started with rectangles.
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00:09:40 --> 00:09:44
And in this case, these
distances, I'm assuming that
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00:09:44 --> 00:09:48
they're all equal, equally
spaced, intervals.
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00:09:48 --> 00:09:50
And I'll always be doing that.
145
00:09:50 --> 00:09:57
And so the spacing, the
bases, the base length,
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00:09:57 --> 00:10:01
is always b / n.
147
00:10:01 --> 00:10:09
All equal intervals.
148
00:10:09 --> 00:10:11
So that's the base length.
149
00:10:11 --> 00:10:15
And next, I need the heights.
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And in order to keep track of
the heights, I'm going to draw
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00:10:17 --> 00:10:22
a little table here, with x and
f ( x), and plug in a
152
00:10:22 --> 00:10:27
few values just to see
what the pattern is.
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The first place here,
after 0, is b / n.
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00:10:34 --> 00:10:36
So here's b / n,
that's an x value.
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00:10:36 --> 00:10:40
And the f ( x) value
is the height there.
156
00:10:40 --> 00:10:43
And that's just, I value
it f(x), f)x) = x^2.
157
00:10:44 --> 00:10:47
And that's (b / n)^2.
158
00:10:49 --> 00:10:56
And similarly, the
next one is 2b / n.
159
00:10:56 --> 00:10:59
And the value here
is (2b / n^2.
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00:11:01 --> 00:11:02
That's this.
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00:11:02 --> 00:11:07
This height here is 2b / n.
162
00:11:07 --> 00:11:14
That's the second rectangle.
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00:11:14 --> 00:11:16
And I'll write down one more.
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00:11:16 --> 00:11:18
3b / n, that's the third one.
165
00:11:18 --> 00:11:20
And the height is (3b / n^2.
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00:11:20 --> 00:11:23
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00:11:23 --> 00:11:29
And so forth.
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00:11:29 --> 00:11:34
Well, my next job is to
add up these areas.
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00:11:34 --> 00:11:36
And I've already prepared
that by finding out what
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00:11:36 --> 00:11:39
the base and the height is.
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00:11:39 --> 00:11:50
So the total area, or the sum
of the areas, let's say, of
172
00:11:50 --> 00:11:57
these rectangles, is - well,
the first one is (b
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00:11:57 --> 00:12:00
/ n) ( b / n)^2.
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00:12:01 --> 00:12:07
The second one is 2b / n - I'm
sorry, is (b / n)( 2b / n)^2.
175
00:12:08 --> 00:12:11
And it just keeps on going.
176
00:12:11 --> 00:12:17
And the last one is
(b / n)( nb / n)^2.
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00:12:17 --> 00:12:20
So it's very important
to figure out what the
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00:12:20 --> 00:12:22
general formula is.
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00:12:22 --> 00:12:25
And here we have a base.
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00:12:25 --> 00:12:28
And here we have a height, and
here we have the same kind of
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00:12:28 --> 00:12:31
base, but we have a new height.
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00:12:31 --> 00:12:32
And so forth.
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00:12:32 --> 00:12:36
And the pattern is that the
coefficient here is 1, then 2,
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00:12:36 --> 00:12:43
then 3, all the way up to n.
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00:12:43 --> 00:12:45
The rectangles are getting
taller and taller, and this
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00:12:45 --> 00:12:50
one, the last one
is the biggest.
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00:12:50 --> 00:12:57
OK, this is a very complicated
gadget. and the first thing I
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00:12:57 --> 00:12:59
want to do is simplify it
and then I'm actually
189
00:12:59 --> 00:13:00
going to evaluate it.
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00:13:00 --> 00:13:03
But actually I'm not going
to evaluate it exactly.
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00:13:03 --> 00:13:04
I'm just going to
evaluate the limit.
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00:13:04 --> 00:13:07
Turns out, limits
are always easier.
193
00:13:07 --> 00:13:09
The point about calculus
here is that these
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00:13:09 --> 00:13:10
rectangles are hard.
195
00:13:10 --> 00:13:13
But the limiting value
is an easy value.
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00:13:13 --> 00:13:16
So what we're heading for is
the simple formula, as opposed
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00:13:16 --> 00:13:19
to the complicated one.
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00:13:19 --> 00:13:22
Alright, so the first thing
I'm going to do is factor
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00:13:22 --> 00:13:25
out all these b / n factors.
200
00:13:25 --> 00:13:26
There's a b / n here, and
there's a (b / n)^2.
201
00:13:27 --> 00:13:29
So all told, we
have a (b / n)^3.
202
00:13:31 --> 00:13:33
As a common factor.
203
00:13:33 --> 00:13:36
And then the first term is
1, and the second term,
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00:13:36 --> 00:13:38
what's left over, is 2^2.
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2^2.
206
00:13:41 --> 00:13:43
And then the third term
would be 3^2, although
207
00:13:43 --> 00:13:46
I haven't written it.
208
00:13:46 --> 00:13:49
In the last term, there's
an extra factor of n^2.
209
00:13:51 --> 00:14:05
In the numerator.
210
00:14:05 --> 00:14:09
OK, is everybody with me here?
211
00:14:09 --> 00:14:23
Now, what I'd like to do is
to eventually take the limit
212
00:14:23 --> 00:14:26
as n goes to infinity here.
