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PROFESSOR: Now, today I need
to get started by finishing
10
00:00:26 --> 00:00:28
up what I did last time.
11
00:00:28 --> 00:00:31
Namely, talking about
numerical methods.
12
00:00:31 --> 00:00:34
And I want to just
carry out one example.
13
00:00:34 --> 00:00:40
And then I want to fill
in one loose end.
14
00:00:40 --> 00:00:48
And then we'll talk
about the unit overall.
15
00:00:48 --> 00:01:01
We were talking, last time,
about numerical integration.
16
00:01:01 --> 00:01:05
I'm going to illustrate this
just with the simplest
17
00:01:05 --> 00:01:07
example that I can.
18
00:01:07 --> 00:01:14
We're going to look at the
integral from 1 to 2 of dx / x.
19
00:01:14 --> 00:01:18
Which we know perfectly well
already is the log of x
20
00:01:18 --> 00:01:23
evaluated between 1 and
2, which is ln 2 - ln 1.
21
00:01:23 --> 00:01:26
Which is just ln 2.
22
00:01:26 --> 00:01:30
Now, if you punch that into
your calculator, you're going
23
00:01:30 --> 00:01:32
to get something like this.
24
00:01:32 --> 00:01:41
I hope I saved it here.
25
00:01:41 --> 00:01:41
Yeah.
26
00:01:41 --> 00:01:53
It's about 0.693147.
27
00:01:53 --> 00:01:55
That's more digits than
we're going to get in
28
00:01:55 --> 00:01:56
our discussion here.
29
00:01:56 --> 00:02:02
Anyway, that's about how
big this number is.
30
00:02:02 --> 00:02:07
And the numerical integration
methods will give you about
31
00:02:07 --> 00:02:13
as much accuracy as you can
get on the function itself.
32
00:02:13 --> 00:02:15
And, of course, some
functions we may have more
33
00:02:15 --> 00:02:17
trouble approximating.
34
00:02:17 --> 00:02:19
But the function 1 / x, we know
pretty well how to do, because
35
00:02:19 --> 00:02:21
we know how to divide.
36
00:02:21 --> 00:02:25
So since the function that
we're integrating here is 1 /
37
00:02:25 --> 00:02:28
x, it's going to be not too
difficult to get
38
00:02:28 --> 00:02:29
some arithmetic.
39
00:02:29 --> 00:02:32
Nevertheless, I'm going to
do this in the simplest
40
00:02:32 --> 00:02:34
possible case.
41
00:02:34 --> 00:02:44
Namely, just with
two intervals.
42
00:02:44 --> 00:02:47
Now, you really can't expect
things to work so well
43
00:02:47 --> 00:02:48
with two intervals.
44
00:02:48 --> 00:02:52
That's a pretty ridiculous
approximation to your function.
45
00:02:52 --> 00:02:55
When you have two intervals,
that means you're looking at
46
00:02:55 --> 00:02:58
the graph of this hyperbola.
47
00:02:58 --> 00:03:04
And you have 1 here, and you
have 2 here and you have 3/2.
48
00:03:04 --> 00:03:07
And you're really only
keeping track of the values
49
00:03:07 --> 00:03:10
at these three spots.
50
00:03:10 --> 00:03:14
So the idea that you can
approximate the area just by
51
00:03:14 --> 00:03:18
knowing the values of three
places is a little bit of a
52
00:03:18 --> 00:03:21
stretch of the imagination.
53
00:03:21 --> 00:03:24
But we're going to
try it anyway.
54
00:03:24 --> 00:03:33
Now, the trapezoidal rule
is the following formula.
55
00:03:33 --> 00:03:38
It's delta x ( 1/2 the first
value + the second value
56
00:03:38 --> 00:03:42
+ 1/2 the third value).
57
00:03:42 --> 00:03:47
In this case, the pattern is
1/2, 1, 1, 1, 1, 1, 1/2.
58
00:03:47 --> 00:03:52
And in this case, delta
x = 1/2 because this
59
00:03:52 --> 00:03:55
interval's of length 1.
60
00:03:55 --> 00:03:57
The b - a, right.
61
00:03:57 --> 00:03:59
Let's just point that out here.
62
00:03:59 --> 00:04:05
Here, b = 2. a = 1. b - a = 1.
63
00:04:05 --> 00:04:08
And the number n = 2.
64
00:04:08 --> 00:04:16
And so, delta x, which
is b - a / n = 1/2.
65
00:04:16 --> 00:04:17
So here's what we get.
66
00:04:17 --> 00:04:19
And let's just see
what this number is.
67
00:04:19 --> 00:04:23
It's 1/2 of the value at here.
68
00:04:23 --> 00:04:25
Well, so let's just check
what these values are.
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00:04:25 --> 00:04:29
This value is 1, this value
over here is 2/3, and
70
00:04:29 --> 00:04:33
the last value is 1/2.
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00:04:33 --> 00:04:36
Because the function, of
course, was y = 1 / x.
72
00:04:36 --> 00:04:38
And those were the three
values that we have.
73
00:04:38 --> 00:04:47
So y0, this one is y0, this one
is y1, and this one is y2.
74
00:04:47 --> 00:05:00
Now, here we have (1/2*
1 + 2/3 + 1/2 * 1/2).
75
00:05:00 --> 00:05:05
Now, on an exam, I don't expect
you to add up long messes
76
00:05:05 --> 00:05:07
of numbers like this.
77
00:05:07 --> 00:05:10
When you have two numbers, I
expect you to add them up if
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00:05:10 --> 00:05:12
they're reasonable,
or subtract them.
79
00:05:12 --> 00:05:14
Just as we do when we
take antiderivatives.
80
00:05:14 --> 00:05:18
Like, for example, I don't want
you to leave the answer to
81
00:05:18 --> 00:05:19
an integration like
this in this form.
82
00:05:19 --> 00:05:21
I want you to simplify it
at least down to here.
83
00:05:21 --> 00:05:23
And I of course don't
expect you to know the
84
00:05:23 --> 00:05:24
numerical approximation.
85
00:05:24 --> 00:05:27
But I certainly expect you
to be able to do that.
86
00:05:27 --> 00:05:29
On the other hand, when the
arithmetic gets a little bit
87
00:05:29 --> 00:05:32
long, you can relax
a little bit.
88
00:05:32 --> 00:05:35
But I did carry this
out on my calculator.
89
00:05:35 --> 00:05:41
Unless I'm mistaken,
it's about 0.96.
90
00:05:41 --> 00:05:44
It's pretty far off.
91
00:05:44 --> 00:05:47
So remember what it was.
92
00:05:47 --> 00:05:49
It's what you get when you
get these straight lines.
93
00:05:49 --> 00:05:52
And there are these little
extra pieces of junk there.
94
00:05:52 --> 00:05:55
Now, don't trust that
too much, but the point
95
00:05:55 --> 00:06:00
is that it's far off.
96
00:06:00 --> 00:06:09
So now, let's take a
look at Simpson's Rule.
97
00:06:09 --> 00:06:13
And I claim that Simpson's Rule
is surprisingly accurate.
98
00:06:13 --> 00:06:15
In this case, really,
even a little more than
99
00:06:15 --> 00:06:17
it deserves to be.
100
00:06:17 --> 00:06:29
The formula is (delta x
/ 3) ( y0 + 4 y1 + y2).
101
00:06:29 --> 00:06:33
So the pattern is 1, 4, 1, or
1, 4 and then it alternates
102
00:06:33 --> 00:06:39
2's and 4's until 4,
1 at the very end.
103
00:06:39 --> 00:06:44
And if I just plug in the
numbers now, what I get is 1/6,
104
00:06:44 --> 00:06:47
because delta x = 1/2 again.
105
00:06:47 --> 00:06:50
And the value for y0 = 1.
