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PROFESSOR: So we're going
on to the third unit here.
9
00:00:24,910 --> 00:00:31,030
So we're getting
started with Unit 3.
10
00:00:31,030 --> 00:00:38,330
And this is our
intro to integration.
11
00:00:38,330 --> 00:00:44,820
It's basically the
second half of calculus
12
00:00:44,820 --> 00:00:51,320
after differentiation.
13
00:00:51,320 --> 00:00:55,600
Today what I'll talk
about is what are
14
00:00:55,600 --> 00:01:16,450
known as definite integrals.
15
00:01:16,450 --> 00:01:19,380
Actually, it looks
like, are we missing
16
00:01:19,380 --> 00:01:21,580
a bunch of overhead lights?
17
00:01:21,580 --> 00:01:24,260
Is there a reason for that?
18
00:01:24,260 --> 00:01:27,320
Hmm.
19
00:01:27,320 --> 00:01:30,460
Let's see.
20
00:01:30,460 --> 00:01:32,650
Aah.
21
00:01:32,650 --> 00:01:34,360
All right.
22
00:01:34,360 --> 00:01:39,200
OK, that's a little
brighter now.
23
00:01:39,200 --> 00:01:39,700
All right.
24
00:01:39,700 --> 00:01:47,420
So the idea of
definite integrals
25
00:01:47,420 --> 00:01:49,740
can be presented in
a number of ways.
26
00:01:49,740 --> 00:01:53,707
But I will be consistent with
the rest of the presentation
27
00:01:53,707 --> 00:01:54,290
in the course.
28
00:01:54,290 --> 00:01:57,060
We're going to start with
the geometric point of view.
29
00:01:57,060 --> 00:01:59,990
And the geometric
point of view is,
30
00:01:59,990 --> 00:02:13,680
the problem we want to solve is
to find the area under a curve.
31
00:02:13,680 --> 00:02:16,460
The other point of
view that one can take,
32
00:02:16,460 --> 00:02:18,590
and we'll mention that at
the end of this lecture,
33
00:02:18,590 --> 00:02:26,220
is the idea of a cumulative sum.
34
00:02:26,220 --> 00:02:30,687
So keep that in mind that
there's a lot going on here.
35
00:02:30,687 --> 00:02:32,520
And there are many
different interpretations
36
00:02:32,520 --> 00:02:37,930
of what the integral is.
37
00:02:37,930 --> 00:02:41,710
Now, so let's draw
a picture here.
38
00:02:41,710 --> 00:02:44,670
I'll start at a place
a and end at a place b.
39
00:02:44,670 --> 00:02:46,720
And I have some curve here.
40
00:02:46,720 --> 00:02:55,100
And what I have in mind
is to find this area here.
41
00:02:55,100 --> 00:02:56,600
And, of course, in
order to do that,
42
00:02:56,600 --> 00:02:58,985
I need more information
than just where we start
43
00:02:58,985 --> 00:02:59,760
and where we end.
44
00:02:59,760 --> 00:03:01,860
I also need the
bottom and the top.
45
00:03:01,860 --> 00:03:05,170
By convention, the bottom
is the x axis and the top
46
00:03:05,170 --> 00:03:11,670
is the curve that we've
specified, which is y = f(x).
47
00:03:11,670 --> 00:03:15,660
And we have a notation
for this, which
48
00:03:15,660 --> 00:03:18,810
is the notation using
calculus for this as opposed
49
00:03:18,810 --> 00:03:20,490
to some geometric notation.
50
00:03:20,490 --> 00:03:24,180
And that's the
following expression.
51
00:03:24,180 --> 00:03:26,560
It's called an
integral, but now it's
52
00:03:26,560 --> 00:03:29,290
going to have what are
known as limits on it.
53
00:03:29,290 --> 00:03:31,970
It will start at a and end at b.
54
00:03:31,970 --> 00:03:35,720
And we write in the
function f(x) dx.
55
00:03:35,720 --> 00:03:40,830
So this is what's known
as a definite integral.
56
00:03:40,830 --> 00:03:43,620
And it's interpreted
geometrically
57
00:03:43,620 --> 00:03:46,460
as the area under the curve.
58
00:03:46,460 --> 00:03:49,260
The only difference between
this collection of symbols
59
00:03:49,260 --> 00:03:51,690
and what we had before
with indefinite integrals
60
00:03:51,690 --> 00:03:54,960
is that before we didn't
specify where it started
61
00:03:54,960 --> 00:04:03,140
and where it ended.
62
00:04:03,140 --> 00:04:08,670
Now, in order to understand
what to do with this guy,
63
00:04:08,670 --> 00:04:12,330
I'm going to just describe
very abstractly what we do.
64
00:04:12,330 --> 00:04:17,050
And then carry out
one example in detail.
65
00:04:17,050 --> 00:04:24,310
So, to compute this
area, we're going
66
00:04:24,310 --> 00:04:27,270
to follow initially three steps.
67
00:04:27,270 --> 00:04:36,870
First of all, we're going
to divide into rectangles.
68
00:04:36,870 --> 00:04:42,080
And unfortunately, because
it's impossible to divide
69
00:04:42,080 --> 00:04:45,510
a curvy region into rectangles,
we're going to cheat.
70
00:04:45,510 --> 00:04:49,310
So they're only
quote-unquote rectangles.
71
00:04:49,310 --> 00:04:52,720
They're almost rectangles.
72
00:04:52,720 --> 00:05:01,060
And the second thing we're going
to do is to add up the areas.
73
00:05:01,060 --> 00:05:04,430
And the third thing
we're going to do
74
00:05:04,430 --> 00:05:10,970
is to rectify this problem
that we didn't actually
75
00:05:10,970 --> 00:05:12,870
hit the answer on the nose.
76
00:05:12,870 --> 00:05:15,310
That we were missing
some pieces or were
77
00:05:15,310 --> 00:05:17,460
choosing some extra bits.
78
00:05:17,460 --> 00:05:19,570
And the way we'll
rectify that is
79
00:05:19,570 --> 00:05:35,070
by taking the limit as
the rectangles get thin.
80
00:05:35,070 --> 00:05:39,380
Infinitesimally thin, very thin.
81
00:05:39,380 --> 00:05:43,780
Pictorially, again,
that looks like this.
82
00:05:43,780 --> 00:05:46,950
We have a and our b, and
we have our guy here,
83
00:05:46,950 --> 00:05:48,390
this is our curve.
84
00:05:48,390 --> 00:05:51,850
And I'm going to chop it up.
85
00:05:51,850 --> 00:05:57,470
First I'm going to chop up the
x axis into little increments.
86
00:05:57,470 --> 00:06:00,460
And then I'm going to
chop things up here.
87
00:06:00,460 --> 00:06:03,060
And I'll decide on
some rectangle, maybe
88
00:06:03,060 --> 00:06:05,450
some staircase pattern here.
89
00:06:05,450 --> 00:06:12,960
Like this.
90
00:06:12,960 --> 00:06:16,100
Now, I don't care so much.
91
00:06:16,100 --> 00:06:19,140
In some cases the rectangles
overshoot; in some cases
92
00:06:19,140 --> 00:06:20,400
they're underneath.
93
00:06:20,400 --> 00:06:23,720
So the new area that
I'm adding up is off.