213
00:14:26 --> 00:14:29
And the quantity that's
hard to understand is this
214
00:14:29 --> 00:14:33
massive quantity here.
215
00:14:33 --> 00:14:36
And there's one change that
I'd like to make, but
216
00:14:36 --> 00:14:40
it's a very modest one.
217
00:14:40 --> 00:14:41
Extremely minuscule.
218
00:14:41 --> 00:14:43
Which is that I'm going to
write 1, just to see that
219
00:14:43 --> 00:14:45
there's a general pattern here.
220
00:14:45 --> 00:14:46
Going to write 1 as 1^2.
221
00:14:46 --> 00:14:52
222
00:14:52 --> 00:14:59
And let's put in the
3 here, why not.
223
00:14:59 --> 00:15:05
And now I want to use a trick.
224
00:15:05 --> 00:15:10
This trick is not completely
recommended, but I will say
225
00:15:10 --> 00:15:13
a lot more about that when
we get through to the end.
226
00:15:13 --> 00:15:16
I want to understand how
big this quantity is.
227
00:15:16 --> 00:15:19
So I'm going to use a geometric
trick to draw a picture
228
00:15:19 --> 00:15:20
of this quantity.
229
00:15:20 --> 00:15:23
Namely, I'm going to
build a pyramid.
230
00:15:23 --> 00:15:29
And the base of the pyramid is
going to be n by n blocks.
231
00:15:29 --> 00:15:32
So imagine we're in Egypt and
we're building a pyramid.
232
00:15:32 --> 00:15:39
And the next layer is going
to be n - 1 by n - 1.
233
00:15:39 --> 00:15:43
So this next layer in is
n minus 1 by n minus 1.
234
00:15:43 --> 00:15:46
So the total number of blocks
on the bottom is n squared.
235
00:15:46 --> 00:15:50
That's this rightmost
term here.
236
00:15:50 --> 00:15:52
But the next term, which I
didn't write in but maybe I
237
00:15:52 --> 00:15:57
should, the next to the
last term was this one.
238
00:15:57 --> 00:16:00
And that's the second
layer that I've put on.
239
00:16:00 --> 00:16:05
Now, this is, if you
like, the top view.
240
00:16:05 --> 00:16:08
But perhaps we should also
think in terms of a side view.
241
00:16:08 --> 00:16:13
So here's the same picture,
we're starting at n and we
242
00:16:13 --> 00:16:15
build up this layer here.
243
00:16:15 --> 00:16:18
And now we're going to put a
layer on top of it, which
244
00:16:18 --> 00:16:19
is a little shorter.
245
00:16:19 --> 00:16:21
So the first layer
is of length n.
246
00:16:21 --> 00:16:25
And the second layers is of
length n - 1, and then on top
247
00:16:25 --> 00:16:28
of that we have something of
length n - 2, and so forth.
248
00:16:28 --> 00:16:29
And we're going
to pile them up.
249
00:16:29 --> 00:16:31
So we pile them up.
250
00:16:31 --> 00:16:35
All the way to the top,
which is just one
251
00:16:35 --> 00:16:37
giant block of stone.
252
00:16:37 --> 00:16:39
And that's this last one, 1^2.
253
00:16:39 --> 00:16:43
So we're going
backwards in the sum.
254
00:16:43 --> 00:16:46
And so I have to build
this whole thing up.
255
00:16:46 --> 00:16:48
And I get all the way up in
this staircase pattern to
256
00:16:48 --> 00:16:57
this top block, up there.
257
00:16:57 --> 00:17:01
So here's the trick that you
can use to estimate the size
258
00:17:01 --> 00:17:05
of this, and it's sufficient
in the limit as n
259
00:17:05 --> 00:17:06
goes to infinity.
260
00:17:06 --> 00:17:15
The trick is that I can imagine
the solid thing underneath
261
00:17:15 --> 00:17:19
the staircase, like this.
262
00:17:19 --> 00:17:24
That's an ordinary pyramid,
not a staircase pyramid.
263
00:17:24 --> 00:17:26
Which is inside.
264
00:17:26 --> 00:17:28
And this one is inside.
265
00:17:28 --> 00:17:32
And so, but it's an ordinary
pyramid as opposed to
266
00:17:32 --> 00:17:34
a staircase pyramid.
267
00:17:34 --> 00:17:37
And so, we know the formula
for the volume of that.
268
00:17:37 --> 00:17:40
Because we know the formula
for volumes of cones.
269
00:17:40 --> 00:17:50
And the formula for the volume
of this guy, of the inside,
270
00:17:50 --> 00:17:58
is 1/3 base times height.
271
00:17:58 --> 00:18:03
And in that case, the base
here - so that's 1/3, and
272
00:18:03 --> 00:18:06
the base is n by n, right?
273
00:18:06 --> 00:18:07
So the base is n^2.
274
00:18:08 --> 00:18:10
That's the base.
275
00:18:10 --> 00:18:13
And the height, it goes all
the way to the top point.
276
00:18:13 --> 00:18:21
So the height is n.
277
00:18:21 --> 00:18:26
And what we've discovered
here is that this whole sum
278
00:18:26 --> 00:18:30
is bigger than 1/3 n^3.
279
00:18:30 --> 00:18:42
280
00:18:42 --> 00:18:46
Now, I claimed that - this
line, by the way has slope 2.
