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PROFESSOR: One correction
from last time.
9
00:00:23,730 --> 00:00:28,240
Sorry to say, I forgot
a very important factor
10
00:00:28,240 --> 00:00:30,900
when I was telling you
what an average value is.
11
00:00:30,900 --> 00:00:33,770
If you don't put in
that factor, it's
12
00:00:33,770 --> 00:00:37,716
only half off on
the exam problem
13
00:00:37,716 --> 00:00:38,840
that will be given on this.
14
00:00:38,840 --> 00:00:43,130
So I would have gotten half off
for missing out on this factor,
15
00:00:43,130 --> 00:00:44,240
too.
16
00:00:44,240 --> 00:00:46,370
So remember you have
to divide by n here,
17
00:00:46,370 --> 00:00:49,150
certainly when you're
integrating over 0 to n,
18
00:00:49,150 --> 00:00:51,460
the Riemann sum is
the numerator here.
19
00:00:51,460 --> 00:00:52,940
And if I divide
by n on that side,
20
00:00:52,940 --> 00:00:55,250
I've got to divide by
n on the other side.
21
00:00:55,250 --> 00:00:57,770
This was meant to illustrate
this idea that we're
22
00:00:57,770 --> 00:01:00,540
dividing by the total here.
23
00:01:00,540 --> 00:01:04,170
And we are going to be talking
about average value in more
24
00:01:04,170 --> 00:01:05,300
detail.
25
00:01:05,300 --> 00:01:07,690
Not today, though.
26
00:01:07,690 --> 00:01:16,190
So this has to do
with average value.
27
00:01:16,190 --> 00:01:20,730
And we'll discuss it
in considerable detail
28
00:01:20,730 --> 00:01:27,340
in a couple of days, I guess.
29
00:01:27,340 --> 00:01:33,900
Now, today I want to continue.
30
00:01:33,900 --> 00:01:36,120
I didn't have time to
finish my discussion
31
00:01:36,120 --> 00:01:39,751
of the Fundamental
Theorem of Calculus 2.
32
00:01:39,751 --> 00:01:41,500
And anyway it's very
important to write it
33
00:01:41,500 --> 00:01:43,350
down on the board
twice, because you
34
00:01:43,350 --> 00:01:46,410
want to see it at least twice.
35
00:01:46,410 --> 00:01:48,370
And many more times as well.
36
00:01:48,370 --> 00:01:51,630
So let's just remind
you, the second version
37
00:01:51,630 --> 00:01:55,430
of the Fundamental Theorem of
Calculus says the following.
38
00:01:55,430 --> 00:02:01,000
It says that the
derivative of an integral
39
00:02:01,000 --> 00:02:03,630
gives you the
function back again.
40
00:02:03,630 --> 00:02:08,310
So here's the theorem.
41
00:02:08,310 --> 00:02:12,150
And the way I'd like
to use it today,
42
00:02:12,150 --> 00:02:14,500
I started this
discussion last time.
43
00:02:14,500 --> 00:02:16,660
But we didn't get into it.
44
00:02:16,660 --> 00:02:19,920
And this is something that's
on your problem set along
45
00:02:19,920 --> 00:02:23,190
with several other examples.
46
00:02:23,190 --> 00:02:30,770
Is that we can use this to
solve differential equations.
47
00:02:30,770 --> 00:02:34,620
And in particular,
for example, we
48
00:02:34,620 --> 00:02:43,540
can solve the equation y'
= 1 / x with this formula.
49
00:02:43,540 --> 00:02:47,930
Namely, using an integral.
50
00:02:47,930 --> 00:02:54,770
L(x) is the integral
from 1 to x of dt / t.
51
00:02:54,770 --> 00:03:00,410
The function f(t) is just 1 / t.
52
00:03:00,410 --> 00:03:07,070
Now, that formula can be
taken to be the starting place
53
00:03:07,070 --> 00:03:11,830
for the derivation of all the
properties of the logarithm
54
00:03:11,830 --> 00:03:12,510
function.
55
00:03:12,510 --> 00:03:14,090
So what we're going
to do right now
56
00:03:14,090 --> 00:03:22,750
is we're going to take
this to be the definition
57
00:03:22,750 --> 00:03:28,190
of the logarithm.
58
00:03:28,190 --> 00:03:31,870
And if we do that, then I
claim that we can read off
59
00:03:31,870 --> 00:03:33,910
the properties of the
logarithm just about as
60
00:03:33,910 --> 00:03:36,150
easily as we could before.
61
00:03:36,150 --> 00:03:38,730
And so I'll illustrate that now.
62
00:03:38,730 --> 00:03:41,470
And there are a
few other examples
63
00:03:41,470 --> 00:03:45,450
of this where somewhat more
unfamiliar functions come up.
64
00:03:45,450 --> 00:03:50,532
This one is one that in theory
we know something about.
65
00:03:50,532 --> 00:03:51,990
The first property
of this function
66
00:03:51,990 --> 00:03:53,950
is the one that's already given.
67
00:03:53,950 --> 00:03:59,580
Namely, its derivative is 1/x.
68
00:03:59,580 --> 00:04:02,150
And we get a lot of information
just out of the fact
69
00:04:02,150 --> 00:04:04,209
that its derivative is 1/x.
70
00:04:04,209 --> 00:04:05,750
The other thing that
we need in order
71
00:04:05,750 --> 00:04:08,750
to nail down the function,
besides its derivative,
72
00:04:08,750 --> 00:04:10,280
is one value of the function.
73
00:04:10,280 --> 00:04:14,670
Because it's really not
specified by this equation,
74
00:04:14,670 --> 00:04:17,490
only specified up to a
constant by this equation.
75
00:04:17,490 --> 00:04:20,020
But we nail down that
constant when we evaluate it
76
00:04:20,020 --> 00:04:22,640
at this one place, L(1).
77
00:04:22,640 --> 00:04:24,640
And there we're getting
the integral from 1 to 1
78
00:04:24,640 --> 00:04:28,430
of dt / t, which is 0.
79
00:04:28,430 --> 00:04:30,860
And that's the case with all
these definite integrals.
80
00:04:30,860 --> 00:04:32,900
If you evaluate them at
their starting places,
81
00:04:32,900 --> 00:04:34,460
the value will be 0.
82
00:04:34,460 --> 00:04:36,350
And together these
two properties
83
00:04:36,350 --> 00:04:42,800
specify this function
L(x) uniquely.
84
00:04:42,800 --> 00:04:46,200
Now, the next step
is to try to think
85
00:04:46,200 --> 00:04:47,860
about what its properties are.
86
00:04:47,860 --> 00:04:50,980
And the first approach
to that, and this
87
00:04:50,980 --> 00:04:52,660
is the approach
that we always take,
88
00:04:52,660 --> 00:04:55,870
is to maybe graph the function,
to get a feeling for it.
89
00:04:55,870 --> 00:04:57,970
And so I'm going to take
the second derivative.
90
00:04:57,970 --> 00:05:00,440
Now, notice that when
you have a function which
91
00:05:00,440 --> 00:05:02,460
is given as an integral,
its first derivative
92
00:05:02,460 --> 00:05:04,752
is really easy to compute.
93
00:05:04,752 --> 00:05:06,460
And then its second
derivative, well, you
94
00:05:06,460 --> 00:05:08,290
have to differentiate
whatever you get.
95
00:05:08,290 --> 00:05:09,720
So it may or may not be easy.
96
00:05:09,720 --> 00:05:12,230
But anyway, it's a
lot harder in the case
97
00:05:12,230 --> 00:05:15,150
when I start with a function to
get to the second derivative.
98
00:05:15,150 --> 00:05:19,070
Here it's relatively easy.
99
00:05:19,070 --> 00:05:21,650
And these are the properties
that I'm going to use.
100
00:05:21,650 --> 00:05:26,410
I won't really use very much
more about it than that.
101
00:05:26,410 --> 00:05:29,020
And qualitatively,
the conclusions
102
00:05:29,020 --> 00:05:31,760
that we can draw from
this are, first of all,
103
00:05:31,760 --> 00:05:34,100
from this, for example
we see that this thing is
104
00:05:34,100 --> 00:05:38,970
concave down every place.
105
00:05:38,970 --> 00:05:40,920
And then to get
started with the graph,
106
00:05:40,920 --> 00:05:45,700
since I see I have a value
here, which is L(1) = 0,
107
00:05:45,700 --> 00:05:48,560
I'm going to throw in
the value of the slope.
108
00:05:48,560 --> 00:05:50,980
So L'(1), which I
know is 1 over 1,
109
00:05:50,980 --> 00:05:55,180
that's reading off from this
equation here, so that's 1.
110
00:05:55,180 --> 00:05:59,200
And now I'm ready to sketch
at least a part of the curve.
111
00:05:59,200 --> 00:06:07,680
So here's a sketch of the graph.