106
00:06:50 --> 00:06:55
And the value for y1 = 2/3.
107
00:06:55 --> 00:06:59
And the value for y2 = 1/2.
108
00:06:59 --> 00:07:03
So here's the estimate
in this case.
109
00:07:03 --> 00:07:08
And this one I did
carry out carefully.
110
00:07:08 --> 00:07:16
And it came out to 0.69444.
111
00:07:16 --> 00:07:19
Which is actually pretty
impressive, if you
112
00:07:19 --> 00:07:20
think about it.
113
00:07:20 --> 00:07:26
Given what the logarithm is.
114
00:07:26 --> 00:07:30
Now, what's going on
with Simpson's Rule
115
00:07:30 --> 00:07:33
in general is this.
116
00:07:33 --> 00:07:43
If you -- Simpson's
minus the exact answer.
117
00:07:43 --> 00:07:46
In absolute value, is
approximately of the
118
00:07:46 --> 00:07:49
size of (delta x)^ 4.
119
00:07:50 --> 00:07:52
That's really the
way it behaves.
120
00:07:52 --> 00:08:03
Which means that if delta x is
about 1/10, so if we had
121
00:08:03 --> 00:08:07
divided this up into 10,
intervals which we didn't, but
122
00:08:07 --> 00:08:09
if we'd divided it up into 10
intervals, then you could
123
00:08:09 --> 00:08:15
expect that delta x, the error
would be about 10 ^ - 4.
124
00:08:15 --> 00:08:21
In other words, four digits of
accuracy here for this thing.
125
00:08:21 --> 00:08:26
But the exact analysis of this,
a more careful analysis of
126
00:08:26 --> 00:08:27
this, is in your textbook.
127
00:08:27 --> 00:08:30
And I'm not going to do.
128
00:08:30 --> 00:08:33
But I just want to point out
that it is an effective method.
129
00:08:33 --> 00:08:36
It really does give you nice
four-digit with manageable,
130
00:08:36 --> 00:08:38
you could even really
do it by hand.
131
00:08:38 --> 00:08:40
It's so convenient.
132
00:08:40 --> 00:08:42
The Simpson's Rule.
133
00:08:42 --> 00:08:46
Whereas the other rules aren't
really that impressive as
134
00:08:46 --> 00:08:51
far as giving fairly
accurate answers.
135
00:08:51 --> 00:08:55
The last little remark to
make is that the reason
136
00:08:55 --> 00:08:59
is that Simpson's Rule
is matching a parabola.
137
00:08:59 --> 00:09:03
And somehow the parabola
follows this curve better.
138
00:09:03 --> 00:09:05
It's giving the exact answer.
139
00:09:05 --> 00:09:07
So I'll mention this.
140
00:09:07 --> 00:09:27
Simpson's Rule is derived using
the exact answer for all
141
00:09:27 --> 00:09:32
degree 2 polynomials.
142
00:09:32 --> 00:09:36
In other words, parabolas.
143
00:09:36 --> 00:09:38
All parabolas.
144
00:09:38 --> 00:09:40
But even all the ones
of lower degree.
145
00:09:40 --> 00:09:43
So straight lines would
work, and constants
146
00:09:43 --> 00:09:44
would work as well.
147
00:09:44 --> 00:09:47
Whereas the other ones only
work for, say, straight lines.
148
00:09:47 --> 00:09:51
The trapezoidal rule only
works for straight lines.
149
00:09:51 --> 00:09:53
But ther isn't a
weird accident.
150
00:09:53 --> 00:09:55
It turns out that it
also works for cubics.
151
00:09:55 --> 00:09:59
Once you get the formulas,
it works for cubics.
152
00:09:59 --> 00:10:06
So it's also exact for cubics.
153
00:10:06 --> 00:10:12
And that's what explains
the fourth order validity.
154
00:10:12 --> 00:10:15
The last thing that I want
to point out is that this
155
00:10:15 --> 00:10:17
is extremely vague,
that I said there.
156
00:10:17 --> 00:10:20
And you should be a little
bit cautious about it.
157
00:10:20 --> 00:10:30
You need to watch out
for 1 / x for x near 0.
158
00:10:30 --> 00:10:34
All bets are off if the
function is singular.
159
00:10:34 --> 00:10:36
And there's a lot of
area under there.
160
00:10:36 --> 00:10:40
And it's also true that if
the derivative messes up,
161
00:10:40 --> 00:10:41
you're in trouble too.
162
00:10:41 --> 00:10:44
You really need for the
function to be nice and smooth
163
00:10:44 --> 00:10:46
in order for Simpson's
Rule to work.
164
00:10:46 --> 00:10:48
This is woth out.
165
00:10:48 --> 00:10:54
That's a real woth
out, but try to.
166
00:10:54 --> 00:10:55
Watch out.
167
00:10:55 --> 00:10:58
Watch out for
whenever x near 0.
168
00:10:58 --> 00:11:00
Then this thing doesn't work.
169
00:11:00 --> 00:11:03
This thing really depends
on bounds on derivatives.
170
00:11:03 --> 00:11:05
But I'm going to be
relatively vague about that.
171
00:11:05 --> 00:11:12
I'm not attempting to give
you an error analysis here.
172
00:11:12 --> 00:11:16
OK, so if you were doing
this on an exam, how do
173
00:11:16 --> 00:11:21
you remember this strange
pattern of numbers?
174
00:11:21 --> 00:11:27
The one thing that I want to
recommend to you is, as a way
175
00:11:27 --> 00:11:37
of remembering it, so the one
mnemonic device, we'll call it
176
00:11:37 --> 00:11:41
a mnemonic device here, for
remembering what it is that
177
00:11:41 --> 00:11:48
you're doing, is to remind
yourself of what happens for
178
00:11:48 --> 00:11:50
the simplest possible case.
179
00:11:50 --> 00:11:53
Which is f ( x) = 1.
180
00:11:53 --> 00:11:57
It seems very modest, but if it
doesn't give you the exact
181
00:11:57 --> 00:12:02
answer for f (x) = 1, you've
got the wrong weightings.
182
00:12:02 --> 00:12:06
And here, if you check out what
happens in the first formula
183
00:12:06 --> 00:12:13
here, y0 / 2 + y1 +..., well,
we'll go all the way
184
00:12:13 --> 00:12:17
to yn - 1 + yn / 2.
185
00:12:17 --> 00:12:19
If you check that formula
out here, this is the
186
00:12:19 --> 00:12:21
trapezoidal rule.
187
00:12:21 --> 00:12:26
If you check it out for this
case, then what you get is
188
00:12:26 --> 00:12:29
that this is equal to
delta x times what?
189
00:12:29 --> 00:12:30
Well, all of these are 1's.
190
00:12:30 --> 00:12:32
And how many are
there in the middle?
191
00:12:32 --> 00:12:34
There are n - 1 of
them in the middle.
192
00:12:34 --> 00:12:38
So it's 1/2 + n - 1 + 1/2.
193
00:12:38 --> 00:12:40
At the tail end.
194
00:12:40 --> 00:12:43
So all told it's (delta x)( n).
195
00:12:43 --> 00:12:50
And I remind you that
delta x = b - a / n.
196
00:12:50 --> 00:12:57
So, delta x, this thing,
is equal to b - a.
197
00:12:57 --> 00:12:58
And that's just
as it should be.
198
00:12:58 --> 00:13:02
What we just calculated
is an approximation to
199
00:13:02 --> 00:13:04
this integral here.
200
00:13:04 --> 00:13:07
Which is just the area of
the rectangle of base
201
00:13:07 --> 00:13:09
b - a and height 1.
202
00:13:09 --> 00:13:12
Which of course is b - a.
203
00:13:12 --> 00:13:15
So this is the check that
you got your weighted
204
00:13:15 --> 00:13:18
average correct here.