281
00:18:46 --> 00:18:50
So you go 1/2 over each
time you go up 1.
282
00:18:50 --> 00:18:52
And that's why you
get to the top.
283
00:18:52 --> 00:18:57
On the other hand, I can trap
it on the outside, too, by
284
00:18:57 --> 00:19:01
drawing a parallel
line out here.
285
00:19:01 --> 00:19:06
And this will go down 1/2
more on this side and 1/2
286
00:19:06 --> 00:19:08
more on the other side.
287
00:19:08 --> 00:19:14
So the base will be (n + 1) by
(n + 1) of this bigger pyramid.
288
00:19:14 --> 00:19:18
And it'll go up 1 higher.
289
00:19:18 --> 00:19:20
So on the other end, we
get that this is less
290
00:19:20 --> 00:19:24
than 1/3 (n + 1)^3.
291
00:19:25 --> 00:19:34
Again, (n + 1)^2 ( n + 1) again
this is a base times a height.
292
00:19:34 --> 00:19:36
Of this bigger pyramid.
293
00:19:36 --> 00:19:38
Yes, question.
294
00:19:38 --> 00:19:39
STUDENT: [INAUDIBLE]
295
00:19:39 --> 00:19:48
and then equating it to volume.
296
00:19:48 --> 00:19:52
PROFESSOR: The question is, it
seems as if I'm adding up areas
297
00:19:52 --> 00:19:54
and equating it to volume.
298
00:19:54 --> 00:19:58
But I'm actually creating
volumes by making these
299
00:19:58 --> 00:20:00
honest increments here.
300
00:20:00 --> 00:20:07
That is, the base is n
but the height is 1.
301
00:20:07 --> 00:20:09
Thank you for
pointing that out.
302
00:20:09 --> 00:20:11
Each one of these little
staircases here has
303
00:20:11 --> 00:20:14
exactly height 1.
304
00:20:14 --> 00:20:16
So I'm honestly
sticking blocks there.
305
00:20:16 --> 00:20:18
They're sort of square blocks,
and I'm lining them up.
306
00:20:18 --> 00:20:21
And I'm thinking of n by
n cubeds, if you like.
307
00:20:21 --> 00:20:22
Honest cubes, there.
308
00:20:22 --> 00:20:25
And the height is 1.
309
00:20:25 --> 00:20:26
And the base is n^2.
310
00:20:26 --> 00:20:33
311
00:20:33 --> 00:20:36
Alright, so I claim that I've
trapped this guy in between
312
00:20:36 --> 00:20:38
two quantities here.
313
00:20:38 --> 00:20:52
And now I'm ready
to take the limit.
314
00:20:52 --> 00:20:55
If you look at what our goal
is, we want to have an
315
00:20:55 --> 00:20:57
expression like this.
316
00:20:57 --> 00:21:00
And I'm going to - this was the
massive expression that we had.
317
00:21:00 --> 00:21:03
And actually, I'm going
to write it differently.
318
00:21:03 --> 00:21:11
I'll write it as b^3(
1^2 + 2^2 + n^2 / n^3).
319
00:21:12 --> 00:21:15
I'm going to combine
all the n's together.
320
00:21:15 --> 00:21:18
Alright, so the right
thing to do is to divide
321
00:21:18 --> 00:21:20
what I had up there.
322
00:21:20 --> 00:21:28
Divide by n^3 in this set
of inequalities there.
323
00:21:28 --> 00:21:38
And what I get here is 1/3 <
(1 + 2^2 + 3^2 + n^2 / n^3)
324
00:21:38 --> 00:21:43
< 1/3 ( n + 1)^3 / n^3.
325
00:21:45 --> 00:21:50
And that's 1/3( 1 + (1 / n))^3.
326
00:21:50 --> 00:21:56
327
00:21:56 --> 00:21:59
And now, I claim we're done.
328
00:21:59 --> 00:22:05
Because this is 1/3, and the
limit, as n goes to infinity,
329
00:22:05 --> 00:22:08
of this quantity here, is
easily seen to be, this is, as
330
00:22:08 --> 00:22:10
n goes to infinity,
this goes to 0.
331
00:22:10 --> 00:22:14
So this also goes to 1/3.
332
00:22:14 --> 00:22:29
And so our total here, so our
total area, under x^2, which
333
00:22:29 --> 00:22:36
we sometimes might write the
integral from 0 to b x^2 / dx,
334
00:22:36 --> 00:22:40
is going to be equal to - well,
it's this 1/3 that I've got.
335
00:22:40 --> 00:22:43
But then there was
also a b^3 there.
336
00:22:43 --> 00:22:45
So there's this
extra b cubed here.
337
00:22:45 --> 00:22:49
So it's 1/3 b^3.
338
00:22:49 --> 00:22:54
That's the result from
this whole computation.
339
00:22:54 --> 00:22:55
Yes, question.
340
00:22:55 --> 00:22:57
STUDENT: [INAUDIBLE]
341
00:22:57 --> 00:23:05
PROFESSOR: So that was
a very good question.
342
00:23:05 --> 00:23:08
The question is, why did
we leave the b / n^3
343
00:23:08 --> 00:23:11
out, for this step.
344
00:23:11 --> 00:23:16
And a part of the answer
is malice aforethought.
345
00:23:16 --> 00:23:19
In other words, we know
what we're heading for.