94
00:06:23,720 --> 00:06:28,260
It's not quite the same as
the area under the curve.
95
00:06:28,260 --> 00:06:32,720
It's this region here.
96
00:06:32,720 --> 00:06:38,110
But it includes these
extra bits here.
97
00:06:38,110 --> 00:06:42,250
And then it's missing
this little guy here.
98
00:06:42,250 --> 00:06:47,950
This little bit
there is missing.
99
00:06:47,950 --> 00:06:51,310
And, as I say, these
little pieces up here,
100
00:06:51,310 --> 00:06:55,740
this a little bit
up here is extra.
101
00:06:55,740 --> 00:06:58,406
So that's why we're
not really dividing up
102
00:06:58,406 --> 00:06:59,530
the region into rectangles.
103
00:06:59,530 --> 00:07:01,300
We're just taking rectangles.
104
00:07:01,300 --> 00:07:05,060
And then the idea is that as
these get thinner and thinner,
105
00:07:05,060 --> 00:07:09,080
the little itty-bitty amounts
that we miss by are going to 0.
106
00:07:09,080 --> 00:07:10,830
And they're going
to be negligible.
107
00:07:10,830 --> 00:07:13,840
Already, you can see it's
kind of a thin piece of area,
108
00:07:13,840 --> 00:07:15,840
so we're not missing by much.
109
00:07:15,840 --> 00:07:19,150
And as these get thinner and
thinner, the problem goes away
110
00:07:19,150 --> 00:07:27,040
and we get the answer on
the nose in the limit.
111
00:07:27,040 --> 00:07:35,050
So here's our first example.
112
00:07:35,050 --> 00:07:41,270
I'll take the first interesting
curve, which is f(x) = x^2.
113
00:07:41,270 --> 00:07:45,150
I don't want to do anything more
complicated than one example,
114
00:07:45,150 --> 00:07:48,030
because this is a
real labor here,
115
00:07:48,030 --> 00:07:50,910
what we're going to go through.
116
00:07:50,910 --> 00:07:52,710
And to make things
easier for myself,
117
00:07:52,710 --> 00:07:55,930
I'm going to start at a = 0.
118
00:07:55,930 --> 00:07:58,400
But in order to see
what the pattern is,
119
00:07:58,400 --> 00:08:11,180
I'm going to allow
b to be arbitrary.
120
00:08:11,180 --> 00:08:15,230
Let's draw the graph and
start breaking things up.
121
00:08:15,230 --> 00:08:18,015
So here's the
parabola, and there's
122
00:08:18,015 --> 00:08:19,640
this piece that we
want, which is going
123
00:08:19,640 --> 00:08:24,170
to stop at this place, b, here.
124
00:08:24,170 --> 00:08:37,380
And the first step is
to divide into n pieces.
125
00:08:37,380 --> 00:08:40,840
That means, well, graphically,
I'll just mark the first three.
126
00:08:40,840 --> 00:08:44,450
And maybe there are
going to be many of them.
127
00:08:44,450 --> 00:08:47,480
And then I'll draw
some rectangles here,
128
00:08:47,480 --> 00:08:51,420
and I'm going to choose
to make the rectangles all
129
00:08:51,420 --> 00:08:53,310
the way from the right.
130
00:08:53,310 --> 00:08:55,545
That is, I'll make us
this staircase pattern
131
00:08:55,545 --> 00:08:58,760
here, like this.
132
00:08:58,760 --> 00:09:00,160
That's my choice.
133
00:09:00,160 --> 00:09:02,360
I get to choose
whatever level I want,
134
00:09:02,360 --> 00:09:04,500
and I'm going to
choose the right ends
135
00:09:04,500 --> 00:09:07,490
as the shape of the staircase.
136
00:09:07,490 --> 00:09:17,190
So I'm overshooting
with each rectangle.
137
00:09:17,190 --> 00:09:19,910
And now I have to
write down formulas
138
00:09:19,910 --> 00:09:23,800
for what these areas are.
139
00:09:23,800 --> 00:09:26,840
Now, there's one big advantage
that rectangles have.
140
00:09:26,840 --> 00:09:28,890
And this is the starting place.
141
00:09:28,890 --> 00:09:33,130
Which is that it's easy
to find their areas.
142
00:09:33,130 --> 00:09:35,260
All you need to know is
the base and the height,
143
00:09:35,260 --> 00:09:37,230
and you multiply,
and you get the area.
144
00:09:37,230 --> 00:09:40,820
That's the reason why we can
get started with rectangles.
145
00:09:40,820 --> 00:09:43,750
And in this case,
these distances,
146
00:09:43,750 --> 00:09:46,350
I'm assuming that they're
all equal, equally
147
00:09:46,350 --> 00:09:48,350
spaced, intervals.
148
00:09:48,350 --> 00:09:50,260
And I'll always be doing that.
149
00:09:50,260 --> 00:10:01,360
And so the spacing, the bases,
the base length, is always b/n.
150
00:10:01,360 --> 00:10:09,590
All equal intervals.
151
00:10:09,590 --> 00:10:11,260
So that's the base length.
152
00:10:11,260 --> 00:10:15,370
And next, I need the heights.
153
00:10:15,370 --> 00:10:17,400
And in order to keep
track of the heights,
154
00:10:17,400 --> 00:10:21,310
I'm going to draw a little
table here, with x and f(x),
155
00:10:21,310 --> 00:10:27,460
and plug in a few values just
to see what the pattern is.
156
00:10:27,460 --> 00:10:34,100
The first place here,
after 0, is b/n.
157
00:10:34,100 --> 00:10:36,810
So here's b/n,
that's an x-value.
158
00:10:36,810 --> 00:10:40,590
And the f(x) value
is the height there.
159
00:10:40,590 --> 00:10:44,980
And that's just, I
evaluate f(x), f(x) is x^2.
160
00:10:44,980 --> 00:10:49,450
And that's (b/n)^2.
161
00:10:49,450 --> 00:10:53,350
And similarly, the
next one is 2b/n.
162
00:10:56,230 --> 00:10:58,560
And the value here is (2b/n)^2.
163
00:11:01,530 --> 00:11:02,950
That's this.
164
00:11:02,950 --> 00:11:07,600
This height here is 2b/n.
165
00:11:07,600 --> 00:11:14,070
That's the second rectangle.
166
00:11:14,070 --> 00:11:16,960
And I'll write down one more.
167
00:11:16,960 --> 00:11:18,600
3b/n, that's the third one.
168
00:11:18,600 --> 00:11:20,090
And the height is (3b/n)^2.
169
00:11:23,250 --> 00:11:29,940
And so forth.
170
00:11:29,940 --> 00:11:34,537
Well, my next job is
to add up these areas.
171
00:11:34,537 --> 00:11:36,620
And I've already prepared
that by finding out what
172
00:11:36,620 --> 00:11:39,020
the base and the height is.
173
00:11:39,020 --> 00:11:50,030
So the total area, or the
sum of the areas, let's say,
174
00:11:50,030 --> 00:12:01,240
of these rectangles, is-- Well,
the first one is (b/n) (b/n)^2.
175
00:12:01,240 --> 00:12:06,380
The second one is 2b/n --
I'm sorry, is (b/n) (2b/n)^2.
176
00:12:08,920 --> 00:12:11,070
And it just keeps on going.