112
00:06:07,680 --> 00:06:13,000
Here's the point (1, 0),
that is, x = 1, y = 0.
113
00:06:13,000 --> 00:06:17,420
And the tangent line,
I know, has slope 1.
114
00:06:17,420 --> 00:06:20,540
And the curve is concave down.
115
00:06:20,540 --> 00:06:27,530
So it's going to look
something like this.
116
00:06:27,530 --> 00:06:31,230
Incidentally, it's
also increasing.
117
00:06:31,230 --> 00:06:34,359
And that's an
important property,
118
00:06:34,359 --> 00:06:35,400
it's strictly increasing.
119
00:06:35,400 --> 00:06:39,570
That's because
L'(x) is positive.
120
00:06:39,570 --> 00:06:44,220
And so, we can get from this the
following important definition.
121
00:06:44,220 --> 00:06:46,610
Which, again, is working
backwards from this definition.
122
00:06:46,610 --> 00:06:48,880
We can get to where
we started with a log
123
00:06:48,880 --> 00:06:50,620
in our previous discussion.
124
00:06:50,620 --> 00:06:57,870
Namely, if I take the
level here, which is y = 1,
125
00:06:57,870 --> 00:06:59,730
then that crosses
the axis someplace.
126
00:06:59,730 --> 00:07:04,180
And this point is what
we're going to define as e.
127
00:07:04,180 --> 00:07:11,970
So the definition
of e is that it's
128
00:07:11,970 --> 00:07:20,220
the value such that L(e) = 1.
129
00:07:20,220 --> 00:07:22,800
And again, the fact that
there's exactly one such place
130
00:07:22,800 --> 00:07:25,370
just comes from the fact
that this L' is positive,
131
00:07:25,370 --> 00:07:29,340
so that L is increasing.
132
00:07:29,340 --> 00:07:32,910
Now, there's just one
other feature of this graph
133
00:07:32,910 --> 00:07:35,882
that I'm going to
emphasize to you.
134
00:07:35,882 --> 00:07:38,090
There's one other thing
which I'm not going to check,
135
00:07:38,090 --> 00:07:40,060
which you would
ordinarily do with graphs.
136
00:07:40,060 --> 00:07:42,143
Once it's increasing there
are no critical points,
137
00:07:42,143 --> 00:07:44,230
so the only other interesting
thing is the ends.
138
00:07:44,230 --> 00:07:46,400
And it turns out that the
limit as you go down to 0
139
00:07:46,400 --> 00:07:47,150
is minus infinity.
140
00:07:47,150 --> 00:07:49,840
As you go over to the right
here it's plus infinity.
141
00:07:49,840 --> 00:07:53,340
It does get arbitrarily
high; it doesn't level off.
142
00:07:53,340 --> 00:07:55,820
But I'm not going to
discuss that here.
143
00:07:55,820 --> 00:07:57,400
Instead, I'm going
to just remark
144
00:07:57,400 --> 00:08:01,920
on one qualitative feature of
the graph, which is this remark
145
00:08:01,920 --> 00:08:06,850
that the part which is to
the left of 1 is below 0.
146
00:08:06,850 --> 00:08:17,292
So I just want to remark, why
is L(x) negative for x < 1.
147
00:08:17,292 --> 00:08:19,750
Maybe I don't have room for
that, so I'll just put in here:
148
00:08:19,750 --> 00:08:23,550
x < 1.
149
00:08:23,550 --> 00:08:25,320
I want to give you two reasons.
150
00:08:25,320 --> 00:08:27,980
Again, we're only working from
very first principles here.
151
00:08:27,980 --> 00:08:33,500
Just that-- the property
that L' = 1/x, and L(1) = 0.
152
00:08:33,500 --> 00:08:39,650
So our first reason is
that, well, I just said it.
153
00:08:39,650 --> 00:08:41,430
L(1) = 0.
154
00:08:41,430 --> 00:08:46,910
And L is increasing.
155
00:08:46,910 --> 00:08:49,920
And if you read that backwards,
if it gets up to 0 here,
156
00:08:49,920 --> 00:08:54,160
it must have been
negative before 0.
157
00:08:54,160 --> 00:08:57,960
So this is one way of seeing
that L(x) is negative.
158
00:08:57,960 --> 00:09:02,410
There's a second way of seeing
it, which is equally important.
159
00:09:02,410 --> 00:09:07,470
And it has to do with just
manipulation of integrals.
160
00:09:07,470 --> 00:09:11,430
Here I'm going to start out
with L(x), and its definition.
161
00:09:11,430 --> 00:09:15,730
Which is the integral
from 1 to x, dt / t.
162
00:09:15,730 --> 00:09:19,250
And now I'm going to reverse
the order of integration.
163
00:09:19,250 --> 00:09:21,500
This is the same, by our
definition of our properties
164
00:09:21,500 --> 00:09:24,010
of integrals, as the
integral from x to 1
165
00:09:24,010 --> 00:09:30,010
with a minus sign dt / t.
166
00:09:30,010 --> 00:09:34,140
Now, I can tell that this
quantity is negative.
167
00:09:34,140 --> 00:09:36,400
And the reason
that I can tell is
168
00:09:36,400 --> 00:09:41,840
that this chunk of it
here, this piece of it,
169
00:09:41,840 --> 00:09:44,010
is a positive number.
170
00:09:44,010 --> 00:09:46,290
This part is positive.
171
00:09:46,290 --> 00:09:51,290
And this part is
positive because x < 1.
172
00:09:51,290 --> 00:09:53,494
So the lower limit is
less than the upper limit,
173
00:09:53,494 --> 00:09:55,785
and so this is interpreted
- the thing in the green box
174
00:09:55,785 --> 00:09:58,070
is interpreted - as an area.
175
00:09:58,070 --> 00:09:58,970
It's an area.
176
00:09:58,970 --> 00:10:02,690
And so negative a positive
quantity is negative,
177
00:10:02,690 --> 00:10:08,280
minus a positive
quantity's negative.
178
00:10:08,280 --> 00:10:13,870
So both of these work perfectly
well as interpretations.
179
00:10:13,870 --> 00:10:16,010
And it's just to
illustrate what we can do.
180
00:10:16,010 --> 00:10:18,280
Now, there's one
more manipulation
181
00:10:18,280 --> 00:10:23,380
of integrals that gives us the
fanciest property of the log.
182
00:10:23,380 --> 00:10:26,410
And that's the last one
that I'm going to do.
183
00:10:26,410 --> 00:10:29,450
And you have a similar
thing on your homework.
184
00:10:29,450 --> 00:10:32,090
So I'm going to
prove that-- This
185
00:10:32,090 --> 00:10:34,970
is, as I say, the fanciest
property of the log.
186
00:10:34,970 --> 00:10:40,200
On your homework, by the way,
you're going to check that
187
00:10:40,200 --> 00:10:43,160
L(1/x) = -L(x).
188
00:10:43,160 --> 00:10:46,680
189
00:10:46,680 --> 00:10:52,070
But we'll do this one.
190
00:10:52,070 --> 00:10:56,430
The idea is just to plug in the
formula and see what it gives.
191
00:10:56,430 --> 00:11:02,830
On the left-hand side,
I have 1 to ab, dt / t.
192
00:11:02,830 --> 00:11:06,040
That's L(ab).
193
00:11:06,040 --> 00:11:08,900
And then that's certainly
equal to the left-hand side.
194
00:11:08,900 --> 00:11:12,650
And then I'm going to now
split this into two pieces.
195
00:11:12,650 --> 00:11:14,600
Again, this is a
property of integrals.
196
00:11:14,600 --> 00:11:18,790
That if you have an integral
from one place to another,
197
00:11:18,790 --> 00:11:20,620
you can break it up into pieces.
198
00:11:20,620 --> 00:11:26,540
So I'm going to start
at 1 but then go to a.
199
00:11:26,540 --> 00:11:33,950
And then I'm going to
continue from a to ab.
200
00:11:33,950 --> 00:11:36,130
So this is the
question that we have.
201
00:11:36,130 --> 00:11:38,010
We haven't proved this.
202
00:11:38,010 --> 00:11:41,050
Well, this one is actually true.
203
00:11:41,050 --> 00:11:42,700
If we want this to
be true, we know
204
00:11:42,700 --> 00:11:44,900
by definition L(ab) is this.
205
00:11:44,900 --> 00:11:48,460
We know, we can see
it, that L(a) is this.
206
00:11:48,460 --> 00:11:52,320
So the question that
this boils down to
207
00:11:52,320 --> 00:11:54,960
is, we want to know that
these two things are equal.
208
00:11:54,960 --> 00:12:01,320
We want to know that L(b) is
that other integral there.
209
00:12:01,320 --> 00:12:04,774
So let's check it.
210
00:12:04,774 --> 00:12:06,190
I'm going to rewrite
the integral.