205
00:13:18 --> 00:13:20
You've put the correct
weightings on everything.
206
00:13:20 --> 00:13:23
And you can do this same
thing with Simpson's Rule.
207
00:13:23 --> 00:13:31
And match up those quantities.
208
00:13:31 --> 00:13:36
There was a question in
the room at some point.
209
00:13:36 --> 00:13:38
No, OK.
210
00:13:38 --> 00:13:45
So now, the next thing I want
to do for you is the loose
211
00:13:45 --> 00:13:49
end which I left hanging.
212
00:13:49 --> 00:13:54
Namely, I want to compute
that mysterious constant
213
00:13:54 --> 00:13:56
square root of pi / 2.
214
00:13:56 --> 00:14:00
This is really one of the
most famous computations
215
00:14:00 --> 00:14:02
in calculus.
216
00:14:02 --> 00:14:04
And it's a very,
very clever trick.
217
00:14:04 --> 00:14:09
I certainly don't expect you
to come up with this trick.
218
00:14:09 --> 00:14:11
I certainly wouldn't
have myself.
219
00:14:11 --> 00:14:15
But it's an important
thing to calculate.
220
00:14:15 --> 00:14:17
And it's just very useful.
221
00:14:17 --> 00:14:19
So I'm going to
tell you about it.
222
00:14:19 --> 00:14:22
And it's just on the subject
that we're dealing with in
223
00:14:22 --> 00:14:28
this unit; namely, slicing.
224
00:14:28 --> 00:14:30
Or adding up.
225
00:14:30 --> 00:14:36
So the first step, which is
just something that we already
226
00:14:36 --> 00:14:50
did, was that we found the
volume under this curve.
227
00:14:50 --> 00:14:54
This bell-shaped
curve, e ^ - r ^2.
228
00:14:54 --> 00:15:00
But rotated around an axis.
229
00:15:00 --> 00:15:09
Rotated around this axis.
230
00:15:09 --> 00:15:10
Around this way.
231
00:15:10 --> 00:15:12
So we figured that out.
232
00:15:12 --> 00:15:15
And that was a relatively
short computation.
233
00:15:15 --> 00:15:18
I'm just going to remind
you, it goes by shells.
234
00:15:18 --> 00:15:23
We integrate the whole
range from 0 to infinity.
235
00:15:23 --> 00:15:29
And we have 2 pi r e^ - r ^2
dr. So this again is the
236
00:15:29 --> 00:15:31
circumference of the shell.
237
00:15:31 --> 00:15:32
This is the height of the
shell, and this is the
238
00:15:32 --> 00:15:34
thickness of the shell.
239
00:15:34 --> 00:15:45
Circumference,
height, thickness.
240
00:15:45 --> 00:15:47
So we're just taking a
little piece here and
241
00:15:47 --> 00:15:48
sweeping it around.
242
00:15:48 --> 00:15:52
And then adding up.
243
00:15:52 --> 00:15:58
And then this antiderivative
is pi, - pi e^ - r ^2,
244
00:15:58 --> 00:16:00
evaluated at 0 and infinity.
245
00:16:00 --> 00:16:03
And we worked this
out last time.
246
00:16:03 --> 00:16:06
This is pi.
247
00:16:06 --> 00:16:12
It's pi (1 - 0).
248
00:16:12 --> 00:16:13
Which is pi.
249
00:16:13 --> 00:16:15
So the conclusion
is that V = pi.
250
00:16:15 --> 00:16:20
We already know that.
251
00:16:20 --> 00:16:23
Now, the problem that we want
to deal with now is the problem
252
00:16:23 --> 00:16:31
not of a volume, but an area.
253
00:16:31 --> 00:16:32
And this looks quite different.
254
00:16:32 --> 00:16:37
And of course the answer
is going to be different.
255
00:16:37 --> 00:16:38
But let's do it.
256
00:16:38 --> 00:16:40
So this is this
question mark here.
257
00:16:40 --> 00:16:42
And I'm going to do
the one from minus
258
00:16:42 --> 00:16:47
infinity to infinity.
259
00:16:47 --> 00:16:52
And I'll relate it to what we
talked about earlier in this
260
00:16:52 --> 00:16:57
unit, in just a couple of
minutes when I show you the
261
00:16:57 --> 00:17:00
procedure that we're
going to follow.
262
00:17:00 --> 00:17:04
So here's the quantity and now,
what this is interpreted as is
263
00:17:04 --> 00:17:07
the area under this bell curve.
264
00:17:07 --> 00:17:22
This time, Q is really an area.
265
00:17:22 --> 00:17:26
Now, what's going to turn
out to happen, is this.
266
00:17:26 --> 00:17:27
This is the trick.
267
00:17:27 --> 00:17:31
We're going to compute
V in a different way.
268
00:17:31 --> 00:17:35
And you'll see it laid
out in just a second.
269
00:17:35 --> 00:17:42
We will compute V by slices.
270
00:17:42 --> 00:17:45
We're going to slice it like
a piece of bread here.
271
00:17:45 --> 00:17:48
We're going to solve for
that same thing here.
272
00:17:48 --> 00:17:51
And then, amazingly, what's
going to happen is that we
273
00:17:51 --> 00:17:57
will discover that V = Q ^2.
274
00:17:57 --> 00:18:03
That's going to be what's
going to come out.
275
00:18:03 --> 00:18:07
And that's the end of the
computation that we want.
276
00:18:07 --> 00:18:08
Because actually we
already know what V is.
277
00:18:08 --> 00:18:11
We don't want to read
this equation forward.
278
00:18:11 --> 00:18:12
We want to read it
the other way.
279
00:18:12 --> 00:18:18
We want to say Q^2 = V, which
we already know is pi.
280
00:18:18 --> 00:18:29
And so Q = the
square root of pi.
281
00:18:29 --> 00:18:33
I haven't shown this yet,
this is the weird part.
282
00:18:33 --> 00:18:36
And I'm going to put it in a
little box so that we know that
283
00:18:36 --> 00:18:38
this is what we need to check.
284
00:18:38 --> 00:18:43
We need to check
this fact here.
285
00:18:43 --> 00:18:45
We haven't done that yet.
286
00:18:45 --> 00:18:51
Now, let me connect this with
what we did a few days ago.
287
00:18:51 --> 00:18:56
With what I called one of
the important functions of
288
00:18:56 --> 00:19:02
mathematics besides the
ones you already know.
289
00:19:02 --> 00:19:05
And so the function that we
were faced with, and that we
290
00:19:05 --> 00:19:15
discussed, was this one.
291
00:19:15 --> 00:19:19
And then, we were interested
in the value at infinity.
292
00:19:19 --> 00:19:23
We were interested in this.
293
00:19:23 --> 00:19:28
Which, if you draw a picture of
it, and you draw the same bell
294
00:19:28 --> 00:19:32
curve, that's the area
under half. of it.
295
00:19:32 --> 00:19:35
That's the area starting from
0 and going to infinity.
296
00:19:35 --> 00:19:37
That's the area under half.
297
00:19:37 --> 00:19:46
So this chunk is F of infinity.
298
00:19:46 --> 00:19:50
And now I hope that this
part of the connection is
299
00:19:50 --> 00:19:52
not meant to be fancy.
300
00:19:52 --> 00:19:58
The idea here is that
Q = 2 F(infinity).
301
00:19:58 --> 00:20:00
This number here.
302
00:20:00 --> 00:20:05
And so F of infinity = to the
square root of pi / 2, if we
303
00:20:05 --> 00:20:10
believe what we said
on the last panel.
304
00:20:10 --> 00:20:14
And that was the thing that I
drew a picture of on the board.
305
00:20:14 --> 00:20:18
Namely, the graph of
F looked like this.