177
00:12:11,070 --> 00:12:13,540
And the last one is
(b/n) (nb / n)^2.
178
00:12:17,810 --> 00:12:19,870
So it's very important
to figure out
179
00:12:19,870 --> 00:12:22,720
what the general formula is.
180
00:12:22,720 --> 00:12:25,000
And here we have a base.
181
00:12:25,000 --> 00:12:27,170
And here we have a
height, and here we
182
00:12:27,170 --> 00:12:31,930
have the same kind of base,
but we have a new height.
183
00:12:31,930 --> 00:12:32,880
And so forth.
184
00:12:32,880 --> 00:12:36,390
And the pattern is that the
coefficient here is 1, then 2,
185
00:12:36,390 --> 00:12:42,976
then 3, all the way up to n.
186
00:12:42,976 --> 00:12:44,850
The rectangles are
getting taller and taller,
187
00:12:44,850 --> 00:12:50,940
and this one, the last
one is the biggest.
188
00:12:50,940 --> 00:12:55,300
OK, this is a very
complicated gadget.
189
00:12:55,300 --> 00:12:58,800
and the first thing I
want to do is simplify it
190
00:12:58,800 --> 00:13:00,600
and then I'm actually
going to evaluate it.
191
00:13:00,600 --> 00:13:03,129
But actually I'm not going
to evaluate it exactly.
192
00:13:03,129 --> 00:13:04,670
I'm just going to
evaluate the limit.
193
00:13:04,670 --> 00:13:07,060
Turns out, limits
are always easier.
194
00:13:07,060 --> 00:13:10,960
The point about calculus here is
that these rectangles are hard.
195
00:13:10,960 --> 00:13:13,330
But the limiting value
is an easy value.
196
00:13:13,330 --> 00:13:16,440
So what we're heading for is
the simple formula, as opposed
197
00:13:16,440 --> 00:13:19,550
to the complicated one.
198
00:13:19,550 --> 00:13:22,190
Alright, so the first
thing I'm going to do
199
00:13:22,190 --> 00:13:25,130
is factor out all
these b/n factors.
200
00:13:25,130 --> 00:13:27,970
There's a b/n, here, and
there's a (b/n)^2, So all told,
201
00:13:27,970 --> 00:13:31,390
we have a (b/n)^3.
202
00:13:31,390 --> 00:13:33,730
As a common factor.
203
00:13:33,730 --> 00:13:36,830
And then the first term
is 1, and the second term,
204
00:13:36,830 --> 00:13:39,310
what's left over, is 2^2.
205
00:13:39,310 --> 00:13:41,100
2^2.
206
00:13:41,100 --> 00:13:43,670
And then the third
term would be 3^2,
207
00:13:43,670 --> 00:13:46,970
although I haven't written it.
208
00:13:46,970 --> 00:13:51,200
In the last term, there's
an extra factor of n^2.
209
00:13:51,200 --> 00:14:05,050
In the numerator.
210
00:14:05,050 --> 00:14:09,820
OK, is everybody with me here?
211
00:14:09,820 --> 00:14:23,930
Now, what I'd like to do is
to eventually take the limit
212
00:14:23,930 --> 00:14:26,520
as n goes to infinity here.
213
00:14:26,520 --> 00:14:29,010
And the quantity that's
hard to understand
214
00:14:29,010 --> 00:14:33,540
is this massive quantity here.
215
00:14:33,540 --> 00:14:36,690
And there's one change
that I'd like to make,
216
00:14:36,690 --> 00:14:40,160
but it's a very modest one.
217
00:14:40,160 --> 00:14:41,410
Extremely minuscule.
218
00:14:41,410 --> 00:14:43,350
Which is that I'm
going to write 1,
219
00:14:43,350 --> 00:14:45,450
just to see that there's
a general pattern here.
220
00:14:45,450 --> 00:14:46,500
Going to write 1 as 1^2.
221
00:14:52,830 --> 00:14:59,420
And let's put in
the 3 here, why not.
222
00:14:59,420 --> 00:15:05,400
And now I want to use a trick.
223
00:15:05,400 --> 00:15:08,560
This trick is not
completely recommended,
224
00:15:08,560 --> 00:15:12,380
but I will say a
lot more about that
225
00:15:12,380 --> 00:15:13,780
when we get through to the end.
226
00:15:13,780 --> 00:15:16,110
I want to understand how
big this quantity is.
227
00:15:16,110 --> 00:15:18,795
So I'm going to use a
geometric trick to draw
228
00:15:18,795 --> 00:15:20,880
a picture of this quantity.
229
00:15:20,880 --> 00:15:23,660
Namely, I'm going
to build a pyramid.
230
00:15:23,660 --> 00:15:29,510
And the base of the pyramid
is going to be n by n blocks.
231
00:15:29,510 --> 00:15:32,410
So imagine we're in Egypt
and we're building a pyramid.
232
00:15:32,410 --> 00:15:39,190
And the next layer is
going to be n-1 by n-1.
233
00:15:39,190 --> 00:15:43,250
So this next layer
in is n-1 by n-1.
234
00:15:43,250 --> 00:15:46,590
So the total number of blocks
on the bottom is n squared.
235
00:15:46,590 --> 00:15:50,020
That's this rightmost term here.
236
00:15:50,020 --> 00:15:52,460
But the next term, which I
didn't write in but maybe I
237
00:15:52,460 --> 00:15:57,280
should, the next-to-the-last
term was this one.
238
00:15:57,280 --> 00:16:00,360
And that's the second
layer that I've put on.
239
00:16:00,360 --> 00:16:05,500
Now, this is, if you
like, the top view.
240
00:16:05,500 --> 00:16:08,950
But perhaps we should also
think in terms of a side view.
241
00:16:08,950 --> 00:16:12,920
So here's the same picture,
we're starting at n
242
00:16:12,920 --> 00:16:15,920
and we build up this layer here.
243
00:16:15,920 --> 00:16:18,507
And now we're going to put
a layer on top of it, which
244
00:16:18,507 --> 00:16:19,340
is a little shorter.
245
00:16:19,340 --> 00:16:21,600
So the first layer
is of length n.
246
00:16:21,600 --> 00:16:25,460
And the second layers is of
length n-1, and then on top
247
00:16:25,460 --> 00:16:28,007
of that we have something
of length n-2, and so forth.
248
00:16:28,007 --> 00:16:29,340
And we're going to pile them up.
249
00:16:29,340 --> 00:16:31,800
So we pile them up.
250
00:16:31,800 --> 00:16:37,630
All the way to the top, which is
just one giant block of stone.
251
00:16:37,630 --> 00:16:39,490
And that's this last one, 1^2.
252
00:16:39,490 --> 00:16:43,290
So we're going
backwards in the sum.
253
00:16:43,290 --> 00:16:46,140
And so I have to build
this whole thing up.
254
00:16:46,140 --> 00:16:48,350
And I get all the way up
in this staircase pattern
255
00:16:48,350 --> 00:16:57,720
to this top block, up there.
256
00:16:57,720 --> 00:17:00,140
So here's the trick
that you can use
257
00:17:00,140 --> 00:17:02,870
to estimate the size
of this, and it's
258
00:17:02,870 --> 00:17:06,410
sufficient in the limit
as n goes to infinity.