211
00:12:06,190 --> 00:12:09,170
It's the integral from--
sorry, from lower limit a
212
00:12:09,170 --> 00:12:13,940
to upper limit ab of dt / t.
213
00:12:13,940 --> 00:12:16,640
And now, again, to
illustrate properties
214
00:12:16,640 --> 00:12:18,170
of integrals, the
key property here
215
00:12:18,170 --> 00:12:23,740
that we're going to have to
use is change of variables.
216
00:12:23,740 --> 00:12:26,680
This is a kind of a scaled
integral where everything
217
00:12:26,680 --> 00:12:28,710
is multiplied by a
factor of a from what
218
00:12:28,710 --> 00:12:32,010
we want to get to
this L(b) quantity.
219
00:12:32,010 --> 00:12:38,240
And so this suggests that
we write down t = au.
220
00:12:38,240 --> 00:12:40,460
That's going to be our trick.
221
00:12:40,460 --> 00:12:43,440
And if I use that new variable
u, then the change in t,
222
00:12:43,440 --> 00:12:50,440
dt, is a du.
223
00:12:50,440 --> 00:12:53,950
And as a result, I can write
this as equal to an integral
224
00:12:53,950 --> 00:12:58,010
from, let's see, dt = a du.
225
00:12:58,010 --> 00:13:00,080
And t = au.
226
00:13:00,080 --> 00:13:05,740
So I've now substituted
in for the integrand.
227
00:13:05,740 --> 00:13:08,720
But on top of this,
with definite integrals,
228
00:13:08,720 --> 00:13:13,080
we also have to
check the limits.
229
00:13:13,080 --> 00:13:17,140
And the limits work
out as follows.
230
00:13:17,140 --> 00:13:20,180
When t = a, that's
the lower limit.
231
00:13:20,180 --> 00:13:22,460
Let's just take a look. t = au.
232
00:13:22,460 --> 00:13:26,600
So that means that
u is equal to, what?
233
00:13:26,600 --> 00:13:28,710
It's 1.
234
00:13:28,710 --> 00:13:31,880
Because a * 1 = a.
235
00:13:31,880 --> 00:13:34,040
So if t = a, this
is if and only if.
236
00:13:34,040 --> 00:13:36,610
So this lower limit, which
really in disguise was where
237
00:13:36,610 --> 00:13:45,400
t = a, becomes where u = 1.
238
00:13:45,400 --> 00:13:53,920
And similarly,
when t = ab, u = b.
239
00:13:53,920 --> 00:13:56,690
So the upper limit here is b.
240
00:13:56,690 --> 00:14:00,460
And now, if you
notice, we're just
241
00:14:00,460 --> 00:14:02,920
going to cancel these
two factors here.
242
00:14:02,920 --> 00:14:06,810
And now we recognize that
this is just the same
243
00:14:06,810 --> 00:14:09,940
as the definition of L(b).
244
00:14:09,940 --> 00:14:13,490
Because L(x) is over
here in the box.
245
00:14:13,490 --> 00:14:16,270
And the fact that I use the
letter t there is irrelevant;
246
00:14:16,270 --> 00:14:18,020
it works equally well
with the letter u.
247
00:14:18,020 --> 00:14:22,430
So this is just L(b).
248
00:14:22,430 --> 00:14:33,160
Which is what we wanted to show.
249
00:14:33,160 --> 00:14:34,890
So that's an
example, and you have
250
00:14:34,890 --> 00:14:45,380
one in your homework,
which is a little similar.
251
00:14:45,380 --> 00:14:49,300
Now, the last example, that I'm
going to discuss of this type,
252
00:14:49,300 --> 00:14:51,390
I already mentioned last time.
253
00:14:51,390 --> 00:14:54,430
Which is the function F(x),
which is the integral from 0
254
00:14:54,430 --> 00:14:58,150
to x of e^(-t^2) dt.
255
00:14:58,150 --> 00:15:04,130
This one is even more exotic
because unlike the logarithm
256
00:15:04,130 --> 00:15:06,030
it's a new function.
257
00:15:06,030 --> 00:15:09,320
It really is not any
function that you
258
00:15:09,320 --> 00:15:13,870
can express in terms of the
functions that we know already.
259
00:15:13,870 --> 00:15:18,340
And the approach, always,
to these new functions
260
00:15:18,340 --> 00:15:22,000
is to think of what
their properties are.
261
00:15:22,000 --> 00:15:24,380
And the way we think
of functions in order
262
00:15:24,380 --> 00:15:26,570
to understand them is
to maybe sketch them.
263
00:15:26,570 --> 00:15:29,830
And so I'm going to do exactly
the same thing I did over here.
264
00:15:29,830 --> 00:15:31,630
So, what is it that I
can get out of this?
265
00:15:31,630 --> 00:15:35,105
Well, immediately I can figure
out what the derivative is.
266
00:15:35,105 --> 00:15:38,000
I read it off from the
fundamental theorem.
267
00:15:38,000 --> 00:15:41,610
It's this.
268
00:15:41,610 --> 00:15:45,470
I also can figure out the
value at the starting place.
269
00:15:45,470 --> 00:15:48,440
In this case, the
starting place is 0.
270
00:15:48,440 --> 00:15:53,990
And the value is 0.
271
00:15:53,990 --> 00:15:57,600
And I should check the second
derivative, which is also not
272
00:15:57,600 --> 00:15:59,145
so difficult to compute.
273
00:15:59,145 --> 00:16:03,400
The second derivative
is -2x e^(-x^2).
274
00:16:03,400 --> 00:16:06,070
275
00:16:06,070 --> 00:16:10,650
And so now I can see that
this function is increasing,
276
00:16:10,650 --> 00:16:12,890
because this
derivative is positive,
277
00:16:12,890 --> 00:16:14,670
it's always increasing.
278
00:16:14,670 --> 00:16:17,830
And it's going to be concave
down when x is positive
279
00:16:17,830 --> 00:16:20,540
and concave up
when x is negative.
280
00:16:20,540 --> 00:16:24,900
Because there's a minus sign
here, so the sign is negative.
281
00:16:24,900 --> 00:16:30,250
This is less than 0 when x is
positive and greater than 0
282
00:16:30,250 --> 00:16:36,800
when x is negative.
283
00:16:36,800 --> 00:16:40,160
And maybe to get started
I'll remind you F(0) is 0.
284
00:16:40,160 --> 00:16:46,750
It's also true that F'(0)-- that
just comes right out of this,
285
00:16:46,750 --> 00:16:52,450
F'(0) = e^(-0^2), which is 1.
286
00:16:52,450 --> 00:16:55,420
That means the tangent
line again has slope 1.
287
00:16:55,420 --> 00:16:57,090
We do this a lot with functions.
288
00:16:57,090 --> 00:17:00,500
We normalize them so that the
slopes of their tangent lines
289
00:17:00,500 --> 00:17:03,600
are 1 at convenient spots.
290
00:17:03,600 --> 00:17:06,390
So here's the tangent
line of slope 1.
291
00:17:06,390 --> 00:17:10,820
We know this thing is
concave down to the right
292
00:17:10,820 --> 00:17:14,890
and concave up to the left.
293
00:17:14,890 --> 00:17:17,570
And so it's going to
look something like this.
294
00:17:17,570 --> 00:17:20,860
With an inflection point.
295
00:17:20,860 --> 00:17:26,290
Right?
296
00:17:26,290 --> 00:17:32,230
Now, I want to say one
more-- make one more remark
297
00:17:32,230 --> 00:17:33,860
about this function,
or maybe two more
298
00:17:33,860 --> 00:17:36,050
remarks about this
function, before we go on.
299
00:17:36,050 --> 00:17:39,230
Really, you want to know this
graph as well as possible.
300
00:17:39,230 --> 00:17:42,220
And so there are just
a couple more features.
301
00:17:42,220 --> 00:17:44,640
And one is enormously
helpful because it
302
00:17:44,640 --> 00:17:47,780
cuts in half all of
the work that you have.
303
00:17:47,780 --> 00:17:49,750
and that is the
property that turns out
304
00:17:49,750 --> 00:17:51,800
that this function is odd.
305
00:17:51,800 --> 00:17:56,860
Namely, - F(-x) = -F(x).
306
00:17:56,860 --> 00:18:01,050
That's what's known
as an odd function.
307
00:18:01,050 --> 00:18:06,840
Now, the reason why it's odd
is that it's the antiderivative
308
00:18:06,840 --> 00:18:08,500
of something that's even.
309
00:18:08,500 --> 00:18:10,430
This function in here is even.
310
00:18:10,430 --> 00:18:14,860
And we nailed it down
so that it was 0 at 0.
311
00:18:14,860 --> 00:18:17,690
Another way of interpreting
that, and let me show it to you
312
00:18:17,690 --> 00:18:20,560
underneath, is the following.