346
00:23:19 --> 00:23:21
We know, we understand,
this quantity.
347
00:23:21 --> 00:23:23
It's all one thing.
348
00:23:23 --> 00:23:26
This thing is a sum, which is
growing larger and larger.
349
00:23:26 --> 00:23:28
It's not what's called
a closed form.
350
00:23:28 --> 00:23:31
So, the thing that's not known,
or not well understood, is how
351
00:23:31 --> 00:23:33
big is this quantity here.
352
00:23:33 --> 00:23:35
1^2 + 2^2.
353
00:23:35 --> 00:23:37
The sum of the squares.
354
00:23:37 --> 00:23:39
Whereas, this is something
that's quite easy
355
00:23:39 --> 00:23:40
to understand.
356
00:23:40 --> 00:23:42
So we factor it out.
357
00:23:42 --> 00:23:47
And we analyze carefully
the piece which we don't
358
00:23:47 --> 00:23:48
know yet, how big it is.
359
00:23:48 --> 00:23:51
And we discovered that it's
very, very similar to n^3.
360
00:23:52 --> 00:23:55
But it's more
similar to 1/3 n^3.
361
00:23:57 --> 00:23:59
It's almost identical
to 1/3 n^3.
362
00:24:00 --> 00:24:02
This extra piece here.
363
00:24:02 --> 00:24:04
So that's what's going on.
364
00:24:04 --> 00:24:05
And then we match that.
365
00:24:05 --> 00:24:08
Since this thing is very
similar to 1/3 n^3 we
366
00:24:08 --> 00:24:24
cancel the n^3's and
we have our result.
367
00:24:24 --> 00:24:28
Let me just mention that
although this may seem odd,
368
00:24:28 --> 00:24:31
in fact this is what you
always do if you analyze
369
00:24:31 --> 00:24:32
these kinds of sum.
370
00:24:32 --> 00:24:34
You always factor out
whatever you can.
371
00:24:34 --> 00:24:37
And then you still are faced
with a sum like this.
372
00:24:37 --> 00:24:40
So this will happen
systematically, every time
373
00:24:40 --> 00:24:45
you're faced with such a sum.
374
00:24:45 --> 00:24:53
OK, now I want to say one
more word about notation.
375
00:24:53 --> 00:25:00
Which is that this notation
is an extreme nuisance here.
376
00:25:00 --> 00:25:04
And it's really sort of too
large for us to deal with.
377
00:25:04 --> 00:25:08
And so, mathematicians
have a shorthand for it.
378
00:25:08 --> 00:25:10
Unfortunately, when you
actually do a computation,
379
00:25:10 --> 00:25:13
you're going to end up
with this collection
380
00:25:13 --> 00:25:15
of stuff anyway.
381
00:25:15 --> 00:25:19
But I want to just show you
this summation notation in
382
00:25:19 --> 00:25:24
order to compress
it a little bit.
383
00:25:24 --> 00:25:31
The idea of summation
notation is the following.
384
00:25:31 --> 00:25:35
So this thing tends, the
ideas are following.
385
00:25:35 --> 00:25:37
I'll illustrate it with
an example first.
386
00:25:37 --> 00:25:45
So, the general notation is
the sum of ai, i = 1 to n
387
00:25:45 --> 00:25:50
= a1 + a2 + ... plus an.
388
00:25:50 --> 00:25:53
So this is the abbreviation.
389
00:25:53 --> 00:26:03
And this is a capital Sigma.
390
00:26:03 --> 00:26:09
And so, this quantity here,
for instance, is (1 / n^3)
391
00:26:09 --> 00:26:15
the sum i^2, i = 1 to n.
392
00:26:15 --> 00:26:17
So that's what this
thing is equal to.
393
00:26:17 --> 00:26:20
And what we just showed is
that that tends to 1/3
394
00:26:20 --> 00:26:23
as n goes to infinity.
395
00:26:23 --> 00:26:30
So this is the way the
summation notation is used.
396
00:26:30 --> 00:26:34
There's a formula for each of
these coefficients, each of
397
00:26:34 --> 00:26:37
these entries here,
or summands.
398
00:26:37 --> 00:26:39
And then this is just
an abbreviation for
399
00:26:39 --> 00:26:40
what the sum is.
400
00:26:40 --> 00:26:44
And this is the reason why I
stuck in that 1^2 at the
401
00:26:44 --> 00:26:46
beginning, so that you could
see that the pattern worked
402
00:26:46 --> 00:26:47
all the way down to i = 1.
403
00:26:47 --> 00:26:50
It isn't an exception
to the rule.
404
00:26:50 --> 00:26:54
It's the same as
all of the others.
405
00:26:54 --> 00:26:59
Now, over here, in this board,
we also had one of these
406
00:26:59 --> 00:27:02
extremely long sums.
407
00:27:02 --> 00:27:06
And this one can be written
in the following way.
408
00:27:06 --> 00:27:10
And I hope you agree, this
is rather hard to scan.
409
00:27:10 --> 00:27:15
But one way of writing it is,
it's the sum from i = 1 to n
410
00:27:15 --> 00:27:18
of, now I have to write
down the formula for
411
00:27:18 --> 00:27:19
the general term.
412
00:27:19 --> 00:27:24
Which is (b / n)( ib / n)^2.