259
00:17:06,410 --> 00:17:14,560
The trick is that I can
imagine the solid thing
260
00:17:14,560 --> 00:17:19,560
underneath the
staircase, like this.
261
00:17:19,560 --> 00:17:24,390
That's an ordinary pyramid,
not a staircase pyramid.
262
00:17:24,390 --> 00:17:26,550
Which is inside.
263
00:17:26,550 --> 00:17:28,940
And this one is inside.
264
00:17:28,940 --> 00:17:32,150
And so, but it's an
ordinary pyramid as opposed
265
00:17:32,150 --> 00:17:34,530
to a staircase pyramid.
266
00:17:34,530 --> 00:17:37,690
And so, we know the formula
for the volume of that.
267
00:17:37,690 --> 00:17:40,580
Because we know the formula
for volumes of cones.
268
00:17:40,580 --> 00:17:50,090
And the formula for the volume
of this guy, of the inside,
269
00:17:50,090 --> 00:17:58,150
is 1/3 base times height.
270
00:17:58,150 --> 00:18:03,500
And in that case, the
base here-- so that's 1/3,
271
00:18:03,500 --> 00:18:06,180
and the base is n by n, right?
272
00:18:06,180 --> 00:18:08,540
So the base is n^2.
273
00:18:08,540 --> 00:18:10,050
That's the base.
274
00:18:10,050 --> 00:18:13,340
And the height, it goes all
the way to the top point.
275
00:18:13,340 --> 00:18:21,290
So the height is n.
276
00:18:21,290 --> 00:18:27,130
And what we've discovered
here is that this whole sum is
277
00:18:27,130 --> 00:18:30,910
bigger than 1/3 n^3.
278
00:18:42,680 --> 00:18:46,050
Now, I claimed that - this
line, by the way has slope 2.
279
00:18:46,050 --> 00:18:50,140
So you go 1/2 over
each time you go up 1.
280
00:18:50,140 --> 00:18:52,960
And that's why you
get to the top.
281
00:18:52,960 --> 00:18:56,950
On the other hand, I can
trap it on the outside,
282
00:18:56,950 --> 00:19:01,840
too, by drawing a
parallel line out here.
283
00:19:01,840 --> 00:19:07,100
And this will go down 1/2
more on this side and 1/2 more
284
00:19:07,100 --> 00:19:08,230
on the other side.
285
00:19:08,230 --> 00:19:14,870
So the base will be n+1 by
n+1 of this bigger pyramid.
286
00:19:14,870 --> 00:19:18,120
And it'll go up 1 higher.
287
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.
288
00:19:25,690 --> 00:19:34,290
Again, (n+1)^2 times n+1, again,
this is a base times a height.
289
00:19:34,290 --> 00:19:36,920
Of this bigger pyramid.
290
00:19:36,920 --> 00:19:38,110
Yes, question.
291
00:19:38,110 --> 00:19:47,860
STUDENT: [INAUDIBLE] and
then equating it to volume.
292
00:19:47,860 --> 00:19:49,360
PROFESSOR: The
question is, it seems
293
00:19:49,360 --> 00:19:54,360
as if I'm adding up areas
and equating it to volume.
294
00:19:54,360 --> 00:19:57,160
But I'm actually
creating volumes
295
00:19:57,160 --> 00:20:00,060
by making these honest
increments here.
296
00:20:00,060 --> 00:20:07,750
That is, the base is
n but the height is 1.
297
00:20:07,750 --> 00:20:09,380
Thank you for pointing that out.
298
00:20:09,380 --> 00:20:11,220
Each one of these
little staircases
299
00:20:11,220 --> 00:20:14,150
here has exactly height 1.
300
00:20:14,150 --> 00:20:15,950
So I'm honestly
sticking blocks there.
301
00:20:15,950 --> 00:20:18,200
They're sort of square blocks,
and I'm lining them up.
302
00:20:18,200 --> 00:20:21,450
And I'm thinking of n
by n cubes, if you like.
303
00:20:21,450 --> 00:20:22,960
Honest cubes, there.
304
00:20:22,960 --> 00:20:25,447
And the height is 1.
305
00:20:25,447 --> 00:20:26,280
And the base is n^2.
306
00:20:33,360 --> 00:20:36,760
Alright, so I claim that I've
trapped this guy in between two
307
00:20:36,760 --> 00:20:38,520
quantities here.
308
00:20:38,520 --> 00:20:52,510
And now I'm ready
to take the limit.
309
00:20:52,510 --> 00:20:55,120
If you look at what
our goal is, we
310
00:20:55,120 --> 00:20:57,060
want to have an
expression like this.
311
00:20:57,060 --> 00:21:00,840
And I'm going to-- This was the
massive expression that we had.
312
00:21:00,840 --> 00:21:03,310
And actually, I'm going
to write it differently.
313
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,
314
00:21:10,230 --> 00:21:12,070
divided by n^3.
315
00:21:12,070 --> 00:21:15,520
I'm going to combine
all the n's together.
316
00:21:15,520 --> 00:21:17,420
Alright, so the
right thing to do
317
00:21:17,420 --> 00:21:20,480
is to divide what
I had up there.
318
00:21:20,480 --> 00:21:28,280
Divide by n^3 in this set
of inequalities there.
319
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
320
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.
321
00:21:45,490 --> 00:21:49,230
And that's 1/3 + (1 + 1/n)^3.
322
00:21:56,230 --> 00:21:59,530
And now, I claim we're done.
323
00:21:59,530 --> 00:22:03,170
Because this is
1/3, and the limit,
324
00:22:03,170 --> 00:22:06,450
as n goes to infinity,
of this quantity here,
325
00:22:06,450 --> 00:22:09,350
is easily seen to be, this
is, as n goes to infinity,
326
00:22:09,350 --> 00:22:10,470
this goes to 0.
327
00:22:10,470 --> 00:22:14,790
So this also goes to 1/3.
328
00:22:14,790 --> 00:22:28,810
And so our total here, so
our total area, under x^2,
329
00:22:28,810 --> 00:22:33,040
which we sometimes might write
the integral from 0 to b x^2
330
00:22:33,040 --> 00:22:38,210
dx, is going to be
equal to - well,
331
00:22:38,210 --> 00:22:40,350
it's this 1/3 that I've got.
332
00:22:40,350 --> 00:22:43,400
But then there was
also a b^3 there.
333
00:22:43,400 --> 00:22:45,040
So there's this
extra b cubed here.
334
00:22:45,040 --> 00:22:49,240
So it's 1/3 b^3.
335
00:22:49,240 --> 00:22:54,630
That's the result from
this whole computation.
336
00:22:54,630 --> 00:22:55,620
Yes, question.
337
00:22:55,620 --> 00:22:57,060
STUDENT: [INAUDIBLE]
338
00:22:57,060 --> 00:23:05,370
PROFESSOR: So that was
a very good question.
339
00:23:05,370 --> 00:23:08,960
The question is, why did
we leave the b/n^3 out,
340
00:23:08,960 --> 00:23:11,190
for this step.
341
00:23:11,190 --> 00:23:16,290
And a part of the answer
is malice aforethought.
342
00:23:16,290 --> 00:23:19,290
In other words, we know
what we're heading for.
343
00:23:19,290 --> 00:23:21,670
We know, we understand,
this quantity.
344
00:23:21,670 --> 00:23:23,570
It's all one thing.