313
00:18:20,560 --> 00:18:24,440
When we look at its derivative,
its derivative, course,
314
00:18:24,440 --> 00:18:25,370
is the function e^x.
315
00:18:25,370 --> 00:18:28,190
316
00:18:28,190 --> 00:18:30,040
Sorry, e^(-x^2).
317
00:18:30,040 --> 00:18:37,430
So that's this shape here.
318
00:18:37,430 --> 00:18:41,530
And you can see the slope is
0, but-- fairly close to 0,
319
00:18:41,530 --> 00:18:42,610
but positive along here.
320
00:18:42,610 --> 00:18:44,560
It's getting, this is
its steepest point.
321
00:18:44,560 --> 00:18:45,965
This is the highest point here.
322
00:18:45,965 --> 00:18:47,340
And then it's
leveling off again.
323
00:18:47,340 --> 00:18:50,670
The slope is going
down, always positive.
324
00:18:50,670 --> 00:18:56,860
This is the graph
of F' = e^(-x^2).
325
00:18:56,860 --> 00:19:01,670
Now, the interpretation of
the function that's up above
326
00:19:01,670 --> 00:19:08,510
is that the value here
is the area from 0 to x.
327
00:19:08,510 --> 00:19:12,690
So this is area F(x).
328
00:19:12,690 --> 00:19:16,570
Maybe I'll color it in,
decorate it a little bit.
329
00:19:16,570 --> 00:19:25,120
So this area here is F(x).
330
00:19:25,120 --> 00:19:28,840
Now, I want to show
you this odd property,
331
00:19:28,840 --> 00:19:30,900
by using this symmetry.
332
00:19:30,900 --> 00:19:34,590
The graph here is even,
so in other words,
333
00:19:34,590 --> 00:19:39,290
what's back here is exactly
the same as what's forward.
334
00:19:39,290 --> 00:19:42,580
But now there's a reversal.
335
00:19:42,580 --> 00:19:44,650
Because we're keeping
track of the area
336
00:19:44,650 --> 00:19:46,330
starting from 0 going forward.
337
00:19:46,330 --> 00:19:47,090
That's positive.
338
00:19:47,090 --> 00:19:49,940
If we go backwards,
it's counted negatively.
339
00:19:49,940 --> 00:19:51,810
So if we went
backwards to -x, we'd
340
00:19:51,810 --> 00:19:54,970
get exactly the same as
that green patch over there.
341
00:19:54,970 --> 00:19:56,810
We'd get a red patch over here.
342
00:19:56,810 --> 00:20:01,280
But it would be
counted negatively.
343
00:20:01,280 --> 00:20:04,210
And that's the
property that it's odd.
344
00:20:04,210 --> 00:20:07,150
You can also check this
by properties of integrals
345
00:20:07,150 --> 00:20:08,920
directly.
346
00:20:08,920 --> 00:20:16,190
That would be just
like this process here.
347
00:20:16,190 --> 00:20:20,280
So it's completely analogous
to checking this formula
348
00:20:20,280 --> 00:20:25,410
over there.
349
00:20:25,410 --> 00:20:29,670
So that's one of the comments
I wanted to make about this.
350
00:20:29,670 --> 00:20:31,900
And why does this
save us a lot of time,
351
00:20:31,900 --> 00:20:33,100
if we know this is odd?
352
00:20:33,100 --> 00:20:35,510
Well, we know that the
shape of this branch
353
00:20:35,510 --> 00:20:37,840
is exactly the reverse,
or the reflection,
354
00:20:37,840 --> 00:20:39,940
if you like, of the
shape of this one.
355
00:20:39,940 --> 00:20:41,950
What we want to do is
flip it under the axis
356
00:20:41,950 --> 00:20:44,960
and then reflect
it over that way.
357
00:20:44,960 --> 00:20:53,590
And that's the symmetry
property of the graph of F(x).
358
00:20:53,590 --> 00:20:56,930
Now, the last property
that I want to mention
359
00:20:56,930 --> 00:21:00,750
is what's happening
with the ends.
360
00:21:00,750 --> 00:21:03,570
And at the end
there's an asymptote,
361
00:21:03,570 --> 00:21:05,820
there's a limit here.
362
00:21:05,820 --> 00:21:10,532
So this is an asymptote.
363
00:21:10,532 --> 00:21:11,990
And the same thing
down here, which
364
00:21:11,990 --> 00:21:13,781
will be exactly because
of the odd feature,
365
00:21:13,781 --> 00:21:15,790
this'll be exactly negative.
366
00:21:15,790 --> 00:21:19,570
The opposite value over here.
367
00:21:19,570 --> 00:21:23,570
And you might ask yourself,
what level is this, exactly.
368
00:21:23,570 --> 00:21:26,850
Now, that level turns out to
be a very important quantity.
369
00:21:26,850 --> 00:21:28,910
It's interpreted
down here as the area
370
00:21:28,910 --> 00:21:31,620
under this whole
infinite stretch.
371
00:21:31,620 --> 00:21:36,670
It's all the way
out to infinity.
372
00:21:36,670 --> 00:21:38,960
So, let's see.
373
00:21:38,960 --> 00:21:48,740
What do you think it is?
374
00:21:48,740 --> 00:21:49,750
You're all clueless.
375
00:21:49,750 --> 00:21:52,460
Well, maybe not all of you,
you're just afraid to say.
376
00:21:52,460 --> 00:21:54,140
So it's obvious.
377
00:21:54,140 --> 00:21:57,040
It's the square root of pi/2.
378
00:21:57,040 --> 00:22:00,170
That was right on the tip
of your tongue, wasn't it?
379
00:22:00,170 --> 00:22:01,610
STUDENT: Ah, yes.
380
00:22:01,610 --> 00:22:04,100
PROFESSOR: Right, so this
is actually very un-obvious,
381
00:22:04,100 --> 00:22:06,290
but it's a very
important quantity.
382
00:22:06,290 --> 00:22:08,750
And it's an amazing
fact that this thing
383
00:22:08,750 --> 00:22:10,720
approaches this number.
384
00:22:10,720 --> 00:22:18,060
And it's something that people
worried about for many years
385
00:22:18,060 --> 00:22:22,860
before actually nailing down.
386
00:22:22,860 --> 00:22:24,680
And so what I just
claimed here is
387
00:22:24,680 --> 00:22:28,540
that the limit as x
approaches infinity of F(x)
388
00:22:28,540 --> 00:22:33,300
is equal to the square
root of pi over 2.
389
00:22:33,300 --> 00:22:35,340
And similarly, if you
do it to minus infinity,
390
00:22:35,340 --> 00:22:38,370
you'll get minus square
root of pi over 2.
391
00:22:38,370 --> 00:22:42,090
And for this reason, people
introduced a new function
392
00:22:42,090 --> 00:22:44,010
because they like the number 1.
393
00:22:44,010 --> 00:22:49,740
This function is erf,
short for error function.
394
00:22:49,740 --> 00:22:54,470
And it's 2 over the square root
of pi times the integral from 0
395
00:22:54,470 --> 00:22:57,630
to x, e^(-t^2) dt.
396
00:22:57,630 --> 00:22:59,650
In other words, it's
just our original,
397
00:22:59,650 --> 00:23:03,110
our previous function
multiplied by 2
398
00:23:03,110 --> 00:23:08,240
over the square root of pi.
399
00:23:08,240 --> 00:23:10,880
And that's the function which
gets tabulated quite a lot.
400
00:23:10,880 --> 00:23:13,260
You'll see it on the
internet everywhere,
401
00:23:13,260 --> 00:23:15,856
and it's a very
important function.
402
00:23:15,856 --> 00:23:17,730
There are other
normalizations that are used,
403
00:23:17,730 --> 00:23:20,640
and the discussions of
the other normalizations
404
00:23:20,640 --> 00:23:23,500
are in your problems.
405
00:23:23,500 --> 00:23:27,540
This is one of them, and another
one is in your exercises.
406
00:23:27,540 --> 00:23:31,130
The standard normal
distribution.
407
00:23:31,130 --> 00:23:33,200
There are tons of
functions like this,
408
00:23:33,200 --> 00:23:35,920
which are new functions
that we can get at once we
409
00:23:35,920 --> 00:23:37,650
have the tool of integrals.
410
00:23:37,650 --> 00:23:40,460
And I'll write down just
one or two more, just so
411
00:23:40,460 --> 00:23:42,760
that you'll see the variety.
412
00:23:42,760 --> 00:23:49,080
Here's one which is
called a Fresnel integral.
413
00:23:49,080 --> 00:23:51,420
On your problem set
next week, we'll
414
00:23:51,420 --> 00:23:57,090
do the other Fresnel integral,
we'll look at this one.
415
00:23:57,090 --> 00:24:01,990
These functions cannot be
expressed in elementary terms.