413
00:27:24 --> 00:27:29
414
00:27:29 --> 00:27:34
So that's a way of abbreviating
this massive formula into one
415
00:27:34 --> 00:27:36
which is just a lot shorter.
416
00:27:36 --> 00:27:40
And now, the manipulation that
I performed with it, which is
417
00:27:40 --> 00:27:45
to factor out this (b / n)^3,
is something that I'm perfectly
418
00:27:45 --> 00:27:49
well allowed to do
also over here.
419
00:27:49 --> 00:27:51
This is the distributive law.
420
00:27:51 --> 00:27:56
This, if I factor out b^3 /
n^3, I'm left with the sum
421
00:27:56 --> 00:28:00
i = 1 to n of i^2, right?
422
00:28:00 --> 00:28:06
So these notations make it
a little bit more compact.
423
00:28:06 --> 00:28:10
What we're dealing with.
424
00:28:10 --> 00:28:14
The conceptual phenomenon
is still the same.
425
00:28:14 --> 00:28:18
And the mess is really still
just hiding under the rug.
426
00:28:18 --> 00:28:23
But the notation is at
least fits with fewer
427
00:28:23 --> 00:28:32
symbols, anyway.
428
00:28:32 --> 00:28:39
So let's continue here.
429
00:28:39 --> 00:28:41
I've giving you
one calculation.
430
00:28:41 --> 00:28:51
And now I want to fit
it into a pattern.
431
00:28:51 --> 00:28:54
And here's the thing that
I'd like to calculate.
432
00:28:54 --> 00:28:59
So, first of all let's try
the case, so I'm going
433
00:28:59 --> 00:29:02
to do two more examples.
434
00:29:02 --> 00:29:04
I'll do two more examples,
but they're going to
435
00:29:04 --> 00:29:05
be much, much easier.
436
00:29:05 --> 00:29:09
And then things are going to
get much easier from now on.
437
00:29:09 --> 00:29:19
So, the second example is going
to be the function f(x) = x.
438
00:29:19 --> 00:29:23
If I draw that, that's this
function here, that's
439
00:29:23 --> 00:29:26
the line with slope 1.
440
00:29:26 --> 00:29:29
And here's b.
441
00:29:29 --> 00:29:33
And so this area here is the
same as the area of the
442
00:29:33 --> 00:29:36
triangle with base
b and height b.
443
00:29:36 --> 00:29:44
So the area is equal to 1/2
b * b, so this is the base.
444
00:29:44 --> 00:29:45
And this is the height.
445
00:29:45 --> 00:29:49
We also know how to find
the area of triangles.
446
00:29:49 --> 00:29:52
And so, the formula is 1/2 b^2.
447
00:29:52 --> 00:29:57
448
00:29:57 --> 00:30:04
And the third example, notice,
by the way, I didn't have to do
449
00:30:04 --> 00:30:06
this elaborate summing to do
that, because we happen
450
00:30:06 --> 00:30:07
to know this area.
451
00:30:07 --> 00:30:13
The third example is going to
be even easier. f(x) = 1.
452
00:30:13 --> 00:30:17
By far the most important
example, remarkably, when you
453
00:30:17 --> 00:30:20
get to 18.02 and multivariable
calculus, you will
454
00:30:20 --> 00:30:22
forget this calculation.
455
00:30:22 --> 00:30:23
Somehow.
456
00:30:23 --> 00:30:26
And I don't know why, but
it happens to everybody.
457
00:30:26 --> 00:30:30
So, the function is just
horizontal, like this.
458
00:30:30 --> 00:30:31
Right?
459
00:30:31 --> 00:30:32
It's the constant 1.
460
00:30:32 --> 00:30:37
And if we stop it at b, then
the area we're interested in
461
00:30:37 --> 00:30:42
is just this, from 0 to b.
462
00:30:42 --> 00:30:47
And we know that this is
height 1, so this is area,
463
00:30:47 --> 00:30:51
is base, which is b * 1.
464
00:30:51 --> 00:31:03
So it's b.
465
00:31:03 --> 00:31:13
Let's look now at the pattern.
466
00:31:13 --> 00:31:19
We're going to look at the
pattern of the function, and
467
00:31:19 --> 00:31:23
it's the area under the curve,
which is this, has this
468
00:31:23 --> 00:31:27
elaborate formula in terms
of, so this is just the
469
00:31:27 --> 00:31:34
area under the curve.
470
00:31:34 --> 00:31:40
Between 0 and b.
471
00:31:40 --> 00:31:47
And we have x^2, which
turned out to be b^3 / 3.
472
00:31:47 --> 00:31:50
And we have x, which turned out
to be - well, let me write them
473
00:31:50 --> 00:31:53
over just a bit more to give
myself some room. x, which
474
00:31:53 --> 00:31:57
turns out to be b^2/ 2.
475
00:31:57 --> 00:32:07
And then we have 1, which
turned out to be b.
476
00:32:07 --> 00:32:10
So this, I claim,
is suggestive.
477
00:32:10 --> 00:32:16
If you can figure out the
pattern, one way of making
478
00:32:16 --> 00:32:20
it a little clearer is
to see that x = x^ 1.
479
00:32:22 --> 00:32:27
And 1 = x ^ 0 .
480
00:32:27 --> 00:32:30
So this is the
case, 0, 1 and 2.
481
00:32:30 --> 00:32:40
And b = b ^ 1 / 1.