345
00:23:23,570 --> 00:23:26,080
This thing is a sum, which
is growing larger and larger.
346
00:23:26,080 --> 00:23:28,430
It's not what's
called a closed form.
347
00:23:28,430 --> 00:23:31,440
So, the thing that's not
known, or not well understood,
348
00:23:31,440 --> 00:23:33,320
is how big is this
quantity here.
349
00:23:33,320 --> 00:23:37,050
1^2 + 2^2, the sum
of the squares.
350
00:23:37,050 --> 00:23:38,620
Whereas, this is
something that's
351
00:23:38,620 --> 00:23:40,740
quite easy to understand.
352
00:23:40,740 --> 00:23:42,570
So we factor it out.
353
00:23:42,570 --> 00:23:47,740
And we analyze carefully the
piece which we don't know yet,
354
00:23:47,740 --> 00:23:48,920
how big it is.
355
00:23:48,920 --> 00:23:52,560
And we discovered that it's
very, very similar to n^3.
356
00:23:52,560 --> 00:23:57,030
But it's more
similar to 1/3 n^3.
357
00:23:57,030 --> 00:24:00,500
It's almost
identical to 1/3 n^3.
358
00:24:00,500 --> 00:24:02,080
This extra piece here.
359
00:24:02,080 --> 00:24:04,120
So that's what's going on.
360
00:24:04,120 --> 00:24:05,110
And then we match that.
361
00:24:05,110 --> 00:24:09,230
Since this thing is very
similar to 1/3 n^3 we cancel
362
00:24:09,230 --> 00:24:24,780
the n^3's and we
have our result.
363
00:24:24,780 --> 00:24:28,230
Let me just mention that
although this may seem odd,
364
00:24:28,230 --> 00:24:30,760
in fact this is what
you always do if you
365
00:24:30,760 --> 00:24:32,630
analyze these kinds of sum.
366
00:24:32,630 --> 00:24:34,810
You always factor
out whatever you can.
367
00:24:34,810 --> 00:24:37,250
And then you still are
faced with a sum like this.
368
00:24:37,250 --> 00:24:40,100
So this will happen
systematically, every time
369
00:24:40,100 --> 00:24:45,860
you're faced with such a sum.
370
00:24:45,860 --> 00:24:53,450
OK, now I want to say one
more word about notation.
371
00:24:53,450 --> 00:25:00,000
Which is that this notation
is an extreme nuisance here.
372
00:25:00,000 --> 00:25:04,170
And it's really sort of too
large for us to deal with.
373
00:25:04,170 --> 00:25:08,440
And so, mathematicians
have a shorthand for it.
374
00:25:08,440 --> 00:25:10,930
Unfortunately, when you
actually do a computation,
375
00:25:10,930 --> 00:25:15,280
you're going to end up with
this collection of stuff anyway.
376
00:25:15,280 --> 00:25:19,530
But I want to just show you
this summation notation in order
377
00:25:19,530 --> 00:25:24,980
to compress it a little bit.
378
00:25:24,980 --> 00:25:31,600
The idea of summation
notation is the following.
379
00:25:31,600 --> 00:25:35,380
So this thing tends-- The
ideas are the following.
380
00:25:35,380 --> 00:25:37,920
I'll illustrate it
with an example first.
381
00:25:37,920 --> 00:25:45,810
So, the general notation is
the sum of a_i, i = 1 to n,
382
00:25:45,810 --> 00:25:50,510
is a_1 plus a_2 plus
dot dot dot plus a_n.
383
00:25:50,510 --> 00:25:53,700
So this is the abbreviation.
384
00:25:53,700 --> 00:26:03,950
And this is a capital sigma.
385
00:26:03,950 --> 00:26:06,720
And so, this quantity
here, for instance,
386
00:26:06,720 --> 00:26:15,380
is 1/n^3 times the
sum i^2, i = 1 to n.
387
00:26:15,380 --> 00:26:17,350
So that's what this
thing is equal to.
388
00:26:17,350 --> 00:26:20,590
And what we just showed
is that that tends to 1/3
389
00:26:20,590 --> 00:26:23,910
as n goes to infinity.
390
00:26:23,910 --> 00:26:30,630
So this is the way the
summation notation is used.
391
00:26:30,630 --> 00:26:34,330
There's a formula for each
of these coefficients,
392
00:26:34,330 --> 00:26:37,360
each of these entries
here, or summands.
393
00:26:37,360 --> 00:26:39,460
And then this is
just an abbreviation
394
00:26:39,460 --> 00:26:40,710
for what the sum is.
395
00:26:40,710 --> 00:26:43,407
And this is the reason
why I stuck in that 1^2
396
00:26:43,407 --> 00:26:45,740
at the beginning, so that you
could see that the pattern
397
00:26:45,740 --> 00:26:47,505
worked all the
way down to i = 1.
398
00:26:47,505 --> 00:26:50,600
It isn't an exception
to the rule.
399
00:26:50,600 --> 00:26:54,160
It's the same as
all of the others.
400
00:26:54,160 --> 00:26:59,020
Now, over here, in
this board, we also
401
00:26:59,020 --> 00:27:02,680
had one of these
extremely long sums.
402
00:27:02,680 --> 00:27:06,770
And this one can be written
in the following way.
403
00:27:06,770 --> 00:27:10,830
And I hope you agree, this
is rather hard to scan.
404
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
405
00:27:16,385 --> 00:27:19,720
- now I have to write down the
formula for the general term.
406
00:27:19,720 --> 00:27:23,130
Which is b/n (ib/n)^2.
407
00:27:29,550 --> 00:27:34,350
So that's a way of abbreviating
this massive formula into one
408
00:27:34,350 --> 00:27:36,980
which is just a lot shorter.
409
00:27:36,980 --> 00:27:40,090
And now, the manipulation
that I performed with it,
410
00:27:40,090 --> 00:27:43,940
which is to factor
out this (b/n)^3,
411
00:27:43,940 --> 00:27:47,560
is something that I'm perfectly
well allowed to do also over
412
00:27:47,560 --> 00:27:49,130
here.
413
00:27:49,130 --> 00:27:51,480
This is the distributive law.
414
00:27:51,480 --> 00:27:56,506
This, if I factor out b^3 /
n^3, I'm left with the sum,
415
00:27:56,506 --> 00:28:00,880
i = 1 to n, of i^2, right?
416
00:28:00,880 --> 00:28:06,040
So these notations make it
a little bit more compact.
417
00:28:06,040 --> 00:28:10,870
What we're dealing with.
418
00:28:10,870 --> 00:28:14,940
The conceptual phenomenon
is still the same.
419
00:28:14,940 --> 00:28:18,160
And the mess is really still
just hiding under the rug.
420
00:28:18,160 --> 00:28:23,960
But the notation is-- at
least fits with fewer symbols,
421
00:28:23,960 --> 00:28:32,320
anyway.
422
00:28:32,320 --> 00:28:39,560
So let's continue here.
423
00:28:39,560 --> 00:28:41,460
I've given you one calculation.
424
00:28:41,460 --> 00:28:51,010
And now I want to fit
it into a pattern.
425
00:28:51,010 --> 00:28:54,650
And here's the thing that
I'd like to calculate.
426
00:28:54,650 --> 00:28:59,120
So, first of all let's
try the case-- S I'm
427
00:28:59,120 --> 00:29:02,697
going to do two more examples.