416
00:24:01,990 --> 00:24:11,890
The one on your homework
for this week was this one.
417
00:24:11,890 --> 00:24:14,980
This one comes up
in Fourier analysis.
418
00:24:14,980 --> 00:24:19,910
And I'm going to just tell you
maybe one more such function.
419
00:24:19,910 --> 00:24:22,220
There's a function
which is called
420
00:24:22,220 --> 00:24:30,220
Li(x), logarithmic integral
of x, which is this guy.
421
00:24:30,220 --> 00:24:33,640
The reciprocal of the
logarithm, the natural log.
422
00:24:33,640 --> 00:24:36,640
And the significance
of this one is
423
00:24:36,640 --> 00:24:45,270
that Li(x) is approximately
equal to the number of primes
424
00:24:45,270 --> 00:24:49,680
less than x.
425
00:24:49,680 --> 00:24:54,430
And, in fact, if you can make
this as precise as possible,
426
00:24:54,430 --> 00:24:59,220
you'll be famous for millennia,
because this is known
427
00:24:59,220 --> 00:25:01,830
as the Riemann hypothesis.
428
00:25:01,830 --> 00:25:05,370
Exactly how closely this
approximation occurs.
429
00:25:05,370 --> 00:25:08,470
But it's a hard
problem, and already
430
00:25:08,470 --> 00:25:10,500
a century ago the
prime number theorem,
431
00:25:10,500 --> 00:25:14,890
which established this
connection was extremely
432
00:25:14,890 --> 00:25:18,450
important to progress in math.
433
00:25:18,450 --> 00:25:19,340
Yeah, question.
434
00:25:19,340 --> 00:25:21,350
STUDENT: [INAUDIBLE]
435
00:25:21,350 --> 00:25:23,700
PROFESSOR: Is this stuff
you're supposed to understand.
436
00:25:23,700 --> 00:25:24,840
That's a good question.
437
00:25:24,840 --> 00:25:26,300
I love that question.
438
00:25:26,300 --> 00:25:31,570
The answer is, this is, so we
launched off into something
439
00:25:31,570 --> 00:25:32,080
here.
440
00:25:32,080 --> 00:25:34,310
And let me just
explain it to you.
441
00:25:34,310 --> 00:25:36,560
I'm going to be talking
a fair amount more
442
00:25:36,560 --> 00:25:41,570
about this particular function,
because it's associated
443
00:25:41,570 --> 00:25:43,240
to the normal distribution.
444
00:25:43,240 --> 00:25:45,240
And I'm going to let you
get familiar with it.
445
00:25:45,240 --> 00:25:47,810
What I'm doing here
is purely cultural.
446
00:25:47,810 --> 00:25:51,500
Well, after this panel, what
I'm doing is purely cultural.
447
00:25:51,500 --> 00:25:53,780
Just saying there's
a lot of other beasts
448
00:25:53,780 --> 00:25:55,810
out there in the world.
449
00:25:55,810 --> 00:25:57,930
And one of them
is called C of x--
450
00:25:57,930 --> 00:26:01,460
So we'll have a just a very
passing familiarity with one
451
00:26:01,460 --> 00:26:02,920
or two of these functions.
452
00:26:02,920 --> 00:26:05,750
But there are literally
dozens and dozens of them.
453
00:26:05,750 --> 00:26:08,980
The only thing that you'll
need to do with such functions
454
00:26:08,980 --> 00:26:12,350
is things like understanding
the derivative,
455
00:26:12,350 --> 00:26:14,960
the second derivative,
and tracking
456
00:26:14,960 --> 00:26:16,140
what the function does.
457
00:26:16,140 --> 00:26:18,950
Sketching the same way you
did with any other tool.
458
00:26:18,950 --> 00:26:22,870
So we're going to do this type
of thing with these functions.
459
00:26:22,870 --> 00:26:25,659
And I'll have to
lead you through.
460
00:26:25,659 --> 00:26:27,450
If I wanted to ask you
a question about one
461
00:26:27,450 --> 00:26:30,070
of these functions,
I have to tell you
462
00:26:30,070 --> 00:26:32,470
exactly what I'm aiming for.
463
00:26:32,470 --> 00:26:35,747
Yeah, another question.
464
00:26:35,747 --> 00:26:36,580
STUDENT: [INAUDIBLE]
465
00:26:36,580 --> 00:26:37,538
PROFESSOR: Yeah, I did.
466
00:26:37,538 --> 00:26:43,520
I called these guys
Fresnel integrals.
467
00:26:43,520 --> 00:26:47,460
The guy's name is Fresnel.
468
00:26:47,460 --> 00:26:49,180
It's just named after a person.
469
00:26:49,180 --> 00:26:51,980
But, and this one, Li
is logarithmic integral,
470
00:26:51,980 --> 00:26:53,250
it's not named after a person.
471
00:26:53,250 --> 00:26:56,820
Logarithm is not
somebody's name.
472
00:26:56,820 --> 00:27:01,510
So look, in fact this
will be mentioned also
473
00:27:01,510 --> 00:27:03,284
on a problem set,
but I don't expect
474
00:27:03,284 --> 00:27:04,450
you to remember these names.
475
00:27:04,450 --> 00:27:05,940
In particular,
that you definitely
476
00:27:05,940 --> 00:27:07,600
don't want to try to remember.
477
00:27:07,600 --> 00:27:08,730
Yes, another question.
478
00:27:08,730 --> 00:27:10,020
STUDENT: [INAUDIBLE]
479
00:27:10,020 --> 00:27:15,240
PROFESSOR: The question is,
will we prove this limit.
480
00:27:15,240 --> 00:27:17,470
And the answer is
yes, if we have time.
481
00:27:17,470 --> 00:27:21,482
It'll be in about a week or so.
482
00:27:21,482 --> 00:27:22,690
We're not going to do it now.
483
00:27:22,690 --> 00:27:29,010
It takes us quite a
bit of work to do it.
484
00:27:29,010 --> 00:27:32,630
OK.
485
00:27:32,630 --> 00:27:36,260
I'm going to change gears
now, I'm going to shift gears.
486
00:27:36,260 --> 00:27:41,400
And we're going to go back
to a more standard thing
487
00:27:41,400 --> 00:27:44,650
which has to do with just
setting up integrals.
488
00:27:44,650 --> 00:27:47,820
And this has to do with
understanding where integrals
489
00:27:47,820 --> 00:27:50,590
play a role, and they play
a role in cumulative sums,
490
00:27:50,590 --> 00:27:52,160
in evaluating things.
491
00:27:52,160 --> 00:27:54,150
This is much more
closely associated
492
00:27:54,150 --> 00:27:57,020
with the first
Fundamental Theorem.
493
00:27:57,020 --> 00:27:59,060
That is, we'll
take, today we were
494
00:27:59,060 --> 00:28:02,260
talking about how integrals
are formulas for functions.
495
00:28:02,260 --> 00:28:04,730
Or solutions to
differential equations.
496
00:28:04,730 --> 00:28:08,400
We're going to go back
and talk about integrals
497
00:28:08,400 --> 00:28:11,040
as being the answer to
a question as opposed
498
00:28:11,040 --> 00:28:14,400
to what we've done now.
499
00:28:14,400 --> 00:28:18,470
So in other words,
and the first example,
500
00:28:18,470 --> 00:28:20,370
or most of the examples
for now, are going
501
00:28:20,370 --> 00:28:22,880
to be taken from geometry.
502
00:28:22,880 --> 00:28:27,440
Later on we'll get
to probability.
503
00:28:27,440 --> 00:28:44,550
And the first topic is
just areas between curves.
504
00:28:44,550 --> 00:28:46,890
Here's the idea.
505
00:28:46,890 --> 00:28:50,310
If you have a couple of curves
that look like this and maybe
506
00:28:50,310 --> 00:28:54,140
like this, and you want
to start at a place a
507
00:28:54,140 --> 00:28:59,270
and you want to
end at a place b,
508
00:28:59,270 --> 00:29:06,720
then you can chop it up the same
way we did with Riemann sums.
509
00:29:06,720 --> 00:29:10,685
And take a chunk
that looks like this.
510
00:29:10,685 --> 00:29:12,810
And I'm going to write the
thickness of that chunk.
511
00:29:12,810 --> 00:29:14,710
Well, let's give
these things names.
512
00:29:14,710 --> 00:29:20,510
Let's say the top curve is f(x),
and the bottom curve is g(x).
513
00:29:20,510 --> 00:29:26,980
And then this thickness
is going to be dx.
514
00:29:26,980 --> 00:29:30,020
That's the thickness.
515
00:29:30,020 --> 00:29:32,300
And what is the height?
516
00:29:32,300 --> 00:29:34,150
Well, the height
is the difference
517
00:29:34,150 --> 00:29:38,970
between the top value
and the bottom value.