306
00:20:18 --> 00:20:23
And there was this asymptote,
which was the limit F (x) tends
307
00:20:23 --> 00:20:26
to square root of pi / 2.
308
00:20:26 --> 00:20:27
As x goes to infinity.
309
00:20:27 --> 00:20:30
That was that limiting value.
310
00:20:30 --> 00:20:33
Which is F of infinity.
311
00:20:33 --> 00:20:40
So this is the asymptote.
312
00:20:40 --> 00:20:43
And now I've explained the
connection between what we
313
00:20:43 --> 00:20:48
claimed to be 4, which was
quite mysterious, and what
314
00:20:48 --> 00:20:50
we're actually going to
be able to check now.
315
00:20:50 --> 00:21:07
Concretely, by making
this computation.
316
00:21:07 --> 00:21:09
So how in the world can you
get something like this.
317
00:21:09 --> 00:21:14
What's in that orange box
there, that V = Q ^2.
318
00:21:14 --> 00:21:19
Again, the technique is
to use slices here.
319
00:21:19 --> 00:21:21
And I'm going to have to
draw you a 3-D picture
320
00:21:21 --> 00:21:24
to visualize the slice.
321
00:21:24 --> 00:21:27
Let's do that.
322
00:21:27 --> 00:21:31
I'm going to draw three axes
now, because we're now going to
323
00:21:31 --> 00:21:36
be in three-dimensional space,
and I want you to imagine the x
324
00:21:36 --> 00:21:39
axis as coming out of the
blackboard, the y axis is
325
00:21:39 --> 00:21:42
horizontal, and there's a new
axis, which I'll call the
326
00:21:42 --> 00:21:45
z axis which is going up.
327
00:21:45 --> 00:21:49
So what's happening here is
that I'm thinking of this,
328
00:21:49 --> 00:21:51
this is, if you like,
some kind of side view.
329
00:21:51 --> 00:21:52
And this is a view where
I've tilted things a
330
00:21:52 --> 00:21:57
little bit up to the top.
331
00:21:57 --> 00:22:01
Now, the distribution, or you
could think of this target in
332
00:22:01 --> 00:22:03
the plane, where the most
likely places to hit were in
333
00:22:03 --> 00:22:05
the middle and it died off.
334
00:22:05 --> 00:22:07
As we went down.
335
00:22:07 --> 00:22:11
Now, I want to draw a
picture of this graph.
336
00:22:11 --> 00:22:15
I'm going to draw a picture
of e^ - r squared.
337
00:22:15 --> 00:22:18
And it's basically a hump.
338
00:22:18 --> 00:22:22
So I'm going to take the
first, the slice along y = 0.
339
00:22:22 --> 00:22:23
The y = 0 slice.
340
00:22:23 --> 00:22:26
And I claim that that
goes up like this.
341
00:22:26 --> 00:22:28
And then comes back down.
342
00:22:28 --> 00:22:32
Let me shade this in, so
that you can see what
343
00:22:32 --> 00:22:34
kind of a slice this is.
344
00:22:34 --> 00:22:39
This is supposed to be along
this vertical plane here.
345
00:22:39 --> 00:22:40
Which is coming out of
the blackboard and
346
00:22:40 --> 00:22:42
coming towards you.
347
00:22:42 --> 00:22:43
And that's a slice.
348
00:22:43 --> 00:22:46
Now, I'm going to draw one
more slice so that you
349
00:22:46 --> 00:22:49
can see what's happening.
350
00:22:49 --> 00:22:52
I'm going to draw a
slice at another place.
351
00:22:52 --> 00:22:52
Along here.
352
00:22:52 --> 00:22:55
This will be y = b.
353
00:22:55 --> 00:22:56
Some other level.
354
00:22:56 --> 00:22:58
And now I'm going to
show you what happens.
355
00:22:58 --> 00:23:02
What happens is that the hump
dies down a little bit.
356
00:23:02 --> 00:23:06
So the bump is just
a little bit lower.
357
00:23:06 --> 00:23:10
And it's going to look a
little bit the same way.
358
00:23:10 --> 00:23:11
But it's just going
to be a bit smaller.
359
00:23:11 --> 00:23:15
So there's another slice here.
360
00:23:15 --> 00:23:16
Like that.
361
00:23:16 --> 00:23:20
And I want to give a
name to these slices.
362
00:23:20 --> 00:23:25
I'm going to call this A ( b).
363
00:23:25 --> 00:23:36
That is, the area
of the b slice.
364
00:23:36 --> 00:23:39
Under the surface.
365
00:23:39 --> 00:23:41
OK Yes, question.
366
00:23:41 --> 00:23:47
STUDENT: [INAUDIBLE]
367
00:23:47 --> 00:23:48
PROFESSOR: Yeah, the solid.
368
00:23:48 --> 00:23:48
Yeah.
369
00:23:48 --> 00:23:53
We're trying to figure out this
volume here, which is the one
370
00:23:53 --> 00:23:56
we started out with, by slices.
371
00:23:56 --> 00:23:59
So first I have to think of,
I'm going to visualize.
372
00:23:59 --> 00:24:01
So here I didn't
even visualize.
373
00:24:01 --> 00:24:04
I took a cross section and I
thought about how to spin it
374
00:24:04 --> 00:24:06
around without actually
doing that in
375
00:24:06 --> 00:24:08
three-dimensional space.
376
00:24:08 --> 00:24:10
But now I'm going to take a
different kind of slice.
377
00:24:10 --> 00:24:12
I'm going to take that
same bump, which is a
378
00:24:12 --> 00:24:13
three-dimensional object.
379
00:24:13 --> 00:24:16
I'm going to lay it
down on a plane.
380
00:24:16 --> 00:24:17
Which looks like this.
381
00:24:17 --> 00:24:20
And then it's a bump here.
382
00:24:20 --> 00:24:22
It's a hump.
383
00:24:22 --> 00:24:27
And now I'm going to try to
slice it by various planes.
384
00:24:27 --> 00:24:29
STUDENT: [INAUDIBLE]
385
00:24:29 --> 00:24:36
PROFESSOR: So one way of
defining the bump, as you just
386
00:24:36 --> 00:24:38
suggested, is you take this
curve and you rotate it
387
00:24:38 --> 00:24:39
around this z-axis.
388
00:24:39 --> 00:24:41
So in other words, you make
this the axis of rotation,
389
00:24:41 --> 00:24:43
you spin it around.
390
00:24:43 --> 00:24:44
That's correct.
391
00:24:44 --> 00:24:48
So that shows you that the
peaks as you go down here are
392
00:24:48 --> 00:24:50
going to descend the same way.
393
00:24:50 --> 00:24:54
But I don't want to
draw those lines.
394
00:24:54 --> 00:24:57
I want to imagine what
the parallel slices are.
395
00:24:57 --> 00:24:59
Because I don't want
to get cross slices.
396
00:24:59 --> 00:25:01
I want all slices parallel
to the same thing.
397
00:25:01 --> 00:25:05
STUDENT: [INAUDIBLE]
398
00:25:05 --> 00:25:05
PROFESSOR: OK.
399
00:25:05 --> 00:25:13
This is not particularly
easy to visualize.
400
00:25:13 --> 00:25:17
Now, here's the formula
for volume by slices.
401
00:25:17 --> 00:25:23
The formula for volume by
slices is that you add up
402
00:25:23 --> 00:25:28
the areas of the slices.
403
00:25:28 --> 00:25:31
That's how you do it.
404
00:25:31 --> 00:25:32
You take each slice.
405
00:25:32 --> 00:25:34
You add the cross-sectional
area, and then you take a
406
00:25:34 --> 00:25:38
little thickness, dy, and
then you add all of them up.
407
00:25:38 --> 00:25:41
Because this is extending over
the whole plane, we're going to
408
00:25:41 --> 00:25:43
have to go all the way from
minus infinity to
409
00:25:43 --> 00:25:45
plus infinity.
