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All right, so the past few
weeks,
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00:00:23 --> 00:00:27
we've been looking at double
integrals and the plane,
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00:00:27 --> 00:00:31
line integrals in the plane,
and will we are going to do now
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00:00:31 --> 00:00:34
from now on basically until the
end of the term,
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will be very similar stuff,
but in space.
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00:00:36 --> 00:00:41
So, we are going to learn how
to do triple integrals in space,
13
00:00:41 --> 00:00:43
flux in space,
work in space,
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00:00:43 --> 00:00:46
divergence, curl,
all that.
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00:00:46 --> 00:00:49
So,
that means, basically,
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00:00:49 --> 00:00:52
if you were really on top of
what we've been doing these past
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00:00:52 --> 00:00:55
few weeks,
then it will be just the same
18
00:00:55 --> 00:00:58
with one more coordinate.
And, you will see there are
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00:00:58 --> 00:01:00
some differences.
But, conceptually,
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00:01:00 --> 00:01:04
it's pretty similar.
There are a few tricky things,
21
00:01:04 --> 00:01:06
though.
Now, that also means that if
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00:01:06 --> 00:01:10
there is stuff that you are not
sure about in the plane,
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00:01:10 --> 00:01:14
then I encourage you to review
the material that we've done
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00:01:14 --> 00:01:18
over the past few weeks to make
sure that everything in the
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00:01:18 --> 00:01:22
plane is completely clear to you
because it will be much harder
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00:01:22 --> 00:01:26
to understand stuff in space if
things are still shaky in the
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00:01:26 --> 00:01:30
plane.
OK, so the plan is we're going
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00:01:30 --> 00:01:36
to basically go through the same
stuff, but in space.
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00:01:36 --> 00:01:45
So, it shouldn't be surprising
that we will start today with
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00:01:45 --> 00:01:52
triple integrals.
OK, so the way triple integrals
31
00:01:52 --> 00:01:58
work is if I give you a function
of three variables,
32
00:01:58 --> 00:02:02
x, y, z,
and I give you some region in
33
00:02:02 --> 00:02:07
space,
so, some solid,
34
00:02:07 --> 00:02:15
then I can take the integral
over this region over function f
35
00:02:15 --> 00:02:20
dV where dV stands for the
volume element.
36
00:02:20 --> 00:02:24
OK, so what it means is we will
just take every single little
37
00:02:24 --> 00:02:28
piece of our solid,
take the value of f there,
38
00:02:28 --> 00:02:30
multiply by the small volume of
each little piece,
39
00:02:30 --> 00:02:35
and sum all these things
together.
40
00:02:35 --> 00:02:40
And,
so this volume element here,
41
00:02:40 --> 00:02:44
well, for example,
if you are doing the integral
42
00:02:44 --> 00:02:48
in rectangular coordinates,
that will become dx dy dz or
43
00:02:48 --> 00:02:54
any permutation of that because,
of course, we have lots of
44
00:02:54 --> 00:02:59
possible orders of integration
to choose from.
45
00:02:59 --> 00:03:08
So, rather than bore you with
theory and all sorts of
46
00:03:08 --> 00:03:15
complicated things,
let's just do examples.
47
00:03:15 --> 00:03:18
And, you will see, basically,
if you understand how to set up
48
00:03:18 --> 00:03:21
iterated integrals into
variables,
49
00:03:21 --> 00:03:23
that you basically understand
how to do them in three
50
00:03:23 --> 00:03:27
variables.
You just have to be a bit more
51
00:03:27 --> 00:03:30
careful.
And, there's one more step.
52
00:03:30 --> 00:03:35
OK, so let's take our first
triple integral to be on the
53
00:03:35 --> 00:03:36
region.
So, of course,
54
00:03:36 --> 00:03:37
there's two different things as
always.
55
00:03:37 --> 00:03:39
There is the region of
integration and there's the
56
00:03:39 --> 00:03:42
function we are integrating.
Now, the function we are
57
00:03:42 --> 00:03:44
integrating, well,
it will come in handy when you
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00:03:44 --> 00:03:46
actually try to evaluate the
integral.
59
00:03:46 --> 00:03:49
But, as you can see,
probably, the new part is
60
00:03:49 --> 00:03:52
really hard to set it up.
So, the function won't really
61
00:03:52 --> 00:03:55
matter that much for me.
So, in the examples I'll do
62
00:03:55 --> 00:03:58
today, functions will be kind of
silly.
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00:03:58 --> 00:04:04
So, for example,
let's say that we want to look
64
00:04:04 --> 00:04:13
at the region between two
paraboloids, one given by z = x
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00:04:13 --> 00:04:20
^2 y ^2.
The other is z = 4 - x ^2 - y
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00:04:20 --> 00:04:22
^2.
And, so, I haven't given you,
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00:04:22 --> 00:04:26
yet, the function to integrate.
OK, this is not the function to
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00:04:26 --> 00:04:28
integrate.
This is what describes the
69
00:04:28 --> 00:04:32
region where I will integrate my
function.
70
00:04:32 --> 00:04:38
And, let's say that I just want
to find the volume of this
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00:04:38 --> 00:04:43
region, which is the triple
integral of just one dV.
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00:04:43 --> 00:04:46
OK, similarly,
remember, when we try to find
73
00:04:46 --> 00:04:49
the area of the region in the
plane, we are just integrating
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00:04:49 --> 00:04:51
one dA.
Here we integrate one dV.
75
00:04:51 --> 00:04:55
that will give us the volume.
Now, I know that you can
76
00:04:55 --> 00:04:59
imagine how to actually do this
one as a double integral.
77
00:04:59 --> 00:05:02
But, the goal of the game is to
set up the triple integral.
78
00:05:02 --> 00:05:05
It's not actually to find the
volume.
79
00:05:05 --> 00:05:12
So, what does that look like?
Well, z = x ^2 y ^2,
80
00:05:12 --> 00:05:16
that's one of our favorite
paraboloids.
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00:05:16 --> 00:05:22
That's something that looks
like a parabola with its bottom
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00:05:22 --> 00:05:28
at the origin that you spin
about the z axis.
83
00:05:28 --> 00:05:32
And, z equals four minus x
squared minus y squared,
84
00:05:32 --> 00:05:36
well, that's also a paraboloid.
But, this one is pointing down,
85
00:05:36 --> 00:05:40
and when you take x equals y
equals zero, you get z equals
86
00:05:40 --> 00:05:44
four.
So, it starts at four,
87
00:05:44 --> 00:05:52
and it goes down like that.