518
00:29:38,970 --> 00:29:44,120
So here we have
(f(x) - g(x)) dx.
519
00:29:44,120 --> 00:29:50,120
This is, if you like, base
times-- Whoops, backwards.
520
00:29:50,120 --> 00:29:54,850
This is height, and this is
the base of the rectangle.
521
00:29:54,850 --> 00:29:56,850
And these are
approximately correct.
522
00:29:56,850 --> 00:29:59,690
But of course, only in limit
when this is an infinitesimal,
523
00:29:59,690 --> 00:30:03,890
is it exactly right.
524
00:30:03,890 --> 00:30:10,337
In order to get the whole
area, I have add these guys up.
525
00:30:10,337 --> 00:30:11,920
So I'm going to
integrate from a to b.
526
00:30:11,920 --> 00:30:14,350
That's summing them,
that's adding them up.
527
00:30:14,350 --> 00:30:16,470
And that's going to be my area.
528
00:30:16,470 --> 00:30:27,280
So that's the story here.
529
00:30:27,280 --> 00:30:30,020
Now, let me just say
two things about this.
530
00:30:30,020 --> 00:30:33,360
First of all, on a very
abstract level before we get
531
00:30:33,360 --> 00:30:36,340
started with details of
more complicated problems.
532
00:30:36,340 --> 00:30:39,390
The first one is
that every problem
533
00:30:39,390 --> 00:30:41,800
that I'm going to be
talking about from now
534
00:30:41,800 --> 00:30:45,730
on for several days, involves
the following collection
535
00:30:45,730 --> 00:30:48,700
of-- the following goals.
536
00:30:48,700 --> 00:30:52,090
I want to identify
something to integrate.
537
00:30:52,090 --> 00:30:58,010
That's called an integrand.
538
00:30:58,010 --> 00:31:06,420
And I want to identify what
are known as the limits.
539
00:31:06,420 --> 00:31:10,290
The whole game is
simply to figure out
540
00:31:10,290 --> 00:31:13,240
what a, b, and this
quantity is here.
541
00:31:13,240 --> 00:31:15,480
Whatever it is.
542
00:31:15,480 --> 00:31:18,090
And the minute we have that,
we can calculate the integral
543
00:31:18,090 --> 00:31:19,810
if we like.
544
00:31:19,810 --> 00:31:21,650
We have numerical
procedures or maybe we
545
00:31:21,650 --> 00:31:23,490
have analytic
procedures, but anyway we
546
00:31:23,490 --> 00:31:25,150
can get at the integral.
547
00:31:25,150 --> 00:31:27,580
The goal here is to set them up.
548
00:31:27,580 --> 00:31:31,720
And in order to set them up, you
must know these three things.
549
00:31:31,720 --> 00:31:33,550
The lower limit,
the upper limit,
550
00:31:33,550 --> 00:31:37,920
and what we're integrating.
551
00:31:37,920 --> 00:31:42,280
If you leave one of these out,
it's like the following thing.
552
00:31:42,280 --> 00:31:45,450
I ask you what the
area of this region is.
553
00:31:45,450 --> 00:31:48,740
If I left out this end,
how could I possibly know?
554
00:31:48,740 --> 00:31:51,240
I don't even know where it
starts, so how can I figure out
555
00:31:51,240 --> 00:31:52,990
what this area is if
I haven't identified
556
00:31:52,990 --> 00:31:55,230
what the left side is.
557
00:31:55,230 --> 00:31:58,210
I can't leave out the bottom.
558
00:31:58,210 --> 00:32:00,560
It's sitting here,
in this formula.
559
00:32:00,560 --> 00:32:03,007
Because I need to
know where it is.
560
00:32:03,007 --> 00:32:05,340
And I need to know the top
and I need to know this side.
561
00:32:05,340 --> 00:32:07,600
Those are the four
sides of the figure.
562
00:32:07,600 --> 00:32:10,190
If I don't incorporate
them into the information,
563
00:32:10,190 --> 00:32:11,640
I'll never get anything out.
564
00:32:11,640 --> 00:32:13,620
So I need to know everything.
565
00:32:13,620 --> 00:32:15,360
And I need to know
exactly those things,
566
00:32:15,360 --> 00:32:20,740
in order to have a
formula for the area.
567
00:32:20,740 --> 00:32:23,720
Now, when this gets
carried out in practice,
568
00:32:23,720 --> 00:32:27,960
as we will do now in
our first example,
569
00:32:27,960 --> 00:32:29,650
it's more complicated
than it looks.
570
00:32:29,650 --> 00:32:48,680
So here's our first example:
Find the area between x = y^2
571
00:32:48,680 --> 00:32:57,650
and y = x - 2.
572
00:32:57,650 --> 00:33:00,110
This is our first example.
573
00:33:00,110 --> 00:33:04,950
Let me make sure that I chose
the example that I wanted to.
574
00:33:04,950 --> 00:33:08,420
Yeah.
575
00:33:08,420 --> 00:33:20,300
Now, there's a first step in
figuring these things out.
576
00:33:20,300 --> 00:33:27,514
And this is that you
must draw a picture.
577
00:33:27,514 --> 00:33:28,930
If you don't draw
a picture you'll
578
00:33:28,930 --> 00:33:30,827
never figure out
what this area is,
579
00:33:30,827 --> 00:33:32,410
because you'll never
figure out what's
580
00:33:32,410 --> 00:33:36,290
what between these curves.
581
00:33:36,290 --> 00:33:40,540
The first curve, y =
x^2, is a parabola.
582
00:33:40,540 --> 00:33:42,990
But x is a function of y.
583
00:33:42,990 --> 00:33:45,090
It's pointing this way.
584
00:33:45,090 --> 00:33:47,450
So it's this parabola here.
585
00:33:47,450 --> 00:33:50,690
That's y = x^2.
586
00:33:50,690 --> 00:33:57,910
Whoops, x = y^2.
587
00:33:57,910 --> 00:34:06,700
The second curve is a line,
a straight line of slope 1,
588
00:34:06,700 --> 00:34:09,670
starting at x = 2, y = 0.
589
00:34:09,670 --> 00:34:13,160
It goes through this place
here, which is 2 over
590
00:34:13,160 --> 00:34:20,060
and has slope 1,
so it does this.
591
00:34:20,060 --> 00:34:22,590
And this shape in
here is what we mean
592
00:34:22,590 --> 00:34:24,100
by the area between the curves.
593
00:34:24,100 --> 00:34:27,050
Now that we see what it
is, we have a better idea
594
00:34:27,050 --> 00:34:28,130
of what our goal is.
595
00:34:28,130 --> 00:34:39,310
If you haven't drawn
it, you have no hope.
596
00:34:39,310 --> 00:34:45,310
Now, I'm going to describe two
ways of getting at this area
597
00:34:45,310 --> 00:34:50,660
here.
598
00:34:50,660 --> 00:34:59,970
And the first one is
motivated by the shape
599
00:34:59,970 --> 00:35:02,970
that I just
described right here.
600
00:35:02,970 --> 00:35:07,210
Namely, I'm going to use it
in a straightforward way.
601
00:35:07,210 --> 00:35:12,560
I'm going to chop things up
into these vertical pieces
602
00:35:12,560 --> 00:35:17,160
just as I did right there.
603
00:35:17,160 --> 00:35:19,415
Now, here's the
difficulty with that.
604
00:35:19,415 --> 00:35:27,000
The difficulty is that the
upper curve here has one formula
605
00:35:27,000 --> 00:35:28,990
but the lower curve
shifts from being
606
00:35:28,990 --> 00:35:33,120
a part of the parabola to being
a part of the straight line.
607
00:35:33,120 --> 00:35:35,160
That means that there are
two different formulas
608
00:35:35,160 --> 00:35:36,930
for the lower function.
609
00:35:36,930 --> 00:35:39,200
And the only way
to accommodate that
610
00:35:39,200 --> 00:35:42,780
is to separate this
up into two halves.
611
00:35:42,780 --> 00:35:44,980
Separate it out into two halves.
612
00:35:44,980 --> 00:35:50,320
I'm going to have to
divide it right here.
613
00:35:50,320 --> 00:35:52,870
So we must break
it into two pieces
614
00:35:52,870 --> 00:35:57,280
and find the integral of
one half and the other half.
615
00:35:57,280 --> 00:35:57,780
Question?
616
00:35:57,780 --> 00:36:06,269
STUDENT: [INAUDIBLE]
617
00:36:06,269 --> 00:36:08,060
PROFESSOR: So, you're
one step ahead of me.
618
00:36:08,060 --> 00:36:09,570
We'll also have to be
sure to distinguish
619
00:36:09,570 --> 00:36:12,130
between the top branch and the
bottom branch of the parabola,
620
00:36:12,130 --> 00:36:14,430
which we're about to do.