410
00:25:45 --> 00:25:56
And this is the formula
for volumes by slicing.
411
00:25:56 --> 00:26:00
And now our goal, in order to
do this calculation, we're
412
00:26:00 --> 00:26:04
going to just fix y = some b.
413
00:26:04 --> 00:26:06
We're just going to fix
one of these slices.
414
00:26:06 --> 00:26:13
And we're going to
calculate A ( b).
415
00:26:13 --> 00:26:16
That's what we need to do
in order to make this
416
00:26:16 --> 00:26:23
procedure succeed.
417
00:26:23 --> 00:26:27
This is the only place
where this method works.
418
00:26:27 --> 00:26:32
But it's an important one.
419
00:26:32 --> 00:26:35
In order to make it work,
I'm going to have to again
420
00:26:35 --> 00:26:37
draw the plot from a
different point of view.
421
00:26:37 --> 00:26:43
I'm going to do the top view.
422
00:26:43 --> 00:26:48
So I want to look down
on this x-y plane here.
423
00:26:48 --> 00:26:52
This is the x direction, and
here's the y direction.
424
00:26:52 --> 00:26:56
And then again I want
to draw my slice.
425
00:26:56 --> 00:27:00
My slice is here.
426
00:27:00 --> 00:27:04
At y = b.
427
00:27:04 --> 00:27:07
So we're just right
on top of it.
428
00:27:07 --> 00:27:09
And it's coming up at
some kind of bump.
429
00:27:09 --> 00:27:11
Here, with a little higher
in the middle and going
430
00:27:11 --> 00:27:15
down on the sides.
431
00:27:15 --> 00:27:20
Now, the formula for
the height is this.
432
00:27:20 --> 00:27:25
If I take a distance r here,
the formula for the height
433
00:27:25 --> 00:27:31
of the bump is e ^ - r ^2.
434
00:27:31 --> 00:27:35
I'll store that over
here. e ^ - r ^2 is the
435
00:27:35 --> 00:27:37
height at this place.
436
00:27:37 --> 00:27:41
If this distance to
the origin is r.
437
00:27:41 --> 00:27:43
That's true all the way around.
438
00:27:43 --> 00:27:48
And in terms of b and x,
we can figure out that
439
00:27:48 --> 00:27:49
by this right triangle.
440
00:27:49 --> 00:27:52
This height is b, and
this distance is x.
441
00:27:52 --> 00:28:02
So r ^2 = b ^2 + x ^2.
442
00:28:02 --> 00:28:02
Question.
443
00:28:02 --> 00:28:15
STUDENT: [INAUDIBLE]
444
00:28:15 --> 00:28:19
PROFESSOR: The question is,
is that the x-y plane.
445
00:28:19 --> 00:28:23
So the answer is that over here
I cleverly used the letter r.
446
00:28:23 --> 00:28:27
I avoided using y's
and z's or anything.
447
00:28:27 --> 00:28:28
And over here, this
is the distance r.
448
00:28:28 --> 00:28:31
And you like this,
is z, going up.
449
00:28:31 --> 00:28:33
That's the way to think of it.
450
00:28:33 --> 00:28:36
So that all of the
letters are consistent.
451
00:28:36 --> 00:28:39
So I just avoided
giving it a name.
452
00:28:39 --> 00:28:41
That's good, that's
exactly the point.
453
00:28:41 --> 00:28:43
Alright.
454
00:28:43 --> 00:28:46
So now, I claim I have
a formula for r ^2.
455
00:28:46 --> 00:28:48
And so I can write this down.
456
00:28:48 --> 00:28:58
This is e ^ - (b ^2 + x ^2).
457
00:28:58 --> 00:29:02
But now I'm going to use
the rule of exponents.
458
00:29:02 --> 00:29:07
Which is that this is the same
as (e ^ - b ^2) ( e^ - x^2).
459
00:29:07 --> 00:29:11
And that's going to be the
main way in which we use the
460
00:29:11 --> 00:29:15
particular function that
we're dealing with here.
461
00:29:15 --> 00:29:20
That's really the main
step, amazingly.
462
00:29:20 --> 00:29:33
So now I get to compute
what A ( b) is.
463
00:29:33 --> 00:29:37
A( b) is the area
under a curve.
464
00:29:37 --> 00:29:41
So it's going to be, let me
write it over here, A(b) is the
465
00:29:41 --> 00:29:47
area under this curve here.
466
00:29:47 --> 00:29:51
Which is some constant times,
so if you imagine, call
467
00:29:51 --> 00:29:53
this thing the name c.
468
00:29:53 --> 00:29:57
Under some curve, c e ^ - x ^2.
469
00:29:57 --> 00:30:06
Where the c = e^ - b ^2.
470
00:30:06 --> 00:30:07
That's what our slice is.
471
00:30:07 --> 00:30:09
In fact, it looks
like one of those.
472
00:30:09 --> 00:30:12
It looks like one
of those bumps.
473
00:30:12 --> 00:30:14
Here's its formula again.
474
00:30:14 --> 00:30:19
It's the integral from minus
infinity to infinity of (e^
475
00:30:19 --> 00:30:28
- b ^2)( e ^ - x ^2) dx.
476
00:30:28 --> 00:30:31
We just recopied what
I had up there.
477
00:30:31 --> 00:30:37
And this is the height at each
value of x, with b fixed.
478
00:30:37 --> 00:30:42
And now, so we have a
lot of steps here.
479
00:30:42 --> 00:30:44
But each of them is
very elementary.
480
00:30:44 --> 00:30:47
The first one was just
that law of exponents.
481
00:30:47 --> 00:30:49
That we could split the
two into products.
482
00:30:49 --> 00:30:51
Now I'm going to make that
splitting even further.
483
00:30:51 --> 00:30:53
This is a constant.
484
00:30:53 --> 00:30:55
It's not varying with x.
485
00:30:55 --> 00:30:59
So I'm going to factor
it out of the integral.
486
00:30:59 --> 00:31:02
This is e ^ - b ^2 times the
integral from minus infinity
487
00:31:02 --> 00:31:06
to infinity of e^ -x^2 dx.
488
00:31:06 --> 00:31:10
So this might look frightening,
but actually it's just the
489
00:31:10 --> 00:31:11
property of an integral.
490
00:31:11 --> 00:31:13
All integrals have this
kind of property.
491
00:31:13 --> 00:31:17
You can always factor
out a constant.
492
00:31:17 --> 00:31:20
And now here comes the
remarkable thing.
493
00:31:20 --> 00:31:24
This is e ^ - b^2 times
a number which is
494
00:31:24 --> 00:31:26
now familiar to us.
495
00:31:26 --> 00:31:27
What is this number?
496
00:31:27 --> 00:31:29
This is what we're looking for.
497
00:31:29 --> 00:31:38
This is our unknown, Q.
498
00:31:38 --> 00:31:41
So I've computed A(b), and
now I'm ready to finish
499
00:31:41 --> 00:31:43
the problem off.
500
00:31:43 --> 00:31:47
A (b) = ( e^ - b^2) Q.
501
00:31:47 --> 00:31:49
Q is that strange number
which we don't know yet.
502
00:31:49 --> 00:31:51
What it is.
503
00:31:51 --> 00:31:55
So now I'm going to
compute the whole volume.
504
00:31:55 --> 00:31:58
The whole volume, remember,
it's over there, it's minus
505
00:31:58 --> 00:32:02
infinity to infinity,
A ( y) dy.
506
00:32:02 --> 00:32:04
And now I'm just going
to plug in the formula
507
00:32:04 --> 00:32:06
that we've found for a.
508
00:32:06 --> 00:32:08
Now I'm doing this for
each b, so I'm doing it
509
00:32:08 --> 00:32:09
varying over all b's.