OK, so the solid that we'd like
88
00:05:52 --> 00:05:56
to consider is what's in between
in here.
89
00:05:56 --> 00:06:00
So, it has a curvy top which is
this downward paraboloid,
90
00:06:00 --> 00:06:04
a curvy bottom which is the
other paraboloid.
91
00:06:04 --> 00:06:08
And, what about the sides?
Well, do you have any idea what
92
00:06:08 --> 00:06:11
we get here?
Yeah, it's going to be a circle
93
00:06:11 --> 00:06:15
because entire picture is
invariant by rotation about the
94
00:06:15 --> 00:06:17
z axis.
So, if you look at the picture
95
00:06:17 --> 00:06:20
just, say, in the yz plane,
you get this point and that
96
00:06:20 --> 00:06:24
point.
And, when you rotate everything
97
00:06:24 --> 00:06:30
around the z axis,
you will just get a circle
98
00:06:30 --> 00:06:33
here.
OK, so our goal is to find the
99
00:06:33 --> 00:06:36
volume of this thing,
and there's lots of things I
100
00:06:36 --> 00:06:38
could do to simplify the
calculation,
101
00:06:38 --> 00:06:41
or even not do it as a triple
integral at all.
102
00:06:41 --> 00:06:46
But, I want to actually set it
up as a triple integral just to
103
00:06:46 --> 00:06:50
show how we do that.
OK, so the first thing we need
104
00:06:50 --> 00:06:54
to do is choose an order of
integration.
105
00:06:54 --> 00:06:56
And, here, well,
I don't know if you can see it
106
00:06:56 --> 00:07:00
yet, but hopefully soon that
will be intuitive to you.
107
00:07:00 --> 00:07:04
I claim that I would like to
start by integrating first over
108
00:07:04 --> 00:07:05
z.
What's the reason for that?
109
00:07:05 --> 00:07:09
Well, the reason is if I give
you x and y, then you can find
110
00:07:09 --> 00:07:13
quickly, what's the bottom and
top values of z for that choice
111
00:07:13 --> 00:07:18
of x and y?
OK, so if I have x and y given,
112
00:07:18 --> 00:07:25
then I can find above that:
what is the bottom z and the
113
00:07:25 --> 00:07:33
top z corresponding to the
vertical line above that point?
114
00:07:33 --> 00:07:39
The portion of it that's inside
our solid, so somehow,
115
00:07:39 --> 00:07:45
there's a bottom z and a top z.
And, so the top z is actually
116
00:07:45 --> 00:07:49
on the downward paraboloid.
So, it's four minus x squared
117
00:07:49 --> 00:07:52
minus y squared.
The bottom value of z is x
118
00:07:52 --> 00:07:58
squared plus y squared.
OK, so if I want to start to
119
00:07:58 --> 00:08:04
set this up, I will write the
triple integral.
120
00:08:04 --> 00:08:09
And then, so let's say I'm
going to do it dz first,
121
00:08:09 --> 00:08:11
and then, say,
dy dx.
122
00:08:11 --> 00:08:16
It doesn't really matter.
So then, for a given value of x
123
00:08:16 --> 00:08:19
and y, I claim z goes from the
bottom surface.
124
00:08:19 --> 00:08:23
The bottom face is z equals x
squared plus y squared.
125
00:08:23 --> 00:08:29
The top face is four minus x
squared minus y squared.
126
00:08:29 --> 00:08:35
OK, is that OK with everyone?
Yeah?
127
00:08:35 --> 00:08:43
Any questions so far?
Yes?
128
00:08:43 --> 00:08:45
Why did I start with z?
That's a very good question.
129
00:08:45 --> 00:08:49
So, I can choose whatever order
I want, but let's say I did x
130
00:08:49 --> 00:08:50
first .
Then, to find the inner
131
00:08:50 --> 00:08:53
integral bounds,
I would need to say, OK,
132
00:08:53 --> 00:08:56
I've chosen values of,
see, in the inner integral,
133
00:08:56 --> 00:08:59
you've fixed the two other
variables,
134
00:08:59 --> 00:09:01
and you're just going to vary
that one.
135
00:09:01 --> 00:09:02
And, you need to find bounds
for it.
136
00:09:02 --> 00:09:05
So, if I integrate over x
first, I have to solve,
137
00:09:05 --> 00:09:10
answer the following question.
Say I'm given values of y and z.
138
00:09:10 --> 00:09:14
What are the bounds for x?
So, that would mean I'm slicing
139
00:09:14 --> 00:09:18
my solid by lines that are
parallel to the x axis.
140
00:09:18 --> 00:09:21
And, see, it's kind of hard to
find, what are the values of x
141
00:09:21 --> 00:09:24
at the front and at the back?
I mean, it's possible,
142
00:09:24 --> 00:09:27
but it's easier to actually
first look for z at the top and
143
00:09:27 --> 00:09:33
bottom.
Yes?
144
00:09:33 --> 00:09:36
dy dx, or dx dy?
No, it's completely at random.
145
00:09:36 --> 00:09:39
I mean, you can see x and y
play symmetric roles.
146
00:09:39 --> 00:09:43
So, if you look at it,
it's reasonably clear that z
147
00:09:43 --> 00:09:49
should be the easiest one to set
up first for what comes next.
148
00:09:49 --> 00:09:54
xy or yx, it's the same.
Yes?
149
00:09:54 --> 00:09:56
Yes, it will be easier to use
cylindrical coordinates.
150
00:09:56 --> 00:10:03
I'll get to that just as soon
as I'm done with this one.
151
00:10:03 --> 00:10:07
OK, so let's continue a bit
with that.
152
00:10:07 --> 00:10:11
And, as you mentioned,
actually we don't actually want
153
00:10:11 --> 00:10:14
to do it with xy in the end.
In a few minutes,
154
00:10:14 --> 00:10:16
we will actually switch to
cylindrical coordinates.
155
00:10:16 --> 00:10:18
But, for now,
we don't even know what they
156
00:10:18 --> 00:10:20
are.
OK, so I've done the inner
157
00:10:20 --> 00:10:25
integral by looking at,
you know, if I slice by
158
00:10:25 --> 00:10:28
vertical lines,
what is the top?
159
00:10:28 --> 00:10:31
What is the bottom for a given
value of x and y?
160
00:10:31 --> 00:10:36
So, the bounds in the inner
integral depend on both the
161
00:10:36 --> 00:10:41
middle and outer variables.
Next, I need to figure out what
162
00:10:41 --> 00:10:44
values of x and y I will be
interested in.