428
00:29:02,697 --> 00:29:04,280
I'll do two more
examples, but they're
429
00:29:04,280 --> 00:29:05,650
going to be much, much easier.
430
00:29:05,650 --> 00:29:09,160
And then things are going to
get much easier from now on.
431
00:29:09,160 --> 00:29:19,580
So, the second example is going
to be the function f(x) = x.
432
00:29:19,580 --> 00:29:23,520
If I draw that, that's
this function here,
433
00:29:23,520 --> 00:29:26,500
that's the line with slope 1.
434
00:29:26,500 --> 00:29:29,280
And here's b.
435
00:29:29,280 --> 00:29:32,300
And so this area
here is the same
436
00:29:32,300 --> 00:29:36,440
as the area of the triangle
with base b and height b.
437
00:29:36,440 --> 00:29:44,360
So the area is equal to 1/2
b * b, so this is the base.
438
00:29:44,360 --> 00:29:45,660
And this is the height.
439
00:29:45,660 --> 00:29:49,970
We also know how to find
the area of triangles.
440
00:29:49,970 --> 00:29:52,770
And so, the formula is 1/2 b^2.
441
00:29:57,270 --> 00:30:02,480
And the third example--
Notice, by the way,
442
00:30:02,480 --> 00:30:05,920
I didn't have to do this
elaborate summing to do that,
443
00:30:05,920 --> 00:30:07,770
because we happen
to know this area.
444
00:30:07,770 --> 00:30:13,370
The third example is going
to be even easier. f(x) = 1.
445
00:30:13,370 --> 00:30:16,200
By far the most
important example.
446
00:30:16,200 --> 00:30:20,230
Remarkably, when you get to
18.02, multivariable calculus,
447
00:30:20,230 --> 00:30:22,420
you will forget
this calculation.
448
00:30:22,420 --> 00:30:23,160
Somehow.
449
00:30:23,160 --> 00:30:26,080
And I don't know why, but
it happens to everybody.
450
00:30:26,080 --> 00:30:30,690
So, the function is just
horizontal, like this.
451
00:30:30,690 --> 00:30:31,190
Right?
452
00:30:31,190 --> 00:30:32,610
It's the constant 1.
453
00:30:32,610 --> 00:30:37,390
And if we stop it at b, then
the area we're interested in
454
00:30:37,390 --> 00:30:42,190
is just this, from 0 to b.
455
00:30:42,190 --> 00:30:47,850
And we know that this is
height 1, so this is area
456
00:30:47,850 --> 00:30:51,600
is base, which is b, times 1.
457
00:30:51,600 --> 00:31:03,750
So it's b.
458
00:31:03,750 --> 00:31:13,300
Let's look now at the pattern.
459
00:31:13,300 --> 00:31:17,730
We're going to look at the
pattern of the function,
460
00:31:17,730 --> 00:31:21,080
and it's the area
under the curve, which
461
00:31:21,080 --> 00:31:26,330
is this-- has this
elaborate formula in terms
462
00:31:26,330 --> 00:31:34,990
of-- so this is just the
area under the curve.
463
00:31:34,990 --> 00:31:40,500
Between 0 and b.
464
00:31:40,500 --> 00:31:47,110
And we have x^2, which
turned out to be b^3 / 3.
465
00:31:47,110 --> 00:31:49,824
And we have x, which
turned out to be-- well,
466
00:31:49,824 --> 00:31:52,240
let me write them over just a
bit more to give myself some
467
00:31:52,240 --> 00:31:57,200
room. x, which turns
out to be b^2 / 2.
468
00:31:57,200 --> 00:32:07,090
And then we have 1,
which turned out to be b.
469
00:32:07,090 --> 00:32:10,830
So this, I claim, is suggestive.
470
00:32:10,830 --> 00:32:14,750
If you can figure
out the pattern,
471
00:32:14,750 --> 00:32:19,750
one way of making it a little
clearer is to see that x is
472
00:32:19,750 --> 00:32:22,210
x^1.
473
00:32:22,210 --> 00:32:24,410
And 1 is x^0.
474
00:32:27,250 --> 00:32:30,070
So this is the case, 0, 1 and 2.
475
00:32:30,070 --> 00:32:32,660
And b is b^1 / 1.
476
00:32:40,010 --> 00:32:56,720
So, if you want to guess what
happens when f(x) is x^3,
477
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;
478
00:33:01,030 --> 00:33:04,270
if it's 2, you do b^3 / 3.
479
00:33:04,270 --> 00:33:07,850
So it's reasonable to guess
that this should be b^4 / 4.
480
00:33:11,110 --> 00:33:15,110
That's a reasonable
guess, I would say.
481
00:33:15,110 --> 00:33:24,740
Now, the strange thing is that
in history, Archimedes figured
482
00:33:24,740 --> 00:33:27,550
out the area under a parabola.
483
00:33:27,550 --> 00:33:29,600
So that was a long time ago.
484
00:33:29,600 --> 00:33:30,910
It was after the pyramids.
485
00:33:30,910 --> 00:33:34,340
And he used, actually, a
much more complicated method
486
00:33:34,340 --> 00:33:36,500
than I just described here.
487
00:33:36,500 --> 00:33:40,700
And his method, which is
just fantastically amazing,
488
00:33:40,700 --> 00:33:43,850
was so brilliant that it may
have set back mathematics
489
00:33:43,850 --> 00:33:46,080
by 2,000 years.
490
00:33:46,080 --> 00:33:48,630
Because people were
so-- it was so difficult
491
00:33:48,630 --> 00:33:51,070
that people couldn't
see this pattern.
492
00:33:51,070 --> 00:33:54,312
And couldn't see that, actually,
these kinds of calculations
493
00:33:54,312 --> 00:33:54,812
are easy.
494
00:33:54,812 --> 00:33:56,740
So they couldn't
get to the cubic.
495
00:33:56,740 --> 00:33:58,240
And even when they
got to the cubic,
496
00:33:58,240 --> 00:33:59,690
they were struggling
with everything else.
497
00:33:59,690 --> 00:34:01,550
And it wasn't until
calculus fit everything
498
00:34:01,550 --> 00:34:04,770
together that people were
able to make serious progress
499
00:34:04,770 --> 00:34:06,640
on calculating these areas.
500
00:34:06,640 --> 00:34:09,940
Even though he was the expert on
calculating areas and volumes,
501
00:34:09,940 --> 00:34:12,130
for his time.
502
00:34:12,130 --> 00:34:15,470
So this is really a great
thing that we now can
503
00:34:15,470 --> 00:34:16,810
have easy methods of doing it.
504
00:34:16,810 --> 00:34:21,430
And the main thing that I
want to tell you is that's we
505
00:34:21,430 --> 00:34:25,740
will not have to labor to
build pyramids to calculate
506
00:34:25,740 --> 00:34:27,250
all of these quantities.
507
00:34:27,250 --> 00:34:29,620
We will have a way
faster way of doing it.
508
00:34:29,620 --> 00:34:32,790
This is the slow, laborious way.
509
00:34:32,790 --> 00:34:37,080
And we will be able to do it
so easily that it will happen
510
00:34:37,080 --> 00:34:39,590
as fast as you differentiate.
511
00:34:39,590 --> 00:34:42,360
So that's coming up tomorrow.