482
00:32:40 --> 00:32:56
So, if you want to guess what
happens when f(x) = x^3, well
483
00:32:56 --> 00:32:58
if it's 0, you do b ^ 1 / 1.
484
00:32:58 --> 00:33:01
If it's 1, you do b ^ 2 / 2.
485
00:33:01 --> 00:33:04
If it's 2, you do b ^ 3 / 3.
486
00:33:04 --> 00:33:11
So it's reasonable to guess
that this should be b ^ 4 / 4.
487
00:33:11 --> 00:33:15
That's a reasonable
guess, I would say.
488
00:33:15 --> 00:33:24
Now, the strange thing is that
in history, Archimedes figured
489
00:33:24 --> 00:33:27
out the area under a parabola.
490
00:33:27 --> 00:33:29
So that was a long time ago.
491
00:33:29 --> 00:33:30
It was after the pyramids.
492
00:33:30 --> 00:33:34
And he used, actually, a much
more complicated method
493
00:33:34 --> 00:33:36
than I just described here.
494
00:33:36 --> 00:33:41
And his method, which is just
fantastically amazing, was so
495
00:33:41 --> 00:33:43
brilliant that it may have set
back mathematics
496
00:33:43 --> 00:33:46
by 2,000 years.
497
00:33:46 --> 00:33:49
Because people were so, it was
so difficult that people
498
00:33:49 --> 00:33:51
couldn't see this pattern.
499
00:33:51 --> 00:33:53
And couldn't see that,
actually, these kinds of
500
00:33:53 --> 00:33:54
calculations are easy.
501
00:33:54 --> 00:33:56
So they couldn't
get to the cubic.
502
00:33:56 --> 00:33:59
And even when they got to the
cubic, they were struggling
503
00:33:59 --> 00:33:59
with everything else.
504
00:33:59 --> 00:34:02
And it wasn't until calculus
fit everything together that
505
00:34:02 --> 00:34:04
people were able to make
serious progress on
506
00:34:04 --> 00:34:06
calculating these areas.
507
00:34:06 --> 00:34:09
Even though he was the expert
on calculating areas and
508
00:34:09 --> 00:34:12
volumes, for his time.
509
00:34:12 --> 00:34:16
So this is really a great thing
that we now can have easy
510
00:34:16 --> 00:34:16
methods of doing it.
511
00:34:16 --> 00:34:21
And the main thing that I want
to tell you is that's we will
512
00:34:21 --> 00:34:25
not have to labor to build
pyramids to calculate all
513
00:34:25 --> 00:34:27
of these quantities.
514
00:34:27 --> 00:34:29
We will have a way
faster way of doing it.
515
00:34:29 --> 00:34:32
This is the slow,
laborious way.
516
00:34:32 --> 00:34:37
And we will be able to do it so
easily that it will happen as
517
00:34:37 --> 00:34:39
fast as you differentiate.
518
00:34:39 --> 00:34:42
So that's coming up tomorrow.
519
00:34:42 --> 00:34:45
But I want you to know
that it's going to be.
520
00:34:45 --> 00:34:47
However, we're going to
go through just a little
521
00:34:47 --> 00:34:52
pain before we do it.
522
00:34:52 --> 00:34:59
And I'll just tell you one
more piece of notation here.
523
00:34:59 --> 00:35:03
So you need to have a little
practice just to recognize
524
00:35:03 --> 00:35:04
how much savings
we're going to make.
525
00:35:04 --> 00:35:08
But never again will you have
to face elaborate geometric
526
00:35:08 --> 00:35:16
arguments like this.
527
00:35:16 --> 00:35:25
So let me just add a little
bit of notation for
528
00:35:25 --> 00:35:27
definite integrals.
529
00:35:27 --> 00:35:35
And this goes under the
name of Riemann sums.
530
00:35:35 --> 00:35:44
Named after a mathematician
from the 1800s.
531
00:35:44 --> 00:35:53
So this is the general
procedure for
532
00:35:53 --> 00:36:01
definite integrals.
533
00:36:01 --> 00:36:04
We divide it up into pieces.
534
00:36:04 --> 00:36:07
And how do we do that?
535
00:36:07 --> 00:36:16
Well, so here's our
a and here's our b.
536
00:36:16 --> 00:36:19
And what we're going to do is
break it up into little pieces.
537
00:36:19 --> 00:36:22
And we're going to give a
name to the increment.
538
00:36:22 --> 00:36:28
And we're going to
call that delta x.
539
00:36:28 --> 00:36:30
So we divide up into these.
540
00:36:30 --> 00:36:32
So how many pieces are there?
541
00:36:32 --> 00:36:38
If there are n pieces, then the
general formula is always the
542
00:36:38 --> 00:36:43
delta x is 1 / n times
the total length.
543
00:36:43 --> 00:36:48
So it has to be b - a / n.
544
00:36:48 --> 00:36:51
We will always use these equal
increments, although you don't
545
00:36:51 --> 00:36:53
absolutely have to do it.
546
00:36:53 --> 00:37:01
We will, for these
Riemann sums.
547
00:37:01 --> 00:37:07
And now there's only one
bit of flexibility that
548
00:37:07 --> 00:37:10
we will allow ourselves.
549
00:37:10 --> 00:37:13
Which is this.
550
00:37:13 --> 00:37:26
We're going to pick any
height of f between.