163
00:10:44 --> 00:10:47
And, the answer for that is,
well, the values of x and y
164
00:10:47 --> 00:10:51
that I want to look at are all
those that are in the shade of
165
00:10:51 --> 00:10:53
my region.
So, in fact,
166
00:10:53 --> 00:10:57
to set up the middle and outer
bounds, what I want to do is
167
00:10:57 --> 00:11:04
project my solid.
So, my solid looks like this
168
00:11:04 --> 00:11:09
kind of thing.
And, I don't really know how to
169
00:11:09 --> 00:11:13
call it.
But, what's interesting now is
170
00:11:13 --> 00:11:18
I want to look at the shadow
that it casts in the xy plane.
171
00:11:18 --> 00:11:22
OK, and, of course,
that shadow will just be the
172
00:11:22 --> 00:11:27
disk that's directly below this
disk here that's separating the
173
00:11:27 --> 00:11:34
two halves of the solid.
And so, now I will want to
174
00:11:34 --> 00:11:41
integrate over,
I want to look at all the xy's,
175
00:11:41 --> 00:11:46
x and y, in the shadow.
So, now I'm left with,
176
00:11:46 --> 00:11:48
actually, something we've
already done,
177
00:11:48 --> 00:11:52
namely setting up a double
integral over x and y.
178
00:11:52 --> 00:11:55
So, if it helps,
here, we don't strictly need
179
00:11:55 --> 00:11:59
it, but if it helps,
it could be useful to actually
180
00:11:59 --> 00:12:03
draw a picture of this shadow in
the xy plane.
181
00:12:03 --> 00:12:14
So, here it would just look,
again, like a disk,
182
00:12:14 --> 00:12:18
and set it up.
Now, the question is,
183
00:12:18 --> 00:12:22
how do we find the size of this
disk, the size of the shadow?
184
00:12:22 --> 00:12:28
Well, basically we have to
figure out where our two
185
00:12:28 --> 00:12:37
paraboloids intersect.
There's nothing else.
186
00:12:37 --> 00:12:49
OK, so, one way how to find the
shadow in the xy plane -- --
187
00:12:49 --> 00:12:53
well,
here we actually know the
188
00:12:53 --> 00:12:56
answer a priori,
but even if we didn't,
189
00:12:56 --> 00:12:59
we could just say,
well, our region lives wherever
190
00:12:59 --> 00:13:03
the bottom surface is below the
top surface,
191
00:13:03 --> 00:13:11
OK, so we want to look at
things wherever bottom value of
192
00:13:11 --> 00:13:15
z is less than the top value of
z,
193
00:13:15 --> 00:13:18
I mean, less or less than or
equal, that's the same thing.
194
00:13:18 --> 00:13:24
So, if the bottom value of z is
x squared plus y squared should
195
00:13:24 --> 00:13:28
be less than four minus x
squared minus y squared,
196
00:13:28 --> 00:13:33
and if you solve for that,
then you will get,
197
00:13:33 --> 00:13:34
well, so let's move these guys
over here.
198
00:13:34 --> 00:13:37
You'll get two x squared plus
two y squared less than four.
199
00:13:37 --> 00:13:42
That becomes x squared plus y
squared less than two.
200
00:13:42 --> 00:13:52
So, that means that's a disk of
radius square root of two,
201
00:13:52 --> 00:13:56
OK?
So, we kind of knew in advance
202
00:13:56 --> 00:14:01
it was going to be a disk,
but what we've learned now is
203
00:14:01 --> 00:14:05
that this radius is square root
of two.
204
00:14:05 --> 00:14:08
So, if we want to set up,
if we really want to set it up
205
00:14:08 --> 00:14:13
using dy dx like they started,
then we can do it because we
206
00:14:13 --> 00:14:16
know,
so, for the middle integral,
207
00:14:16 --> 00:14:19
now,
we want to fix a value of x.
208
00:14:19 --> 00:14:21
And, for that fixed value of x,
we want to figure out the
209
00:14:21 --> 00:14:25
bounds for y.
Well, the answer is y goes from
210
00:14:25 --> 00:14:26
here to here.
What's here?
211
00:14:26 --> 00:14:31
Well, here, y is square root of
two minus x squared.
212
00:14:31 --> 00:14:36
And, here it's negative square
root of two minus x squared.
213
00:14:36 --> 00:14:40
So, y will go from negative
square root of two minus x
214
00:14:40 --> 00:14:46
squared to positive square root.
And then, x will go from
215
00:14:46 --> 00:14:52
negative root two to root two.
OK, if that's not completely
216
00:14:52 --> 00:14:55
clear to you,
then I encourage you to go over
217
00:14:55 --> 00:14:58
how we set up double integrals
again.
218
00:14:58 --> 00:15:02
OK, does that make sense,
kind of?
219
00:15:02 --> 00:15:17
Yeah?
Well, so, when we set up,
220
00:15:17 --> 00:15:20
remember, we are setting up a
double integral,
221
00:15:20 --> 00:15:23
dy dx here.
So, when we do it dy dx,
222
00:15:23 --> 00:15:27
it means we slice this region
of a plane by vertical line
223
00:15:27 --> 00:15:30
segments.
So, this middle guy would be
224
00:15:30 --> 00:15:33
what used to be the inner
integral.
225
00:15:33 --> 00:15:36
So, in the inner,
remember, you fix the value of
226
00:15:36 --> 00:15:39
x, and you ask yourself,
what is the range of values of
227
00:15:39 --> 00:15:43
y in my region?
So, y goes from here to here,
228
00:15:43 --> 00:15:46
and what here and here are
depends on the value of x.
229
00:15:46 --> 00:15:48
How?
Well, we have to find the
230
00:15:48 --> 00:15:50
relation between x and y at
these points.
231
00:15:50 --> 00:15:53
These points are on the circle
of radius root two.
232
00:15:53 --> 00:15:56
So, if you want this circle
maybe I should have written,
233
00:15:56 --> 00:15:58
is x squared plus y squared
equals two.
234
00:15:58 --> 00:16:03
And, if you solve for y,
given x, you get plus minus
235
00:16:03 --> 00:16:07
root of two minus x squared,
OK?
236
00:16:07 --> 00:16:11
Yes?
Is there a way to compute this
237
00:16:11 --> 00:16:13
with symmetry?
Well, certainly,
238
00:16:13 --> 00:16:15
yeah, this solid looks
sufficiently symmetric,
239
00:16:15 --> 00:16:17
but actually you could
certainly,
240
00:16:17 --> 00:16:19
if you don't want to do the
whole disk,
241
00:16:19 --> 00:16:21
you could just do quarter
disks,
242
00:16:21 --> 00:16:25
and multiply by four.