512
00:34:42,360 --> 00:34:45,920
But I want you to know that
it's going to be-- However,
513
00:34:45,920 --> 00:34:52,550
we're going to go through just
a little pain before we do it.
514
00:34:52,550 --> 00:34:59,400
And I'll just tell you one
more piece of notation here.
515
00:34:59,400 --> 00:35:01,410
So you need to have a
little practice just
516
00:35:01,410 --> 00:35:04,850
to recognize how much
savings we're going to make.
517
00:35:04,850 --> 00:35:07,340
But never again will
you have to face
518
00:35:07,340 --> 00:35:16,190
elaborate geometric
arguments like this.
519
00:35:16,190 --> 00:35:21,110
So let me just add a
little bit of notation
520
00:35:21,110 --> 00:35:27,910
for definite integrals.
521
00:35:27,910 --> 00:35:35,810
And this goes under the
name of Riemann sums.
522
00:35:35,810 --> 00:35:44,140
Named after a mathematician
from the 1800s.
523
00:35:44,140 --> 00:36:01,150
So this is the general procedure
for definite integrals.
524
00:36:01,150 --> 00:36:04,890
We divide it up into pieces.
525
00:36:04,890 --> 00:36:07,430
And how do we do that?
526
00:36:07,430 --> 00:36:16,120
Well, so here's our
a and here's our b.
527
00:36:16,120 --> 00:36:19,600
And what we're going to do is
break it up into little pieces.
528
00:36:19,600 --> 00:36:22,620
And we're going to give
a name to the increment.
529
00:36:22,620 --> 00:36:28,510
And we're going to
call that delta x.
530
00:36:28,510 --> 00:36:30,380
So we divide up into these.
531
00:36:30,380 --> 00:36:32,110
So how many pieces are there?
532
00:36:32,110 --> 00:36:37,570
If there are n pieces,
then the general formula
533
00:36:37,570 --> 00:36:43,390
is always the delta x is
1/n times the total length.
534
00:36:43,390 --> 00:36:44,880
So it has to be (b-a) / n.
535
00:36:48,170 --> 00:36:50,550
We will always use
these equal increments,
536
00:36:50,550 --> 00:36:53,020
although you don't
absolutely have to do it.
537
00:36:53,020 --> 00:37:01,080
We will, for these Riemann sums.
538
00:37:01,080 --> 00:37:07,610
And now there's only
one bit of flexibility
539
00:37:07,610 --> 00:37:10,560
that we will allow ourselves.
540
00:37:10,560 --> 00:37:13,020
Which is this.
541
00:37:13,020 --> 00:37:29,720
We're going to pick any height
of f between-- in the interval,
542
00:37:29,720 --> 00:37:34,610
in each interval.
543
00:37:34,610 --> 00:37:39,020
So what that means is,
let me just show it
544
00:37:39,020 --> 00:37:43,870
to you on the picture here.
545
00:37:43,870 --> 00:37:47,200
Is, I just pick any
value in between,
546
00:37:47,200 --> 00:37:49,770
I'll call it c_i,
which is in there.
547
00:37:49,770 --> 00:37:51,420
And then I go up here.
548
00:37:51,420 --> 00:37:55,180
And I have the level,
which is f(c_i).
549
00:37:55,180 --> 00:37:58,730
And that's the
rectangle that I choose.
550
00:37:58,730 --> 00:38:01,530
In the case that
we did, we always
551
00:38:01,530 --> 00:38:03,730
chose the right-hand,
which turned out
552
00:38:03,730 --> 00:38:04,930
to be the largest one.
553
00:38:04,930 --> 00:38:07,800
But I could've chosen
some level in between.
554
00:38:07,800 --> 00:38:09,140
Or even the left-hand end.
555
00:38:09,140 --> 00:38:11,223
Which would have meant
that the staircase would've
556
00:38:11,223 --> 00:38:13,580
been quite a bit lower.
557
00:38:13,580 --> 00:38:17,950
So any of these staircases
will work perfectly well.
558
00:38:17,950 --> 00:38:25,650
So that means were picking
f(c_i), and that's a height.
559
00:38:25,650 --> 00:38:33,210
And now we're just going
to add them all up.
560
00:38:33,210 --> 00:38:35,680
And this is the sum of the
areas of the rectangles,
561
00:38:35,680 --> 00:38:37,350
because this is the height.
562
00:38:37,350 --> 00:38:43,700
And this is the base.
563
00:38:43,700 --> 00:38:46,160
This notation is
supposed to be, now,
564
00:38:46,160 --> 00:38:54,640
very suggestive of the
notation that Leibniz used.
565
00:38:54,640 --> 00:38:58,250
Which is that in the limit,
this becomes an integral from a
566
00:38:58,250 --> 00:39:01,010
to b of f(x) dx.
567
00:39:01,010 --> 00:39:05,230
And notice that the delta
x gets replaced by a dx.
568
00:39:05,230 --> 00:39:07,960
So this is what
happens in the limit.
569
00:39:07,960 --> 00:39:10,600
As the rectangles get thin.
570
00:39:10,600 --> 00:39:17,170
So that's as delta x goes to 0.
571
00:39:17,170 --> 00:39:21,780
And these gadgets are
called Riemann sums.
572
00:39:21,780 --> 00:39:29,740
This is called a Riemann sum.
573
00:39:29,740 --> 00:39:31,580
And we already worked
out an example.
574
00:39:31,580 --> 00:39:40,680
This very complicated guy was
an example of a Riemann sum.
575
00:39:40,680 --> 00:39:42,060
So that's a notation.
576
00:39:42,060 --> 00:39:44,182
And we'll give you
a chance to get
577
00:39:44,182 --> 00:39:45,640
used to it a little
more when we do
578
00:39:45,640 --> 00:39:51,680
some numerical work at the end.
579
00:39:51,680 --> 00:39:55,130
Now, the last
thing for today is,
580
00:39:55,130 --> 00:40:05,240
I promised you an example
which was not an area example.
581
00:40:05,240 --> 00:40:10,010
I want to be able to show
you that integrals can be
582
00:40:10,010 --> 00:40:21,630
interpreted as cumulative sums.
583
00:40:21,630 --> 00:40:36,480
Integrals as cumulative sums.
584
00:40:36,480 --> 00:40:39,020
So this is just an example.
585
00:40:39,020 --> 00:40:48,650
And, so here's the way it goes.
586
00:40:48,650 --> 00:40:51,440
So we're going to
consider a function f,
587
00:40:51,440 --> 00:40:55,460
we're going to consider a
variable t, which is time.
588
00:40:55,460 --> 00:40:59,340
In years.
589
00:40:59,340 --> 00:41:02,170
And we'll consider
a function f(t),
590
00:41:02,170 --> 00:41:06,560
which is in dollars per year.
591
00:41:06,560 --> 00:41:09,470
Right, this is a
financial example here.
592
00:41:09,470 --> 00:41:13,250
That's the unit here,
dollars per year.
593
00:41:13,250 --> 00:41:21,500
And this is going to
be a borrowing rate.
594
00:41:21,500 --> 00:41:24,000
Now, the reason why I
want to put units in here
595
00:41:24,000 --> 00:41:27,320
is to show you that
there's a good reason
596
00:41:27,320 --> 00:41:33,920
for this strange dx, which
we append on these integrals.