551
00:37:26 --> 00:37:34
In the interval,
in each interval.
552
00:37:34 --> 00:37:39
So what that means is, let
me just show it to you
553
00:37:39 --> 00:37:43
on the picture here.
554
00:37:43 --> 00:37:47
Is, I just pick any value
in between, I'll call it
555
00:37:47 --> 00:37:49
ci, which is in there.
556
00:37:49 --> 00:37:51
And then I go up here.
557
00:37:51 --> 00:37:55
And I have the level,
which is f( ci).
558
00:37:55 --> 00:37:58
And that's the rectangle
that I choose.
559
00:37:58 --> 00:38:03
In the case that we did, we
always chose the right-hand,
560
00:38:03 --> 00:38:04
which turned out to
be the largest one.
561
00:38:04 --> 00:38:07
But I could've chosen
some level in between.
562
00:38:07 --> 00:38:09
Or even the left-hand end.
563
00:38:09 --> 00:38:10
Which would have meant that
the staircase would've
564
00:38:10 --> 00:38:13
been quite a bit lower.
565
00:38:13 --> 00:38:17
So any of these staircases
will work perfectly well.
566
00:38:17 --> 00:38:25
So that means were picking f
( ci), and that's a height.
567
00:38:25 --> 00:38:33
And now we're just going
to add them all up.
568
00:38:33 --> 00:38:35
And this is the sum of the
areas of the rectangles,
569
00:38:35 --> 00:38:37
because this is the height.
570
00:38:37 --> 00:38:43
And this is the base.
571
00:38:43 --> 00:38:48
This notation is supposed to
be, now, very suggestive of the
572
00:38:48 --> 00:38:54
notation that Leibniz used.
573
00:38:54 --> 00:38:57
Which is that in the limit,
this becomes an integral
574
00:38:57 --> 00:39:01
from a to b of f(x) dx.
575
00:39:01 --> 00:39:05
And notice that the delta
x gets replaced by a dx.
576
00:39:05 --> 00:39:07
So this is what
happens in the limit.
577
00:39:07 --> 00:39:10
As the rectangles get thin.
578
00:39:10 --> 00:39:17
So that's as delta x goes to 0.
579
00:39:17 --> 00:39:21
And these gadgets are
called Riemann sums.
580
00:39:21 --> 00:39:29
This is called a Riemann sum.
581
00:39:29 --> 00:39:31
And we already worked
out an example.
582
00:39:31 --> 00:39:40
This very complicated guy was
an example of a Riemann sum.
583
00:39:40 --> 00:39:42
So that's a notation.
584
00:39:42 --> 00:39:45
And we'll give you a chance to
get used to it a little more
585
00:39:45 --> 00:39:51
when we do some numerical
work at the end.
586
00:39:51 --> 00:39:58
Now, the last thing for today
is, I promised you an example
587
00:39:58 --> 00:40:05
which was not an area example.
588
00:40:05 --> 00:40:10
I want to be able to show
you that integrals can be
589
00:40:10 --> 00:40:21
interpreted as cumulative sums.
590
00:40:21 --> 00:40:36
Integrals as cumulative sums.
591
00:40:36 --> 00:40:39
So this is just an example.
592
00:40:39 --> 00:40:48
And, so here's the way it goes.
593
00:40:48 --> 00:40:52
So we're going to consider a
function f, we're going to
594
00:40:52 --> 00:40:55
consider a variable
t, which is time.
595
00:40:55 --> 00:40:59
In years.
596
00:40:59 --> 00:41:02
And we'll consider a
function f( t), which
597
00:41:02 --> 00:41:06
is in dollars per year.
598
00:41:06 --> 00:41:09
Right, this is a
financial example here.
599
00:41:09 --> 00:41:13
That's the unit here,
dollars per year.
600
00:41:13 --> 00:41:21
And this is going to
be a borrowing rate.
601
00:41:21 --> 00:41:24
Now, the reason why I want to
put units in here is to show
602
00:41:24 --> 00:41:32
you that there's a good reason
for this strange dx, which we
603
00:41:32 --> 00:41:33
append on these integrals.
604
00:41:33 --> 00:41:34
This notation.
605
00:41:34 --> 00:41:37
It allows us to change
variables, it allows this to
606
00:41:37 --> 00:41:39
be consistent with units.
607
00:41:39 --> 00:41:42
And allows us to develop
meaningful formulas, which are
608
00:41:42 --> 00:41:44
consistent across the board.
609
00:41:44 --> 00:41:48
And so I want to emphasize the
units in this when I set up
610
00:41:48 --> 00:41:51
this modeling problem here.
611
00:41:51 --> 00:41:56
Now, you're borrowing money.
612
00:41:56 --> 00:41:59
Let's say, every day.
613
00:41:59 --> 00:42:06
So that means delta t = 1/365.
614
00:42:06 --> 00:42:08
That's almost 1 / infinity,
from the point of view
615
00:42:08 --> 00:42:11
of various purposes.
616
00:42:11 --> 00:42:15
So this is how much
you're borrowing.
617
00:42:15 --> 00:42:17
In each time increment
you're borrowing.
618
00:42:17 --> 00:42:23
And let's say that you borrow,
your rate varies over the year.
619
00:42:23 --> 00:42:27
I mean, sometimes you need more
money sometimes you need less.