You could even just look at the
243
00:16:25 --> 00:16:29
lower half of the solid,
and multiply them by two,
244
00:16:29 --> 00:16:33
so, total by eight.
So, yeah, certainly there's
245
00:16:33 --> 00:16:36
lots of ways to make it slightly
easier by using symmetry.
246
00:16:36 --> 00:16:39
Now, the most spectacular way
to use symmetry here,
247
00:16:39 --> 00:16:41
of course, is to use that we
have this rotation symmetry and
248
00:16:41 --> 00:16:45
switch,
actually, not do this guy in xy
249
00:16:45 --> 00:16:50
coordinates but instead in polar
coordinates.
250
00:16:50 --> 00:17:08
So -- So, the smarter thing to
do would be to use polar
251
00:17:08 --> 00:17:21
coordinates instead of x and y.
Of course, we want to keep z.
252
00:17:21 --> 00:17:23
I mean, we are very happy with
z the way it is.
253
00:17:23 --> 00:17:28
But, we'll just change x and y
to R cos theta,
254
00:17:28 --> 00:17:31
R sine theta,
OK, because,
255
00:17:31 --> 00:17:37
well, let's see actually how we
would evaluate this guy.
256
00:17:37 --> 00:17:46
So, well actually, let's not.
It's kind of boring.
257
00:17:46 --> 00:17:50
So, let me just point out one
small thing here,
258
00:17:50 --> 00:17:54
sorry, before I do that.
So, if you start computing the
259
00:17:54 --> 00:17:59
inner integral,
OK, so let me not do that yet,
260
00:17:59 --> 00:18:03
sorry,
so if you try to compute the
261
00:18:03 --> 00:18:06
inner integral,
you'll be integrating from x
262
00:18:06 --> 00:18:11
squared plus y squared to four
minus x squared minus y squared
263
00:18:11 --> 00:18:15
dz.
Well, that will integrate to z
264
00:18:15 --> 00:18:22
between these two bounds.
So, you will get four minus two
265
00:18:22 --> 00:18:27
x squared minus two y squared.
Now, when you put that into the
266
00:18:27 --> 00:18:33
remaining ones,
you'll get something that's
267
00:18:33 --> 00:18:41
probably not very pleasant of
four minus two x squared minus
268
00:18:41 --> 00:18:49
two y squared dy dx.
And here, you see that to
269
00:18:49 --> 00:18:56
evaluate this,
you would switch to polar
270
00:18:56 --> 00:18:59
coordinates.
Oh, by the way,
271
00:18:59 --> 00:19:04
so if your initial instincts
had been to,
272
00:19:04 --> 00:19:06
given that you just want the
volume,
273
00:19:06 --> 00:19:09
you could also have found the
volume just by doing a double
274
00:19:09 --> 00:19:12
integral of the height between
the top and bottom.
275
00:19:12 --> 00:19:14
Well, you would just have
gotten this, right,
276
00:19:14 --> 00:19:17
because this is the height
between top and bottom.
277
00:19:17 --> 00:19:21
So, it's all the same.
It doesn't really matter.
278
00:19:21 --> 00:19:23
But with this,
of course, we will be able to
279
00:19:23 --> 00:19:26
integrate all sorts of
functions, not just one over the
280
00:19:26 --> 00:19:31
solid.
So, we will be able to do much
281
00:19:31 --> 00:19:35
more than just volumes.
OK, so let's see,
282
00:19:35 --> 00:19:37
how do we do it with polar
coordinates instead?
283
00:19:37 --> 00:19:53
Well, so -- Well,
that would become,
284
00:19:53 --> 00:20:02
so let's see.
So, I want to keep dz.
285
00:20:02 --> 00:20:10
But then, dx dy or dy dx would
become r dr d theta.
286
00:20:10 --> 00:20:13
And, if I try to set up the
bounds, well,
287
00:20:13 --> 00:20:17
I probably shouldn't keep this
x squared plus y squared around.
288
00:20:17 --> 00:20:20
But, x squared plus y squared
is easy in terms of r and theta.
289
00:20:20 --> 00:20:26
That's just r squared.
OK, I mean, in general I could
290
00:20:26 --> 00:20:28
have something that depends also
on theta.
291
00:20:28 --> 00:20:32
That's perfectly legitimate.
But here, it simplifies,
292
00:20:32 --> 00:20:36
and this guy up here,
four minus x squared minus y
293
00:20:36 --> 00:20:40
squared becomes four minus r
squared.
294
00:20:40 --> 00:20:43
And now, the integral that we
have to do over r and theta,
295
00:20:43 --> 00:20:45
well, we look again at the
shadow.
296
00:20:45 --> 00:20:48
The shadow is still a disk of
radius root two.
297
00:20:48 --> 00:20:52
That hasn't changed.
And now, we know how to set up
298
00:20:52 --> 00:20:54
this integral in polar
coordinates.
299
00:20:54 --> 00:21:01
r goes from zero to root two,
and theta goes from zero to two
300
00:21:01 --> 00:21:11
pi.
OK, and now it becomes actually
301
00:21:11 --> 00:21:20
easier to evaluate.
OK, so now we have actually a
302
00:21:20 --> 00:21:24
name for this because we're
doing it in space.
303
00:21:24 --> 00:21:28
So, these are called,
actually, cylindrical
304
00:21:28 --> 00:21:30
coordinates.
So, in fact,
305
00:21:30 --> 00:21:35
you already knew about
cylindrical coordinates even if
306
00:21:35 --> 00:21:40
you did not know the name.
OK, so the idea of cylindrical
307
00:21:40 --> 00:21:45
coordinates is that instead of
x, y, and z, to locate a point
308
00:21:45 --> 00:21:48
in space, you will use three
coordinates.
309
00:21:48 --> 00:22:00
One of them is basically how
high it is above the xy plane.
310
00:22:00 --> 00:22:04
So, that will be z.
And then, you will use polar
311
00:22:04 --> 00:22:08
coordinates for the projection
of your point on the xy plane.
312
00:22:08 --> 00:22:12
So, r will be the distance from
the z axis.
313
00:22:12 --> 00:22:17
And theta will be the angle
from the x axis
314
00:22:17 --> 00:22:21
counterclockwise.
So, the one thing to be careful
315
00:22:21 --> 00:22:24
about is because of the usual
convention, that we make the x
316
00:22:24 --> 00:22:27
axis point toward us.