597
00:41:33,920 --> 00:41:34,890
This notation.
598
00:41:34,890 --> 00:41:36,520
It allows us to
change variables,
599
00:41:36,520 --> 00:41:39,020
it allows this to be
consistent with units.
600
00:41:39,020 --> 00:41:42,360
And allows us to develop
meaningful formulas, which are
601
00:41:42,360 --> 00:41:44,130
consistent across the board.
602
00:41:44,130 --> 00:41:46,020
And so I want to
emphasize the units
603
00:41:46,020 --> 00:41:51,620
in this when I set up this
modeling problem here.
604
00:41:51,620 --> 00:41:59,660
Now, you're borrowing
money, let's say, every day.
605
00:41:59,660 --> 00:42:06,160
So that means delta t = 1/365.
606
00:42:06,160 --> 00:42:08,450
That's almost 1 /
infinity, from the point
607
00:42:08,450 --> 00:42:11,700
of view of various purposes.
608
00:42:11,700 --> 00:42:15,180
So this is how much
you're borrowing.
609
00:42:15,180 --> 00:42:17,820
In each time increment
you're borrowing.
610
00:42:17,820 --> 00:42:23,990
And let's say that you borrow--
your rate varies over the year.
611
00:42:23,990 --> 00:42:27,140
I mean, sometimes you need more
money sometimes you need less.
612
00:42:27,140 --> 00:42:29,650
Certainly any business
would be that way.
613
00:42:29,650 --> 00:42:32,440
And so here you are,
you've got your money.
614
00:42:32,440 --> 00:42:35,070
And you're borrowing
but the rate is varying.
615
00:42:35,070 --> 00:42:36,960
And so how much did you borrow?
616
00:42:36,960 --> 00:42:51,230
Well, in day 45, which
corresponds to t is 45/365,
617
00:42:51,230 --> 00:42:55,210
you borrowed the
following amount.
618
00:42:55,210 --> 00:43:00,770
Here was your borrowing
rate times this quantity.
619
00:43:00,770 --> 00:43:02,900
So, dollars per year.
620
00:43:02,900 --> 00:43:05,500
And so this is, if
you like-- I want
621
00:43:05,500 --> 00:43:11,170
to emphasize the scaling
that comes about here.
622
00:43:11,170 --> 00:43:14,910
You have dollars per year.
623
00:43:14,910 --> 00:43:21,060
And this is this
number of years.
624
00:43:21,060 --> 00:43:23,180
So that comes out
to be in dollars.
625
00:43:23,180 --> 00:43:24,050
This final amount.
626
00:43:24,050 --> 00:43:25,883
This is the amount that
you actually borrow.
627
00:43:25,883 --> 00:43:30,250
So you borrow this amount.
628
00:43:30,250 --> 00:43:37,880
And now, if I want to
add up how much you get--
629
00:43:37,880 --> 00:43:39,920
you've borrowed in
the entire year.
630
00:43:39,920 --> 00:43:50,380
That's this sum. i = 1 to 365
of f of, well, it's (i / 365)
631
00:43:50,380 --> 00:43:50,880
delta t.
632
00:43:50,880 --> 00:43:53,220
Which I'll just leave
as delta t here.
633
00:43:53,220 --> 00:44:01,620
This is total amount borrowed.
634
00:44:01,620 --> 00:44:02,830
This is kind of a messy sum.
635
00:44:02,830 --> 00:44:05,570
In fact, your bank probably
will keep track of it
636
00:44:05,570 --> 00:44:06,830
and they know how to do that.
637
00:44:06,830 --> 00:44:09,679
But when we're modeling things
with strategies, you know,
638
00:44:09,679 --> 00:44:11,220
trading strategies,
of course, you're
639
00:44:11,220 --> 00:44:13,910
really some kind of
financial engineer
640
00:44:13,910 --> 00:44:17,000
and you want to cleverly
optimize how much you borrow.
641
00:44:17,000 --> 00:44:19,610
And how much you spend,
and how much you invest.
642
00:44:19,610 --> 00:44:23,900
This is going to be very,
very similar to the integral
643
00:44:23,900 --> 00:44:29,460
from 0 to 1 of f(t) dt.
644
00:44:29,460 --> 00:44:36,340
At the scale of 1/35,
it's probably-- 365,
645
00:44:36,340 --> 00:44:39,800
it's probably enough
for many purposes.
646
00:44:39,800 --> 00:44:44,942
Now, however,
there's another thing
647
00:44:44,942 --> 00:44:46,150
that you would want to model.
648
00:44:46,150 --> 00:44:47,670
Which is equally important.
649
00:44:47,670 --> 00:44:49,810
This is how much you
borrowed, but there's also
650
00:44:49,810 --> 00:44:53,380
how much you owe the back
at the end of the year.
651
00:44:53,380 --> 00:44:56,680
And the amount that you owe the
bank at the end of the year,
652
00:44:56,680 --> 00:44:58,680
I'm going to do
it in a fancy way.
653
00:44:58,680 --> 00:45:04,950
It's, the interest, we'll say,
is compounded continuously.
654
00:45:04,950 --> 00:45:07,780
So the interest rate,
if you start out with P
655
00:45:07,780 --> 00:45:20,120
as your principal, then after
time t you owe-- So borrow P,
656
00:45:20,120 --> 00:45:30,000
after time t, you owe P e^(rt),
where r is your interest rate.
657
00:45:30,000 --> 00:45:36,070
Say .05 per year.
658
00:45:36,070 --> 00:45:40,320
That would be an example
of an interest rate.
659
00:45:40,320 --> 00:45:45,740
And so, if you want to
understand how much money
660
00:45:45,740 --> 00:45:52,330
you actually owe at
the end of the year.
661
00:45:52,330 --> 00:45:54,380
At the end of the
year what you owe is,
662
00:45:54,380 --> 00:46:02,690
well, you borrowed
these amounts here.
663
00:46:02,690 --> 00:46:04,680
But now you owe more
at the end of the year.
664
00:46:04,680 --> 00:46:10,050
You owe e^r times the amount
of time left in the year.
665
00:46:10,050 --> 00:46:15,270
So the amount of time left
in the year is 1 - i / 365.
666
00:46:15,270 --> 00:46:18,900
Or 365 - i days left.
667
00:46:18,900 --> 00:46:26,600
So this is 1 - i / 365.
668
00:46:26,600 --> 00:46:34,770
And this is what you have to
add up, to see how much you owe.
669
00:46:34,770 --> 00:46:39,540
And that is essentially
the integral from 0 to 1.
670
00:46:39,540 --> 00:46:41,310
The delta t comes out.
671
00:46:41,310 --> 00:46:49,940
And you have here e^(r(1-t)),
so the t is replacing this i /
672
00:46:49,940 --> 00:46:54,880
365, f(t) dt.
673
00:46:54,880 --> 00:46:58,630
And so when you start computing
and thinking about what's
674
00:46:58,630 --> 00:47:04,170
the right strategy, you're faced
with integrals of this type.
675
00:47:04,170 --> 00:47:06,140
So that's just an example.
676
00:47:06,140 --> 00:47:08,930
And see you next time.
677
00:47:08,930 --> 00:47:10,640
Remember to think
about questions
678
00:47:10,640 --> 00:47:12,667
that you'll ask next time.