510
00:32:09 --> 00:32:12
So I have the integral from
minus infinity to infinity.
511
00:32:12 --> 00:32:15
And here I have e^ - y ^2.
512
00:32:15 --> 00:32:17
I've replaced b by y.
513
00:32:17 --> 00:32:19
And now I have Q.
514
00:32:19 --> 00:32:21
And I have dy.
515
00:32:21 --> 00:32:24
I just recopied what I
had over there into the
516
00:32:24 --> 00:32:27
formula for slicing.
517
00:32:27 --> 00:32:30
And now, I'm going to do this
trick of factoring out the
518
00:32:30 --> 00:32:32
constant a second time.
519
00:32:32 --> 00:32:33
This is a constant.
520
00:32:33 --> 00:32:35
It doesn't depend on y.
521
00:32:35 --> 00:32:38
It's the same for all y,
it just will factor out.
522
00:32:38 --> 00:32:40
So this is the same as Q times
the integral from minus
523
00:32:40 --> 00:32:47
infinity to infinity,
e ^ - y ^2 dy.
524
00:32:47 --> 00:32:52
And now, lo and behold,
this expression here.
525
00:32:52 --> 00:32:54
Of course, notice
how I defined Q.
526
00:32:54 --> 00:32:56
Let's go back carefully
to where Q is defined.
527
00:32:56 --> 00:33:01
Here's Q.
528
00:33:01 --> 00:33:02
This t is a dummy variable.
529
00:33:02 --> 00:33:04
It doesn't matter
what I call it.
530
00:33:04 --> 00:33:06
I can call it x, I can call
it u, I can call it v.
531
00:33:06 --> 00:33:10
In this case, I've given
it two different names.
532
00:33:10 --> 00:33:13
At this stage, I called it x.
533
00:33:13 --> 00:33:14
And at this stage
I'm calling it y.
534
00:33:14 --> 00:33:17
But it's the same variable.
535
00:33:17 --> 00:33:20
And so this little chunk is Q
and altogether I have two of
536
00:33:20 --> 00:33:23
them, for Q ^2 being the total.
537
00:33:23 --> 00:33:32
And that's the end
of the argument.
538
00:33:32 --> 00:33:33
It's a real miracle.
539
00:33:33 --> 00:33:45
STUDENT: [INAUDIBLE]
540
00:33:45 --> 00:33:47
PROFESSOR: Great question.
541
00:33:47 --> 00:33:49
The question is, wait a minute.
542
00:33:49 --> 00:33:53
As y changes, doesn't x change.
543
00:33:53 --> 00:33:58
And so then this
wouldn't be a constant.
544
00:33:58 --> 00:34:01
So that's the way in which
we've used the letters x and
545
00:34:01 --> 00:34:04
y in this whole course.
546
00:34:04 --> 00:34:07
When you get to 18.02, you'll
almost never do that.
547
00:34:07 --> 00:34:11
Always y and x will be
different variables.
548
00:34:11 --> 00:34:12
And they won't have to
depend on each other.
549
00:34:12 --> 00:34:15
Now, let me show you where
on this picture the
550
00:34:15 --> 00:34:17
x and the y are.
551
00:34:17 --> 00:34:20
We've got a whole x-y plain,
and here I'm fixing y
552
00:34:20 --> 00:34:23
= b, y isn't varying.
553
00:34:23 --> 00:34:25
Whereas x is changing.
554
00:34:25 --> 00:34:27
So, in other words, I don't
have a relationship between
555
00:34:27 --> 00:34:29
x and y, unless I fix it.
556
00:34:29 --> 00:34:32
In this case I've decided that
y is going to be constant.
557
00:34:32 --> 00:34:35
For all x.
558
00:34:35 --> 00:34:39
Over here, I made
a computation.
559
00:34:39 --> 00:34:41
And I have a Q, which is
just a single number.
560
00:34:41 --> 00:34:44
No matter which b I took,
it didn't matter which.
561
00:34:44 --> 00:34:46
No matter which y = b.
562
00:34:46 --> 00:34:48
Of course, I changed the
name to b so it wouldn't
563
00:34:48 --> 00:34:50
be so jarring to you.
564
00:34:50 --> 00:34:53
But in fact this b
was y all along.
565
00:34:53 --> 00:34:56
It's just that the x
varied completely
566
00:34:56 --> 00:34:57
independently of the y.
567
00:34:57 --> 00:35:00
I could fix the y and
vary the x, I could fix
568
00:35:00 --> 00:35:03
the x and vary the y.
569
00:35:03 --> 00:35:06
So it's a different
use of the letters.
570
00:35:06 --> 00:35:07
From what you're used to.
571
00:35:07 --> 00:35:10
It happens that y is
not a function of x.
572
00:35:10 --> 00:35:11
In this case.
573
00:35:11 --> 00:35:11
Yes.
574
00:35:11 --> 00:35:16
STUDENT: [INAUDIBLE]
575
00:35:16 --> 00:35:17
PROFESSOR: Yes.
576
00:35:17 --> 00:35:23
STUDENT: [INAUDIBLE]
577
00:35:23 --> 00:35:26
PROFESSOR: The question is,
because I'm rotating around
578
00:35:26 --> 00:35:30
the z axis, doesn't x change
exactly as much as y does.
579
00:35:30 --> 00:35:33
What happens is that x and
y are symmetric variables.
580
00:35:33 --> 00:35:36
They can be treated equally.
581
00:35:36 --> 00:35:41
But if I decide to take slices
with respect to y being fixed
582
00:35:41 --> 00:35:45
and x varying, then of course
they're now separated, and I
583
00:35:45 --> 00:35:48
have a separate role for the x
and a separate role for the y.
584
00:35:48 --> 00:35:50
Or if I'd sliced it the
other way, I would have
585
00:35:50 --> 00:35:51
gotten the same answer.
586
00:35:51 --> 00:35:54
I just would have reversed
the roles of x and y.
587
00:35:54 --> 00:35:57
So what's happening is
that x and y are on equal
588
00:35:57 --> 00:35:59
footing with each other.
589
00:35:59 --> 00:36:02
In this picture, and I could've
sliced the other way.
590
00:36:02 --> 00:36:04
It would have gotten
the same answer.
591
00:36:04 --> 00:36:07
That's more or less the
answer to your question.
592
00:36:07 --> 00:36:13
OK.
593
00:36:13 --> 00:36:17
Now I have given you a review
sheet, and I want to run
594
00:36:17 --> 00:36:23
through, briefly, what's
going to be on the exam.
595
00:36:23 --> 00:36:28
And this list of exam
questions is what's
596
00:36:28 --> 00:36:30
going to be on the exam.
597
00:36:30 --> 00:36:33
There are, sorry this is
not displayed correctly.
598
00:36:33 --> 00:36:35
So, exam questions, but now
I'm just going to show
599
00:36:35 --> 00:36:36
you what they are.
600
00:36:36 --> 00:36:38
There are five
questions on the exam.
601
00:36:38 --> 00:36:46
They are completely parallel
to what you got last year.
602
00:36:46 --> 00:36:48
So you should look
at that test.
603
00:36:48 --> 00:36:50
It's worth looking at.
604
00:36:50 --> 00:36:54
And you'll see in the
descriptions on this sheet
605
00:36:54 --> 00:37:00
that what I'm describing
is what's on that test.
606
00:37:00 --> 00:37:04
So what's going to happen is,
and this is also posted on the
607
00:37:04 --> 00:37:10
Web, is that you'll be expected
to calculate some definite
608
00:37:10 --> 00:37:13
integrals using the fundamental
theorem of calculus.
609
00:37:13 --> 00:37:16
Do a numerical approximation.
610
00:37:16 --> 00:37:17
There'll be a Riemann,
a trapezoidal rule
611
00:37:17 --> 00:37:19
and a Simpson's Rule.