Theta equals zero is no longer
317
00:22:27 --> 00:22:30
to the right.
Now, theta equals zero is to
318
00:22:30 --> 00:22:34
the front, and the angel is
measured from the front
319
00:22:34 --> 00:22:39
counterclockwise.
OK, so,
320
00:22:39 --> 00:22:41
and of course,
if you want to know how to
321
00:22:41 --> 00:22:44
convert between x,
y, z and r theta z,
322
00:22:44 --> 00:22:49
well, the formulas are just the
same as in usual polar
323
00:22:49 --> 00:22:52
coordinates.
R cos theta,
324
00:22:52 --> 00:22:56
r sine theta,
and z remain z.
325
00:22:56 --> 00:22:59
OK, so why are these called
cylindrical coordinates,
326
00:22:59 --> 00:23:02
by the way?
Well, let's say that I gave you
327
00:23:02 --> 00:23:07
the equation r equals a,
where a is some constant.
328
00:23:07 --> 00:23:12
Say r equals one, for example.
So, r equals one in 2D,
329
00:23:12 --> 00:23:15
that used to be just a circle
of radius one.
330
00:23:15 --> 00:23:19
Now, in space,
a single equation actually
331
00:23:19 --> 00:23:23
defines a surface,
not just a curve anymore.
332
00:23:23 --> 00:23:26
And, the set of points where r
is a, well, that's all the
333
00:23:26 --> 00:23:29
points that are distance a from
the z axis.
334
00:23:29 --> 00:23:34
So, in fact,
what you get this way is a
335
00:23:34 --> 00:23:41
cylinder of radius a centered on
the z axis.
336
00:23:41 --> 00:23:48
OK, so that's why they are
called cylindrical coordinates.
337
00:23:48 --> 00:23:51
By the way, so now,
similarly, if you look at the
338
00:23:51 --> 00:23:55
equation theta equals some given
value, well, so that used to be
339
00:23:55 --> 00:23:59
just a ray from the origin.
Now, that becomes a vertical
340
00:23:59 --> 00:24:01
half plane.
For example,
341
00:24:01 --> 00:24:04
if I set the value of theta and
let r and z vary,
342
00:24:04 --> 00:24:08
well, r is always positive,
but basically that means I am
343
00:24:08 --> 00:24:13
taking a vertical plane that
comes out in this direction.
344
00:24:13 --> 00:24:18
OK, any questions about
cylindrical coordinates?
345
00:24:18 --> 00:24:27
Yes?
Yeah, so I'm saying when you
346
00:24:27 --> 00:24:30
fix theta, you get only a half
plane, not a full plane.
347
00:24:30 --> 00:24:33
I mean, it goes all the way up
and down, but it doesn't go back
348
00:24:33 --> 00:24:35
to the other side of the z axis.
Why?
349
00:24:35 --> 00:24:39
That's because r is always
positive by convention.
350
00:24:39 --> 00:24:41
So, for example,
here, we say theta is zero.
351
00:24:41 --> 00:24:44
At the back, we say theta is pi.
We don't say theta is zero and
352
00:24:44 --> 00:24:47
r is negative.
We say r is positive and theta
353
00:24:47 --> 00:24:50
is pi.
It's a convention, largely.
354
00:24:50 --> 00:24:52
But, sticking with this
convention really will help you
355
00:24:52 --> 00:24:54
to set up the integrals
properly.
356
00:24:54 --> 00:24:58
I mean, otherwise there is just
too much risk for mistakes.
357
00:24:58 --> 00:25:08
Yes?
Well, so the question is if I
358
00:25:08 --> 00:25:11
were to use symmetry to do this
one, would I multiply by four or
359
00:25:11 --> 00:25:13
by two?
Well, it depends on how much
360
00:25:13 --> 00:25:16
symmetry you are using.
So, I mean, it's your choice.
361
00:25:16 --> 00:25:19
You can multiply by two,
by four, by eight depending on
362
00:25:19 --> 00:25:22
how much you cut it.
So, it depends on what symmetry
363
00:25:22 --> 00:25:24
you use, if you use symmetry
between top and bottom you'd
364
00:25:24 --> 00:25:27
say, well, the volume is twice
the lower half.
365
00:25:27 --> 00:25:30
If you use the left and right
half, you would say it's twice
366
00:25:30 --> 00:25:34
each half.
If you cut it into four pieces,
367
00:25:34 --> 00:25:37
and so on.
So, and again,
368
00:25:37 --> 00:25:41
you don't have to use the
symmetry.
369
00:25:41 --> 00:25:44
If you don't think of using
polar coordinates,
370
00:25:44 --> 00:25:46
then it can save you from
doing,
371
00:25:46 --> 00:25:47
you know, you can just start at
zero here and here,
372
00:25:47 --> 00:26:01
and simplify things a tiny bit.
But, OK, yes?
373
00:26:01 --> 00:26:04
So, to define a vertical full
plane, well, first of all it
374
00:26:04 --> 00:26:07
depends on whether it passes
through the z axis or not.
375
00:26:07 --> 00:26:09
If it doesn't,
then you'd have to remember how
376
00:26:09 --> 00:26:13
you do in polar coordinates.
I mean, basically the answer
377
00:26:13 --> 00:26:16
is, if you have a vertical
plane, so, it doesn't depend on
378
00:26:16 --> 00:26:18
z.
The equation does not involve z.
379
00:26:18 --> 00:26:21
It only involves r and theta.
And, how it involves r and
380
00:26:21 --> 00:26:24
theta is exactly the same as
when you do a line in polar
381
00:26:24 --> 00:26:27
coordinates in the plane.
So, if it's a line passing
382
00:26:27 --> 00:26:29
through the origin,
you say, well,
383
00:26:29 --> 00:26:32
theta is either some value or
the other one.
384
00:26:32 --> 00:26:33
If it's a line that doesn't
passes to the origin,
385
00:26:33 --> 00:26:38
but it's more tricky.
But hopefully you've seen how
386
00:26:38 --> 00:26:49
to do that.
OK, let's move on a bit.
387
00:26:49 --> 00:26:53
So, one thing to know,
I mean, basically,
388
00:26:53 --> 00:26:57
the important thing to remember
is that the volume element in
389
00:26:57 --> 00:27:05
cylindrical coordinates,
well, dx dy dz becomes r dr d
390
00:27:05 --> 00:27:08
theta dz.
And, that shouldn't be
391
00:27:08 --> 00:27:12
surprising because that's just
dx dy becomes r dr d theta.
392
00:27:12 --> 00:27:18
And, dz remains dz.