620
00:42:27 --> 00:42:29
Certainly any business
would be that way.
621
00:42:29 --> 00:42:32
And so here you are,
you've got your money.
622
00:42:32 --> 00:42:35
And you're borrowing but
the rate is varying.
623
00:42:35 --> 00:42:36
And so how much did you borrow?
624
00:42:36 --> 00:42:53
Well, in Day 45, which
is 45/365, you borrowed
625
00:42:53 --> 00:42:55
the following amount.
626
00:42:55 --> 00:43:00
Here was your borrowing
rate times this quantity.
627
00:43:00 --> 00:43:02
So, dollars per year.
628
00:43:02 --> 00:43:06
And so this is, if you like, I
want to emphasize the scaling
629
00:43:06 --> 00:43:11
that comes about here.
630
00:43:11 --> 00:43:14
You have dollars per year.
631
00:43:14 --> 00:43:21
And this is this
number of years.
632
00:43:21 --> 00:43:23
So that comes out
to be in dollars.
633
00:43:23 --> 00:43:24
This final amount.
634
00:43:24 --> 00:43:25
This is the amount that
you actually borrow.
635
00:43:25 --> 00:43:30
So you borrow this amount.
636
00:43:30 --> 00:43:38
And now, if I want to add up
how much you get, you've
637
00:43:38 --> 00:43:39
borrowed in the entire year.
638
00:43:39 --> 00:43:46
That's this sum. i = 1 to
365 of f of, well, it's
639
00:43:46 --> 00:43:50
(i / 365) delta t.
640
00:43:50 --> 00:43:53
Which I'll just leave
as delta t here.
641
00:43:53 --> 00:44:01
This is total amount borrowed.
642
00:44:01 --> 00:44:02
This is kind of a messy sum.
643
00:44:02 --> 00:44:05
In fact, your bank probably
will keep track of it and
644
00:44:05 --> 00:44:06
they know how to do that.
645
00:44:06 --> 00:44:09
But when we're modeling things
with strategies, you know,
646
00:44:09 --> 00:44:12
trading strategies of course,
you're really some kind of
647
00:44:12 --> 00:44:15
financial engineer and you want
to cleverly optimize
648
00:44:15 --> 00:44:17
how much you borrow.
649
00:44:17 --> 00:44:19
And how much you spend,
and how much you invest.
650
00:44:19 --> 00:44:23
This is going to be very,
very similar to the integral
651
00:44:23 --> 00:44:29
from 0 to 1 of f (t) dt.
652
00:44:29 --> 00:44:36
At the scale of 1/35, it's
probably, 365, it's probably
653
00:44:36 --> 00:44:39
enough for many purposes.
654
00:44:39 --> 00:44:45
Now, however, there's another
thing that you would
655
00:44:45 --> 00:44:46
want to model.
656
00:44:46 --> 00:44:47
Which is equally important.
657
00:44:47 --> 00:44:50
This is how much you borrowed,
but there's also how much you
658
00:44:50 --> 00:44:53
owe the back at the
end of the year.
659
00:44:53 --> 00:44:56
And the amount that you owe the
bank at the end of the year,
660
00:44:56 --> 00:44:58
I'm going to do it
in a fancy way.
661
00:44:58 --> 00:45:04
It's, the interest, we'll say,
is compounded continuously.
662
00:45:04 --> 00:45:08
So the interest rate, if you
start out with P as your
663
00:45:08 --> 00:45:20
principal, then after time t,
you owe, so borrow P, after
664
00:45:20 --> 00:45:30
time t, you owe P e ^ rt, where
r is your interest rate.
665
00:45:30 --> 00:45:36
Say, 0.05 per year.
666
00:45:36 --> 00:45:40
That would be an example
of an interest rate.
667
00:45:40 --> 00:45:45
And so, if you want to
understand how much money you
668
00:45:45 --> 00:45:53
actually owe at the end of the
year, at the end of the year
669
00:45:53 --> 00:46:02
what you owe is, well, you
borrowed these amounts here.
670
00:46:02 --> 00:46:04
But now you owe more at
the end of the year.
671
00:46:04 --> 00:46:10
You owe e ^ r times the amount
of time left in the year.
672
00:46:10 --> 00:46:15
So the amount of time left in
the year is 1 - (i / 365).
673
00:46:15 --> 00:46:18
Or 365 - i days left.
674
00:46:18 --> 00:46:26
So this is (1 - i / 365).
675
00:46:26 --> 00:46:33
And this is what you
have to add up, to
676
00:46:33 --> 00:46:34
see how much you owe.
677
00:46:34 --> 00:46:39
And that is essentially
the integral from 0 to 1.
678
00:46:39 --> 00:46:41
The delta t comes out.
679
00:46:41 --> 00:46:49
And you have here e ^ r (1 -
t), so the t is replacing
680
00:46:49 --> 00:46:54
this i / 365, f (t) dt.
681
00:46:54 --> 00:46:58
And so when you start computing
and thinking about what's the
682
00:46:58 --> 00:47:04
right strategy, you're faced
with integrals of this type.
683
00:47:04 --> 00:47:06
So that's just an example.
684
00:47:06 --> 00:47:08
And see you next time.
685
00:47:08 --> 00:47:10
Remember to think about
questions that you'll
686
00:47:10 --> 00:47:12
ask next time.
687
00:47:12 --> 00:47:13