612
00:37:19 --> 00:37:22
Calculate areas and volumes.
613
00:37:22 --> 00:37:25
And then some other
cumulative sum.
614
00:37:25 --> 00:37:29
Either an average value or
probability or perhaps work.
615
00:37:29 --> 00:37:33
And sketch a function
which is given in this
616
00:37:33 --> 00:37:37
form as an integral.
617
00:37:37 --> 00:37:42
So those are the questions,
and you'll see by the
618
00:37:42 --> 00:37:46
example of last year's
exam exactly the style.
619
00:37:46 --> 00:37:48
They're really going
to be very similar.
620
00:37:48 --> 00:37:49
Yes, question.
621
00:37:49 --> 00:38:01
STUDENT: [INAUDIBLE]
622
00:38:01 --> 00:38:02
PROFESSOR: OK, good question.
623
00:38:02 --> 00:38:06
So the question is, for Riemann
sums, what's the difference
624
00:38:06 --> 00:38:19
between upper and lower,
and right and left?
625
00:38:19 --> 00:38:25
So here we have a Riemann sum.
626
00:38:25 --> 00:38:30
And I'm going to give you a
picture which is, maybe this
627
00:38:30 --> 00:38:33
function y = 1 / x, which was
the one that we were
628
00:38:33 --> 00:38:37
discussing earlier.
629
00:38:37 --> 00:38:41
If you take the function y = 1
/ x and you break it up into
630
00:38:41 --> 00:38:45
pieces here, however it doesn't
matter how many pieces, let's
631
00:38:45 --> 00:38:48
just say there are
four of them.
632
00:38:48 --> 00:38:53
Then the lower Riemann sum
is the staircase which
633
00:38:53 --> 00:38:55
fits underneath.
634
00:38:55 --> 00:39:01
So this one is a picture
of the lower sum.
635
00:39:01 --> 00:39:03
It's always less.
636
00:39:03 --> 00:39:09
And in the case of a decreasing
function, it's going to be, so
637
00:39:09 --> 00:39:20
since if you like, since 1 / x
decreases, the lower sum
638
00:39:20 --> 00:39:27
equals the right sum.
639
00:39:27 --> 00:39:29
You can see that visually
on this picture.
640
00:39:29 --> 00:39:33
The values you're going to
select are going to be the
641
00:39:33 --> 00:39:38
right ends of the rectangles.
642
00:39:38 --> 00:39:40
The upper sum is the left one.
643
00:39:40 --> 00:39:44
Now, if the function wiggles up
and down, then you have to pick
644
00:39:44 --> 00:39:45
whichever side is appropriate.
645
00:39:45 --> 00:39:47
Or maybe it'll be a point in
the middle, if the maximum
646
00:39:47 --> 00:39:54
is achieved in the middle.
647
00:39:54 --> 00:39:55
Yeah, another question.
648
00:39:55 --> 00:40:02
STUDENT: [INAUDIBLE]
649
00:40:02 --> 00:40:03
PROFESSOR: Correct.
650
00:40:03 --> 00:40:09
If the function is increasing,
then the lower sum
651
00:40:09 --> 00:40:11
is the left sum.
652
00:40:11 --> 00:40:13
So it just exactly
reverses what's here.
653
00:40:13 --> 00:40:17
So this is decreasing, lower
sum is right-hand sum.
654
00:40:17 --> 00:40:24
Increasing, lower sum
is left-hand sum.
655
00:40:24 --> 00:40:26
STUDENT: [INAUDIBLE]
656
00:40:26 --> 00:40:26
PROFESSOR: Yes.
657
00:40:26 --> 00:40:44
STUDENT: [INAUDIBLE]
658
00:40:44 --> 00:40:45
PROFESSOR: Good question.
659
00:40:45 --> 00:40:47
Suppose you're faced with
a function like this
660
00:40:47 --> 00:40:48
in this last problem.
661
00:40:48 --> 00:40:52
Which, generally, these are
the trickiest problems.
662
00:40:52 --> 00:40:54
And the question is, how are
you ever going to be able to
663
00:40:54 --> 00:40:58
decide on an asymptote, even
whether there is an asymptote.
664
00:40:58 --> 00:41:01
And the answer is, you're not.
665
00:41:01 --> 00:41:05
It's going to be pretty
tricky to get keep track
666
00:41:05 --> 00:41:07
of what's happening as
it goes to infinity.
667
00:41:07 --> 00:41:09
We had an example on the
homework where is was
668
00:41:09 --> 00:41:12
oscillating and it's very
unclear what's going on.
669
00:41:12 --> 00:41:15
You have to do a very
long analysis for that.
670
00:41:15 --> 00:41:21
So in fact, just don't
worry about that now.
671
00:41:21 --> 00:41:23
At the very end of the class,
we'll talk a little bit
672
00:41:23 --> 00:41:24
about these asymptotes.
673
00:41:24 --> 00:41:27
And really, the first issue is
whether they exist or not.
674
00:41:27 --> 00:41:30
And that's even something.
675
00:41:30 --> 00:41:32
That's a serious question
which we'll address at the
676
00:41:32 --> 00:41:33
very end of this course.
677
00:41:33 --> 00:41:36
STUDENT: [INAUDIBLE]
678
00:41:36 --> 00:41:36
PROFESSOR: That's right.
679
00:41:36 --> 00:41:40
It's not going to be
anything that complicated.
680
00:41:40 --> 00:41:42
Other questions?
681
00:41:42 --> 00:41:44
We we still have a five whole
minutes, and I have an example
682
00:41:44 --> 00:41:50
to give, if nobody
has a question.
683
00:41:50 --> 00:41:50
Yeah.
684
00:41:50 --> 00:41:51
STUDENT: [INAUDIBLE]
685
00:41:51 --> 00:41:56
PROFESSOR: The question,
uh, will I tell you which
686
00:41:56 --> 00:42:00
one of what to use?
687
00:42:00 --> 00:42:00
STUDENT: [INAUDIBLE]
688
00:42:00 --> 00:42:02
PROFESSOR: When I tell you the
numeric approximation is,
689
00:42:02 --> 00:42:04
you'll see on the exam.
690
00:42:04 --> 00:42:06
The practice exam
that you have.
691
00:42:06 --> 00:42:09
I will ask you for all three.
692
00:42:09 --> 00:42:11
I will ask you for the Riemann
sum, the trapezoidal rule,
693
00:42:11 --> 00:42:12
and the Simpson's rule.
694
00:42:12 --> 00:42:16
I'm guaranteeing you they'll
all three be on the exam.
695
00:42:16 --> 00:42:18
I'm guaranteeing that every
single thing which is on that
696
00:42:18 --> 00:42:20
piece of paper is on the exam.
697
00:42:20 --> 00:42:23
And you'll see it on the
exam that you've got.
698
00:42:23 --> 00:42:27
It's exactly parallel
to what's there.
699
00:42:27 --> 00:42:31
STUDENT: [INAUDIBLE]
700
00:42:31 --> 00:42:34
PROFESSOR: So with areas and
volume, the question is will I
701
00:42:34 --> 00:42:36
tell you which method to use.
702
00:42:36 --> 00:42:48
So let's discuss that.
703
00:42:48 --> 00:42:58
So with areas and volumes,
there's basically, so this
704
00:42:58 --> 00:43:00
is always true with areas.
705
00:43:00 --> 00:43:07
And it's true with
volumes of revolution.
706
00:43:07 --> 00:43:08
By the way you should
read this sheet.
707
00:43:08 --> 00:43:12
Not everything that's on
here is, have I said.
708
00:43:12 --> 00:43:15
But you should read it.
709
00:43:15 --> 00:43:16
Because it's all relevant.
710
00:43:16 --> 00:43:19
So with volumes of revolution,
you always work your way
711
00:43:19 --> 00:43:25
back to some 2-D diagram.