I mean, so, the way to think
393
00:27:18 --> 00:27:19
about it,
if you want,
394
00:27:19 --> 00:27:25
is that if you take a little
piece of solid in space,
395
00:27:25 --> 00:27:31
so it has some height, delta z,
and it has a base which has
396
00:27:31 --> 00:27:36
some area delta A,
then the small volume, delta v,
397
00:27:36 --> 00:27:41
is equal to the area of a base
times the height.
398
00:27:41 --> 00:27:43
So, now, when you make the
things infinitely small,
399
00:27:43 --> 00:27:51
you will get dV is dA times dz,
and you can use whichever
400
00:27:51 --> 00:27:56
formula you want for area in the
xy plane.
401
00:27:56 --> 00:28:00
OK, now in practice,
you choose which order you
402
00:28:00 --> 00:28:03
integrate in.
As you have probably seen,
403
00:28:03 --> 00:28:07
a favorite of mine is z first
because very often you'll know
404
00:28:07 --> 00:28:10
what the top and bottom of your
solid look like,
405
00:28:10 --> 00:28:13
and then you will reduce to
just something in the xy plane.
406
00:28:13 --> 00:28:18
But, there might be situations
where it's actually easier to
407
00:28:18 --> 00:28:22
start first with dx dy or r dr d
theta, and then save dz for
408
00:28:22 --> 00:28:24
last.
I mean, if you seen how to,
409
00:28:24 --> 00:28:27
in single variable calculus,
the disk and shell methods for
410
00:28:27 --> 00:28:30
finding volumes,
that's exactly the dilemma of
411
00:28:30 --> 00:28:39
shells versus disks.
One of them is you do z first.
412
00:28:39 --> 00:28:49
The other is you do z last.
OK, so what are things we can
413
00:28:49 --> 00:28:56
do now with triple integrals?
Well, we can find the volume of
414
00:28:56 --> 00:29:00
solids by just integrating dV.
And, we've seen that.
415
00:29:00 --> 00:29:06
We can find the mass of a solid.
OK, so if we have a density,
416
00:29:06 --> 00:29:10
delta, which,
remember, delta is basically
417
00:29:10 --> 00:29:17
the mass divided by the volume.
OK, so the small mass element,
418
00:29:17 --> 00:29:23
maybe I should have written
that as dm, the mass element,
419
00:29:23 --> 00:29:27
is density times dV.
So now, this is the real
420
00:29:27 --> 00:29:29
physical density.
If you are given a material,
421
00:29:29 --> 00:29:33
usually, the density will be in
grams per cubic meter or cubic
422
00:29:33 --> 00:29:35
inch, or whatever.
I mean, there is tons of
423
00:29:35 --> 00:29:37
different units.
But, so then,
424
00:29:37 --> 00:29:43
the mass of your solid will be
just the triple integral of
425
00:29:43 --> 00:29:50
density, dV because you just sum
the mass of each little piece.
426
00:29:50 --> 00:29:53
And, of course,
if the density is one,
427
00:29:53 --> 00:29:55
then it just becomes the
volume.
428
00:29:55 --> 00:29:59
OK,
now, it shouldn't be surprising
429
00:29:59 --> 00:30:02
to you that we can also do
classics that we had seen in the
430
00:30:02 --> 00:30:05
plane such as the average value
of a function,
431
00:30:05 --> 00:30:07
the center of mass,
and moment of inertia.
432
00:30:07 --> 00:30:38
433
00:30:38 --> 00:30:47
OK, so the average value of the
function f of x,
434
00:30:47 --> 00:30:51
y, z in the region,
r,
435
00:30:51 --> 00:30:57
that would be f bar,
would be one over the volume of
436
00:30:57 --> 00:31:02
the region times the triple
integral of f dV.
437
00:31:02 --> 00:31:09
Or, if we have a density,
and we want to take a weighted
438
00:31:09 --> 00:31:21
average -- Then we take one over
the mass where the mass is the
439
00:31:21 --> 00:31:32
triple integral of the density
times the triple integral of f
440
00:31:32 --> 00:31:36
density dV.
So, as particular cases,
441
00:31:36 --> 00:31:39
there is, again,
the notion of center of mass of
442
00:31:39 --> 00:31:41
the solid.
So, that's the point that
443
00:31:41 --> 00:31:44
somehow right in the middle of
the solid.
444
00:31:44 --> 00:31:48
That's the point mass by which
there is a point at which you
445
00:31:48 --> 00:31:53
should put point mass so that it
would be equivalent from the
446
00:31:53 --> 00:31:57
point of view of dealing with
forces and translation effects,
447
00:31:57 --> 00:32:03
of course, not for rotation.
But, so the center of mass of a
448
00:32:03 --> 00:32:09
solid is just given by taking
the average values of x,
449
00:32:09 --> 00:32:14
y, and z.
OK, so there is a special case
450
00:32:14 --> 00:32:21
where, so, x bar is one over the
mass times triple integral of x
451
00:32:21 --> 00:32:31
density dV.
And, same thing with y and z.
452
00:32:31 --> 00:32:33
And, of course,
very often, you can use
453
00:32:33 --> 00:32:37
symmetry to not have to compute
all three of them.
454
00:32:37 --> 00:32:39
For example,
if you look at this solid that
455
00:32:39 --> 00:32:41
we had, well,
I guess I've erased it now.
456
00:32:41 --> 00:32:43
But, if you remember what it
looked, well,
457
00:32:43 --> 00:32:45
it was pretty obvious that the
center of mass would be in the z
458
00:32:45 --> 00:32:47
axis.
So, no need to waste time
459
00:32:47 --> 00:32:49
considering x bar and y bar.
460
00:32:49 --> 00:33:03
461
00:33:03 --> 00:33:08
And, in fact,
you can also find z bar by
462
00:33:08 --> 00:33:16
symmetry between the top and
bottom, and let you figure that
463
00:33:16 --> 00:33:18
out.
Of course, symmetry only works,
464
00:33:18 --> 00:33:20
I should say,
symmetry only works if the
465
00:33:20 --> 00:33:25
density is also symmetric.
If I had taken my guy to be
466
00:33:25 --> 00:33:31
heavier at the front than at the
back, then it would no longer be
467
00:33:31 --> 00:33:37
true that x bar would be zero.
OK, next on the list is moment
468
00:33:37 --> 00:33:40
of inertia.
Actually, in a way,
469
00:33:40 --> 00:33:45
moment of inertia in 3D is
easier conceptually than in 2D.
470
00:33:45 --> 00:33:49
So, why is that?