621
00:36:14,430 --> 00:36:17,870
Now, in order to
distinguish what's going on
622
00:36:17,870 --> 00:36:22,000
I actually have to
use multi colors here.
623
00:36:22,000 --> 00:36:24,680
And so we will do that.
624
00:36:24,680 --> 00:36:29,720
First there's the top
part, which is orange.
625
00:36:29,720 --> 00:36:32,200
That's the top part.
626
00:36:32,200 --> 00:36:33,990
I'll call it top.
627
00:36:33,990 --> 00:36:41,600
And then there's the bottom
part, which has two halves.
628
00:36:41,600 --> 00:36:53,280
They are pink, and I
guess this is blue.
629
00:36:53,280 --> 00:37:01,240
All right, so now let's
see what's happening.
630
00:37:01,240 --> 00:37:07,110
The most important two points
that I have to figure out
631
00:37:07,110 --> 00:37:08,580
in order to get started here.
632
00:37:08,580 --> 00:37:10,480
Well, really I'm going to have
to figure out three points,
633
00:37:10,480 --> 00:37:11,010
I claim.
634
00:37:11,010 --> 00:37:13,460
I'm going to have to figure
out where this point is.
635
00:37:13,460 --> 00:37:17,650
Where this point is,
and where that point is.
636
00:37:17,650 --> 00:37:20,220
If I know where these
three points are,
637
00:37:20,220 --> 00:37:23,990
then I have a chance of knowing
where to start, where to end,
638
00:37:23,990 --> 00:37:25,390
and so forth.
639
00:37:25,390 --> 00:37:26,170
Another question.
640
00:37:26,170 --> 00:37:27,490
STUDENT: [INAUDIBLE]
641
00:37:27,490 --> 00:37:31,110
PROFESSOR: Could you speak up?
642
00:37:31,110 --> 00:37:37,539
STUDENT: [INAUDIBLE]
643
00:37:37,539 --> 00:37:39,080
PROFESSOR: The
question is, why do we
644
00:37:39,080 --> 00:37:41,450
need to split up the area.
645
00:37:41,450 --> 00:37:44,170
And I think in order to
answer that question further,
646
00:37:44,170 --> 00:37:46,930
I'm going to have to go into
the details of the method,
647
00:37:46,930 --> 00:37:51,480
and then you'll see
where it's necessary.
648
00:37:51,480 --> 00:37:54,440
So the first step is that
I'm going to figure out
649
00:37:54,440 --> 00:37:57,160
what these three points are.
650
00:37:57,160 --> 00:38:02,460
This one is kind of easy;
it's the point (0, 0).
651
00:38:02,460 --> 00:38:04,880
This point down here
and this point up here
652
00:38:04,880 --> 00:38:08,010
are intersections
of the two curves.
653
00:38:08,010 --> 00:38:11,150
I can identify them by
the following equation.
654
00:38:11,150 --> 00:38:21,330
I need to see where
these curves intersect.
655
00:38:21,330 --> 00:38:26,690
At what, well, if I plug in
x = y^2, I get y = y^2 - 2.
656
00:38:26,690 --> 00:38:29,010
And then I can solve
this quadratic equation.
657
00:38:29,010 --> 00:38:33,840
y^2 - y - 2 = 0.
658
00:38:33,840 --> 00:38:36,620
So (y - 2) (y+1) = 0.
659
00:38:36,620 --> 00:38:39,360
= 0.
660
00:38:39,360 --> 00:38:47,960
And this is telling me
that y = 2 or y = -1.
661
00:38:47,960 --> 00:38:52,630
662
00:38:52,630 --> 00:38:54,250
So I've found y = -1.
663
00:38:54,250 --> 00:39:00,430
That means this point down
here has second entry -1.
664
00:39:00,430 --> 00:39:04,770
Its first entry, its x-value,
I can get from this formula
665
00:39:04,770 --> 00:39:06,890
here or the other formula.
666
00:39:06,890 --> 00:39:10,800
I have to square,
this, -1^2 = 1.
667
00:39:10,800 --> 00:39:15,020
So that's the formula
for this point.
668
00:39:15,020 --> 00:39:20,180
And the other point
has second entry 2.
669
00:39:20,180 --> 00:39:22,590
And, again, with
his formula y = x^2,
670
00:39:22,590 --> 00:39:31,790
I have to square y to
get x, so this is 4.
671
00:39:31,790 --> 00:39:37,340
Now, I claim I have enough
data to get started.
672
00:39:37,340 --> 00:39:41,160
But maybe I'll identify
one more thing.
673
00:39:41,160 --> 00:39:48,890
I need the top, the bottom
left, and the bottom right.
674
00:39:48,890 --> 00:39:54,770
The top is the formula for
this branch of x = y^2,
675
00:39:54,770 --> 00:39:57,880
which is in the
positive y region.
676
00:39:57,880 --> 00:40:05,130
And that is y is equal
to square root of x.
677
00:40:05,130 --> 00:40:09,390
The bottom curve,
part of the parabola,
678
00:40:09,390 --> 00:40:21,980
so this is the bottom left, is
y equals minus square root x.
679
00:40:21,980 --> 00:40:24,210
That's the other branch
of the square root.
680
00:40:24,210 --> 00:40:26,561
And this is exactly what
you were asking before.
681
00:40:26,561 --> 00:40:28,810
And this is, we have to
distinguish between these two.
682
00:40:28,810 --> 00:40:31,510
And the point is, these
formulas really are different.
683
00:40:31,510 --> 00:40:34,400
They're not the same.
684
00:40:34,400 --> 00:40:37,860
Now, the last bit is the
bottom right chunk here,
685
00:40:37,860 --> 00:40:39,910
which is this pink part.
686
00:40:39,910 --> 00:40:44,330
Bottom right.
687
00:40:44,330 --> 00:40:49,060
And that one is the
formula for the line.
688
00:40:49,060 --> 00:40:55,990
And that's y = x - 2.
689
00:40:55,990 --> 00:41:03,450
Now I'm ready to find the area.
690
00:41:03,450 --> 00:41:06,070
It's going to be in two chunks.
691
00:41:06,070 --> 00:41:15,060
This is the left part,
plus the right part.
692
00:41:15,060 --> 00:41:18,210
And the left part, and I want
to set it up as an integral,
693
00:41:18,210 --> 00:41:21,350
I want there to be a dx and here
I want to set up an integral
694
00:41:21,350 --> 00:41:23,740
and I want it to be dx.
695
00:41:23,740 --> 00:41:26,011
I need to figure out
what the range of x is.
696
00:41:26,011 --> 00:41:27,510
So, first I'm going
to-- well, let's
697
00:41:27,510 --> 00:41:37,640
leave ourselves a little
more room than that.
698
00:41:37,640 --> 00:41:39,310
Just to be safe.
699
00:41:39,310 --> 00:41:45,740
OK, here's the right.
700
00:41:45,740 --> 00:41:48,490
So here we have our dx.
701
00:41:48,490 --> 00:41:53,830
Now, I need to figure out the
starting place and the ending
702
00:41:53,830 --> 00:41:54,960
place.
703
00:41:54,960 --> 00:41:57,425
So the starting place
is the leftmost place.
704
00:41:57,425 --> 00:42:00,000
The leftmost place is over here.
705
00:42:00,000 --> 00:42:02,890
And x = 0 there.
706
00:42:02,890 --> 00:42:05,900
So we're going to travel
from this vertical line
707
00:42:05,900 --> 00:42:08,420
to the green line.
708
00:42:08,420 --> 00:42:09,070
Over here.
709
00:42:09,070 --> 00:42:13,100
And that's from 0 to 1.
710
00:42:13,100 --> 00:42:18,400
And the difference between the
orange curve and the blue curve
711
00:42:18,400 --> 00:42:22,330
is what I call top and
bottom left, over there.
712
00:42:22,330 --> 00:42:32,280
So that is square root of x
minus minus square root of x.
713
00:42:32,280 --> 00:42:41,170
Again, this is what I call
top, and this was bottom.
714
00:42:41,170 --> 00:42:48,850
But only the left.
715
00:42:48,850 --> 00:42:51,960
I claim that's giving me
the left half of this,
716
00:42:51,960 --> 00:42:55,650
the left section
of this diagram.
717
00:42:55,650 --> 00:42:59,780
Now I'm going to do the
right section of the diagram.
718
00:42:59,780 --> 00:43:02,040
I start at 1.
719
00:43:02,040 --> 00:43:04,160
The lower limit is 1.
720
00:43:04,160 --> 00:43:08,490
And I go all the way
to this point here.
721
00:43:08,490 --> 00:43:11,160
Which is the last bit.
722
00:43:11,160 --> 00:43:13,070
And that's going to be x = 4.
723
00:43:13,070 --> 00:43:19,130
The upper limit here is 4.