712
00:43:25 --> 00:43:31
So there's some 2-D diagram
which is always two-dimensional
713
00:43:31 --> 00:43:34
diagram, which is always
connected with these problems.
714
00:43:34 --> 00:43:37
I mean, something this hard
is really just too hard
715
00:43:37 --> 00:43:39
to do on an exam, right?
716
00:43:39 --> 00:43:40
I mean, I'm not going to
ask you something this
717
00:43:40 --> 00:43:42
complicated on the exam.
718
00:43:42 --> 00:43:44
Because this involves
a three-dimensional
719
00:43:44 --> 00:43:47
visualization.
720
00:43:47 --> 00:43:50
But once you're down to
2-D, you're supposed to
721
00:43:50 --> 00:43:52
be able to handle it.
722
00:43:52 --> 00:43:54
Now, what's the main
issue after you've
723
00:43:54 --> 00:43:56
got your 2-D diagram?
724
00:43:56 --> 00:43:59
The main issue is, do you
want to integrate with
725
00:43:59 --> 00:44:06
respect to dx or dy?
726
00:44:06 --> 00:44:13
And the answer is
that it will depend.
727
00:44:13 --> 00:44:18
And if there's one that's going
to cause you incredible
728
00:44:18 --> 00:44:23
difficulty, and I feel that
you're not able to dodge it,
729
00:44:23 --> 00:44:26
then I might give you a hint
and say you'd better use
730
00:44:26 --> 00:44:30
shells, or you'd better
use disks or washers or
731
00:44:30 --> 00:44:31
something like that.
732
00:44:31 --> 00:44:35
But if I feel that you're grown
up enough to figure out which
733
00:44:35 --> 00:44:37
one it is, because one of them
is so ridiculous you say
734
00:44:37 --> 00:44:39
forget it, immediately.
735
00:44:39 --> 00:44:40
After thinking about it.
736
00:44:40 --> 00:44:43
Then I won't tell
you which one.
737
00:44:43 --> 00:44:45
Because I figure, in other
words, I don't want you
738
00:44:45 --> 00:44:47
to waste your time.
739
00:44:47 --> 00:44:50
But I'm willing to waste a
minute or two of your time
740
00:44:50 --> 00:44:57
on a wild goose chase.
741
00:44:57 --> 00:45:01
So let me give you
an example of this.
742
00:45:01 --> 00:45:14
Suppose you're looking at
the curve y < 0 < - x^3.
743
00:45:14 --> 00:45:17
So this is some kind of lump.
744
00:45:17 --> 00:45:18
Like this.
745
00:45:18 --> 00:45:22
It goes from 0 to 1, because
the right-hand side
746
00:45:22 --> 00:45:26
is 0 at 0 and 1 here.
747
00:45:26 --> 00:45:26
It's some kind of thing.
748
00:45:26 --> 00:45:29
And there are these
two possibilities.
749
00:45:29 --> 00:45:33
One of them is to do shells.
750
00:45:33 --> 00:45:36
And then, so this is
supposed to be rotated
751
00:45:36 --> 00:45:41
around the y axis.
752
00:45:41 --> 00:45:45
In this case.
753
00:45:45 --> 00:45:49
And the same would apply,
actually, to the area problem.
754
00:45:49 --> 00:45:50
So I'm doing a slightly
more complicated problem.
755
00:45:50 --> 00:45:52
But you could ask for the
area underneath this.
756
00:45:52 --> 00:45:53
So forth.
757
00:45:53 --> 00:45:54
OK.
758
00:45:54 --> 00:45:57
So we can integrate this dx,
or we can integrate this dy.
759
00:45:57 --> 00:46:00
This indicates that I'm
deciding that this is going
760
00:46:00 --> 00:46:02
to be of thickness dx,
and I'm integrating dx.
761
00:46:02 --> 00:46:04
So that's a choice
that I'm making.
762
00:46:04 --> 00:46:08
Now, the minute I made
that choice I know
763
00:46:08 --> 00:46:10
that these are shells.
764
00:46:10 --> 00:46:12
Because they sweep around this
way and that makes them shells.
765
00:46:12 --> 00:46:15
Cylindrical shells.
766
00:46:15 --> 00:46:19
And if I do that,
the setup is this.
767
00:46:19 --> 00:46:25
It's 2 pi x ( x - x ^3) dx.
768
00:46:25 --> 00:46:28
Now, I claim that when you
get to this point, you
769
00:46:28 --> 00:46:30
already know you've won.
770
00:46:30 --> 00:46:32
Because this is an easy
integral to calculate.
771
00:46:32 --> 00:46:34
So you're done here.
772
00:46:34 --> 00:46:37
You're happy.
773
00:46:37 --> 00:46:42
Now, if you happened to say,
oh gee, I hate to do this.
774
00:46:42 --> 00:46:48
I want to do something clever,
you could try to do it
775
00:46:48 --> 00:46:54
with cutting this way.
776
00:46:54 --> 00:46:56
Let's do this.
777
00:46:56 --> 00:47:01
And this would be
the dy thickness.
778
00:47:01 --> 00:47:05
And then when you sweep
this around, you get
779
00:47:05 --> 00:47:10
what we call a washer.
780
00:47:10 --> 00:47:14
Which is really just the
difference of two disks.
781
00:47:14 --> 00:47:24
So the shape here is this
thing swung around this axis.
782
00:47:24 --> 00:47:26
And it looks like this.
783
00:47:26 --> 00:47:29
So it's going to be the
difference of radii.
784
00:47:29 --> 00:47:32
So what's the formula for this?
785
00:47:32 --> 00:47:37
It's some integral of pi times
the right end, which I'll
786
00:47:37 --> 00:47:40
call x2, and here the left
end, which I'll call x1.
787
00:47:40 --> 00:47:47
So this is pi (x2
^2 - x1 ^2) dy.
788
00:47:47 --> 00:47:50
Now, already at this stage,
you think to yourself
789
00:47:50 --> 00:47:54
this is more complicated
than the other method.
790
00:47:54 --> 00:47:56
So you've already abandoned it.
791
00:47:56 --> 00:47:59
But I'm just going to go one
step further into this one
792
00:47:59 --> 00:48:01
to see what it is
that's happening.
793
00:48:01 --> 00:48:08
If you try to figure out what
these values x1 and x2 are,
794
00:48:08 --> 00:48:14
that corresponds to solving
for x1 and x2 in terms of y.
795
00:48:14 --> 00:48:21
So that's the following
equation. x1 and x2 solve
796
00:48:21 --> 00:48:32
the equation that the
curve, x - x^3 = y.
797
00:48:32 --> 00:48:33
Now, look at this equation.
798
00:48:33 --> 00:48:42
That's the equation x ^3 -
sorry, x ^3 - x + y, I guess.
799
00:48:42 --> 00:48:43
Let's see.
800
00:48:43 --> 00:48:46
Yeah, that's right,
is equal to 0.
801
00:48:46 --> 00:48:53
This is a cubic equation.
802
00:48:53 --> 00:48:55
Although there is a
formula for this.
803
00:48:55 --> 00:48:57
You've never been taught the
formula for this equation.
804
00:48:57 --> 00:49:00
So therefore, you will never,
ever be able to get a formula
805
00:49:00 --> 00:49:02
for x2 and x1 as
a function of y.
806
00:49:02 --> 00:49:05
And you'll never be able
to compute this one.
807
00:49:05 --> 00:49:08
This is more than just a dead
end, it's like crash, burn,
808
00:49:08 --> 00:49:11
and, you know self-destruct.
809
00:49:11 --> 00:49:14
So there may be such a
thing, so do the other way.
810
00:49:14 --> 00:49:17
Good luck, folks.
811
00:49:17 --> 00:49:18