Well, because now the various
471
00:33:49 --> 00:33:53
flavors that we had come
together in a nice way.
472
00:33:53 --> 00:33:56
So, the moment of inertia of an
axis,
473
00:33:56 --> 00:33:58
sorry, with respect to an axis
would be,
474
00:33:58 --> 00:34:06
again, given by the triple
integral of the distance to the
475
00:34:06 --> 00:34:11
axis squared times density,
times dV.
476
00:34:11 --> 00:34:15
And, in particular,
we have our solid.
477
00:34:15 --> 00:34:19
And, we might skewer it using
any of the coordinate axes and
478
00:34:19 --> 00:34:22
then try to rotate it about one
of the axes.
479
00:34:22 --> 00:34:24
So, we have three different
possibilities,
480
00:34:24 --> 00:34:27
of course, the x,
y, or z axis.
481
00:34:27 --> 00:34:29
And, so now,
rotating about the z axis
482
00:34:29 --> 00:34:34
actually corresponds to when we
were just doing things for flat
483
00:34:34 --> 00:34:38
objects in the xy plane.
That corresponded to rotating
484
00:34:38 --> 00:34:40
about the origin.
So, secretly,
485
00:34:40 --> 00:34:42
we were saying we were rotating
about the point.
486
00:34:42 --> 00:34:44
But actually,
it was just rotating about the
487
00:34:44 --> 00:34:47
z axis.
Just I didn't want to introduce
488
00:34:47 --> 00:34:51
the z coordinate that we didn't
actually need at the time.
489
00:34:51 --> 00:35:18
490
00:35:18 --> 00:35:23
So -- [APPLAUSE]
OK, so moment of inertia about
491
00:35:23 --> 00:35:28
the z axis, so,
what's the distance to the z
492
00:35:28 --> 00:35:31
axis?
Well, we've said that's exactly
493
00:35:31 --> 00:35:34
r.
That's the cylindrical
494
00:35:34 --> 00:35:39
coordinate, r.
So, the square of a distance is
495
00:35:39 --> 00:35:44
just r squared.
Now, if you didn't want to do
496
00:35:44 --> 00:35:49
it in cylindrical coordinates
then, of course,
497
00:35:49 --> 00:35:55
r squared is just x squared
plus y squared.
498
00:35:55 --> 00:35:58
Square of distance from the z
axis is just x squared plus y
499
00:35:58 --> 00:36:00
squared.
Similarly, now,
500
00:36:00 --> 00:36:04
if you want the distance from
the x axis, well,
501
00:36:04 --> 00:36:07
that will be y squared plus z
squared.
502
00:36:07 --> 00:36:09
OK, try to convince yourselves
of the picture,
503
00:36:09 --> 00:36:13
or else just argue by symmetry:
you know, if you change the
504
00:36:13 --> 00:36:18
positions of the axis.
So, moment of inertia about the
505
00:36:18 --> 00:36:25
x axis is the double integral of
y squared plus z squared delta
506
00:36:25 --> 00:36:29
dV.
And moment of inertia about the
507
00:36:29 --> 00:36:34
y axis is the same thing,
but now with x squared plus z
508
00:36:34 --> 00:36:36
squared.
And so, now,
509
00:36:36 --> 00:36:39
if you try to apply these
things for flat solids that are
510
00:36:39 --> 00:36:42
in the xy plane,
so where there's no z to look
511
00:36:42 --> 00:36:45
at,
well, you see these formulas
512
00:36:45 --> 00:36:48
become the old formulas that we
had.
513
00:36:48 --> 00:36:56
But now, they all fit together
in a more symmetric way.
514
00:36:56 --> 00:37:04
OK, any questions about that?
No?
515
00:37:04 --> 00:37:08
OK, so these are just formulas
to remember.
516
00:37:08 --> 00:37:21
So, OK, let's do an example.
Was there a question that I
517
00:37:21 --> 00:37:24
missed?
No?
518
00:37:24 --> 00:37:34
OK, so let's find the moment of
inertia about the z axis of a
519
00:37:34 --> 00:37:44
solid cone -- -- between z
equals a times r and z equals b.
520
00:37:44 --> 00:37:47
So, just to convince you that
it's a cone, so,
521
00:37:47 --> 00:37:52
z equals a times r means the
height is proportional to the
522
00:37:52 --> 00:37:57
distance from the z axis.
So, let's look at what we get
523
00:37:57 --> 00:38:00
if we just do it in the plane of
a blackboard.
524
00:38:00 --> 00:38:04
So, if I go to the right here,
r is just the distance from the
525
00:38:04 --> 00:38:06
x axis.
The height should be
526
00:38:06 --> 00:38:09
proportional with
proportionality factor A.
527
00:38:09 --> 00:38:14
So, that means I take a line
with slope A.
528
00:38:14 --> 00:38:16
If I'm on the left,
well, it's the same story
529
00:38:16 --> 00:38:19
except distance to the z axis is
still positive.
530
00:38:19 --> 00:38:22
So, I get the symmetric thing.
And, in fact,
531
00:38:22 --> 00:38:25
it doesn't matter which
vertical plane I do it in.
532
00:38:25 --> 00:38:28
This is the same if I rotate
about.
533
00:38:28 --> 00:38:32
See, there's no theta in here.
So, it's the same in all
534
00:38:32 --> 00:38:36
directions.
So, I claim it's a cone where
535
00:38:36 --> 00:38:43
the slope of the rays is A.
OK, and z equals b.
536
00:38:43 --> 00:38:50
Well, that just means we stop
in our horizontal plane at
537
00:38:50 --> 00:38:53
height b.
OK, so that's solid cone really
538
00:38:53 --> 00:38:58
just looks like this.
That's our solid.
539
00:38:58 --> 00:39:02
OK, so it has a flat top,
that circular top,
540
00:39:02 --> 00:39:08
and then the point is at v.
The tip of it is at the origin.
541
00:39:08 --> 00:39:12
So, let's try to compute its
moment of inertia about the z
542
00:39:12 --> 00:39:14
axis.
So, that means maybe this is
543
00:39:14 --> 00:39:16
like the top that you are going
to spin.
544
00:39:16 --> 00:39:21
And, it tells you how hard it
is to actually spin that top.
545
00:39:21 --> 00:39:24
Actually, that's also useful if
you're going to do mechanical
546
00:39:24 --> 00:39:27
engineering because if you are
trying to design gears,
547
00:39:27 --> 00:39:28
and things like that that will
rotate,
548
00:39:28 --> 00:39:31
you might want to know exactly
how much effort you'll have to
549
00:39:31 --> 00:39:33
put to actually get them to
spin,
550
00:39:33 --> 00:39:37
and whether you're actually
going to have a strong enough
551
00:39:37 --> 00:39:39
engine, or whatever,
to do it.