724
00:43:19,130 --> 00:43:21,410
And now I have to take the
difference between the top
725
00:43:21,410 --> 00:43:22,570
and the bottom again.
726
00:43:22,570 --> 00:43:24,800
The top is square root
of x all over again.
727
00:43:24,800 --> 00:43:26,570
But the bottom has changed.
728
00:43:26,570 --> 00:43:28,660
The bottom is now
the quantity x - 2.
729
00:43:28,660 --> 00:43:31,430
730
00:43:31,430 --> 00:43:32,990
Please don't forget
your parenthesis.
731
00:43:32,990 --> 00:43:44,000
There's going to be minus
signs and cancellations.
732
00:43:44,000 --> 00:43:46,420
Now, this is almost
the end of the problem.
733
00:43:46,420 --> 00:43:48,270
The rest of it is routine.
734
00:43:48,270 --> 00:43:52,440
We would just have to
evaluate these integrals.
735
00:43:52,440 --> 00:43:57,500
And, fortunately, I'm
going to spare you that.
736
00:43:57,500 --> 00:43:59,330
We're not going to
bother to do it.
737
00:43:59,330 --> 00:44:01,050
That's the easy part.
738
00:44:01,050 --> 00:44:03,466
We're not going to do it.
739
00:44:03,466 --> 00:44:05,840
But I'm going to show you that
there's a much quicker way
740
00:44:05,840 --> 00:44:07,320
with this integral.
741
00:44:07,320 --> 00:44:09,980
And with this area calculation.
742
00:44:09,980 --> 00:44:10,910
Right now.
743
00:44:10,910 --> 00:44:18,030
The quicker way is what you see
when you see how long this is.
744
00:44:18,030 --> 00:44:20,310
And you see that
there's another device
745
00:44:20,310 --> 00:44:24,180
that you can use that looks
similar in principle to this,
746
00:44:24,180 --> 00:44:28,250
but reverses the
roles of x and y.
747
00:44:28,250 --> 00:44:35,530
And the other device, which I'll
draw over here, schematically.
748
00:44:35,530 --> 00:44:47,130
No, maybe I'll draw it
on this blackboard here.
749
00:44:47,130 --> 00:44:55,380
So, Method 2, if you
like, this was Method 1,
750
00:44:55,380 --> 00:45:04,180
and we should call
it the hard way.
751
00:45:04,180 --> 00:45:10,970
Method 2, which is
better in this case,
752
00:45:10,970 --> 00:45:24,710
is to use horizontal slices.
753
00:45:24,710 --> 00:45:33,100
Let me draw the picture,
at least schematically.
754
00:45:33,100 --> 00:45:35,230
Here's our picture
that we had before.
755
00:45:35,230 --> 00:45:38,460
And now instead of
slicing it vertically,
756
00:45:38,460 --> 00:45:40,530
I'm going to slice
it horizontally.
757
00:45:40,530 --> 00:45:44,910
Like this.
758
00:45:44,910 --> 00:45:49,740
Now, the dimensions
have different names.
759
00:45:49,740 --> 00:45:51,990
But the principle is similar.
760
00:45:51,990 --> 00:45:55,170
The width, we now call dy.
761
00:45:55,170 --> 00:45:58,600
Because it's the change in y.
762
00:45:58,600 --> 00:46:05,700
And this distance here, from
the left end to the right end,
763
00:46:05,700 --> 00:46:09,950
we have to figure out what the
formulas for those things are.
764
00:46:09,950 --> 00:46:16,500
So on the left, maybe I'll
draw them color coded again.
765
00:46:16,500 --> 00:46:19,630
So here's a left.
766
00:46:19,630 --> 00:46:23,400
And, whoops, orange
is right, I guess.
767
00:46:23,400 --> 00:46:24,580
So here we go.
768
00:46:24,580 --> 00:46:30,470
So we have the left -
which is this green -
769
00:46:30,470 --> 00:46:46,090
is x = y^2 And the right,
which is orange, is y = x - 2.
770
00:46:46,090 --> 00:46:49,820
And now in order
to use this, it's
771
00:46:49,820 --> 00:46:51,865
going to turn out that
we want to write x
772
00:46:51,865 --> 00:46:53,170
as-- we want to reverse roles.
773
00:46:53,170 --> 00:46:57,820
So we want to write this
as x is a function of y.
774
00:46:57,820 --> 00:47:04,560
So we'll use it in this form.
775
00:47:04,560 --> 00:47:11,950
And now I want to set
up the integral for you.
776
00:47:11,950 --> 00:47:21,200
This time, the area is equal to
an integral in the dy variable.
777
00:47:21,200 --> 00:47:26,350
And its starting
place is down here.
778
00:47:26,350 --> 00:47:28,960
And its ending
place is up there.
779
00:47:28,960 --> 00:47:31,900
This is the lowest value of y,
and this is the top value of y.
780
00:47:31,900 --> 00:47:34,940
And we've already
computed those things.
781
00:47:34,940 --> 00:47:39,830
The lowest level of y is -1.
782
00:47:39,830 --> 00:47:42,200
So this is y = -1.
783
00:47:42,200 --> 00:47:45,660
And this top value is y = 2.
784
00:47:45,660 --> 00:47:51,490
So this goes from -1 to 2.
785
00:47:51,490 --> 00:47:56,690
And now the difference
is this distance here,
786
00:47:56,690 --> 00:47:58,870
the distance between
the rightmost point
787
00:47:58,870 --> 00:47:59,880
and the leftmost point.
788
00:47:59,880 --> 00:48:02,020
Those are the two dimensions.
789
00:48:02,020 --> 00:48:04,590
So again, it's a rectangle
but its horizontal is long
790
00:48:04,590 --> 00:48:07,570
and its vertical is very short.
791
00:48:07,570 --> 00:48:08,320
And what are they?
792
00:48:08,320 --> 00:48:11,530
It's the difference between
the right and the left.
793
00:48:11,530 --> 00:48:21,760
The right-hand is y + 2,
and the right-hand is y^2.
794
00:48:21,760 --> 00:48:26,990
So this is the formula.
795
00:48:26,990 --> 00:48:35,750
STUDENT: [INAUDIBLE]
796
00:48:35,750 --> 00:48:39,090
PROFESSOR: What
was the question?
797
00:48:39,090 --> 00:48:41,630
Why is it right minus left?
798
00:48:41,630 --> 00:48:42,770
That's very important.
799
00:48:42,770 --> 00:48:44,540
Why is it right minus left?
800
00:48:44,540 --> 00:48:47,230
And that's actually the point
that I was about to make.
801
00:48:47,230 --> 00:48:48,400
Which is this.
802
00:48:48,400 --> 00:48:53,860
That y + 2, which is the
right, is bigger than y^2,
803
00:48:53,860 --> 00:48:55,480
which is the left.
804
00:48:55,480 --> 00:49:00,320
So that means that y
+ 2 - y^2 is positive.
805
00:49:00,320 --> 00:49:02,830
If you do it backwards, you'll
always get a negative number
806
00:49:02,830 --> 00:49:07,300
and you'll always
get the wrong answer.
807
00:49:07,300 --> 00:49:10,860
So this is the right-hand
end minus the left-hand end
808
00:49:10,860 --> 00:49:14,300
gives you a positive number.
809
00:49:14,300 --> 00:49:16,760
And it's not obvious,
actually, where you are.
810
00:49:16,760 --> 00:49:19,030
There's another
double-check, by the way.
811
00:49:19,030 --> 00:49:22,630
When you look at this quantity,
you see that the ends pinch.
812
00:49:22,630 --> 00:49:25,320
And that's exactly
the crossover points.
813
00:49:25,320 --> 00:49:30,950
That is, when y =
-1, y + 2 - y^2 = 0.
814
00:49:30,950 --> 00:49:36,940
And when y = 2, y + 2 - y^2 = 0.
815
00:49:36,940 --> 00:49:38,740
And that's not an
accident, that's
816
00:49:38,740 --> 00:49:44,600
exactly the geometry of the
shape that we picked out there.
817
00:49:44,600 --> 00:49:46,560
So this is the technique.
818
00:49:46,560 --> 00:49:51,200
Now, this is a much
more routine integral.
819
00:49:51,200 --> 00:49:54,080
I'm not going to carry it out,
I'll just do one last step.
820
00:49:54,080 --> 00:50:01,400
Which is that this is
y^2 / 2 + 2y - y^3 / 3,
821
00:50:01,400 --> 00:50:03,600
evaluated at -1 and 2.
822
00:50:03,600 --> 00:50:08,310
Which, if you work
it out, is 9/2.
823
00:50:08,310 --> 00:50:10,230
So we're done for today.
824
00:50:10,230 --> 00:50:12,960
And tomorrow we'll do
more volumes, more things,
825
00:50:12,960 --> 00:50:15,070
including three dimensions.
826
00:50:15,070 --> 00:50:16,148