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