552
00:39:39 --> 00:39:41
OK, so what's the moment of
inertia of this guy?
553
00:39:41 --> 00:39:44
Well, that's the triple
integral of, well,
554
00:39:44 --> 00:39:49
we have to choose x squared
plus y squared or r squared.
555
00:39:49 --> 00:39:52
Let's see, I think I want to
use cylindrical coordinates to
556
00:39:52 --> 00:39:57
do that, given the shape.
So, we use r squared.
557
00:39:57 --> 00:40:02
I might have a density that
let's say the density is one.
558
00:40:02 --> 00:40:06
So, I don't have density.
I still have dV.
559
00:40:06 --> 00:40:13
Now, it will be my choice to
choose between doing the dz
560
00:40:13 --> 00:40:17
first or doing r dr d theta
first.
561
00:40:17 --> 00:40:20
Just to show you how it goes
the other way around,
562
00:40:20 --> 00:40:23
let me do it r dr d theta dz
this time.
563
00:40:23 --> 00:40:29
Then you can decide on a
case-by-case basis which one you
564
00:40:29 --> 00:40:33
like best.
OK, so if we do it in this
565
00:40:33 --> 00:40:36
direction, it means that in the
inner and middle integrals,
566
00:40:36 --> 00:40:40
we've fixed a value of z.
And, for that particular value
567
00:40:40 --> 00:40:45
of z, we'll be actually slicing
our solid by a horizontal plane,
568
00:40:45 --> 00:40:47
and looking at what we get,
OK?
569
00:40:47 --> 00:40:54
So, what does that look like?
Well, I fixed a value of z,
570
00:40:54 --> 00:40:59
and I slice my solid by a
horizontal plane.
571
00:40:59 --> 00:41:04
Well, I'm going to get a circle
certainly.
572
00:41:04 --> 00:41:07
What's the radius,
well, a disk actually,
573
00:41:07 --> 00:41:11
what's the radius of the disk?
Yeah, the radius of the disk
574
00:41:11 --> 00:41:14
should be z over a because the
equation of that cone,
575
00:41:14 --> 00:41:19
we said it's z equals ar.
So, if you flip it around,
576
00:41:19 --> 00:41:24
so, maybe I should switch to
another blackboard.
577
00:41:24 --> 00:41:33
So, the equation of a cone is z
equals ar, or equivalently r
578
00:41:33 --> 00:41:40
equals z over a.
So, for a given value of z,
579
00:41:40 --> 00:41:49
I will get, this guy will be a
disk of radius z over a.
580
00:41:49 --> 00:41:55
OK, so, moment of inertia is
going to be, well,
581
00:41:55 --> 00:41:59
we said r squared,
r dr d theta dz.
582
00:41:59 --> 00:42:02
Now, so, to set up the inner
and middle integrals,
583
00:42:02 --> 00:42:06
I just set up a double integral
over this disk of radius z over
584
00:42:06 --> 00:42:07
a.
So, it's easy.
585
00:42:07 --> 00:42:14
r goes from zero to z over a.
Theta goes from zero to 2pi.
586
00:42:14 --> 00:42:17
OK, and then,
well, if I set up the bounds
587
00:42:17 --> 00:42:20
for z, now it's my outer
variable.
588
00:42:20 --> 00:42:24
So, the question I have to ask
is what is the first slice?
589
00:42:24 --> 00:42:28
What is the last slice?
So, the bottommost value of z
590
00:42:28 --> 00:42:33
would be zero,
and the topmost would be b.
591
00:42:33 --> 00:42:37
And so, that's it I get.
So, exercise,
592
00:42:37 --> 00:42:42
it's not very hard.
Try to set it up the other way
593
00:42:42 --> 00:42:47
around with dz first and then r
dr d theta.
594
00:42:47 --> 00:42:49
It's pretty much the same level
of difficulty.
595
00:42:49 --> 00:42:52
I'm sure you can do both of
them.
596
00:42:52 --> 00:42:57
So, and also,
if you want to practice
597
00:42:57 --> 00:43:03
calculations,
you should end up getting pi b
598
00:43:03 --> 00:43:10
to the five over 10a to the four
if I got it right.
599
00:43:10 --> 00:43:14
OK, let me finish with one more
example.
600
00:43:14 --> 00:43:17
I'm trying to give you plenty
of practice because in case you
601
00:43:17 --> 00:43:19
haven't noticed,
Monday is a holiday.
602
00:43:19 --> 00:43:22
So, you don't have recitation
on Monday, which is good.
603
00:43:22 --> 00:43:25
But it means that there will be
lots of stuff to cover on
604
00:43:25 --> 00:43:47
Wednesday.
So -- Thank you.
605
00:43:47 --> 00:43:58
OK, so third example,
let's say that I want to just
606
00:43:58 --> 00:44:09
set up a triple integral for the
region where z is bigger than
607
00:44:09 --> 00:44:18
one minus y inside the unit ball
centered at the origin.
608
00:44:18 --> 00:44:24
So, the unit ball is just,
you know, well,
609
00:44:24 --> 00:44:30
stay inside of the unit sphere.
So, its equation,
610
00:44:30 --> 00:44:33
if you want,
would be x squared plus y
611
00:44:33 --> 00:44:35
squared plus z squared less than
one.
612
00:44:35 --> 00:44:37
OK, so that's one thing you
should remember.
613
00:44:37 --> 00:44:40
The equation of a sphere
centered at the origin is x
614
00:44:40 --> 00:44:44
squared plus y squared plus z
squared equals radius squared.
615
00:44:44 --> 00:44:48
And now, we are going to take
this plane, z equals one minus
616
00:44:48 --> 00:44:50
y.
So, if you think about it,
617
00:44:50 --> 00:44:52
it's parallel to the x axis
because there's no x in its
618
00:44:52 --> 00:44:55
coordinate in its equation.
At the origin,
619
00:44:55 --> 00:44:59
the height is one.
So, it starts right here at one.
620
00:44:59 --> 00:45:04
And, it slopes down with y with
slope one.
621
00:45:04 --> 00:45:07
OK, so it's a plane that comes
straight out here,
622
00:45:07 --> 00:45:10
and it intersects the sphere,
so here and here,
623
00:45:10 --> 00:45:13
but also at other points in
between.
624
00:45:13 --> 00:45:18
Any idea what kind of shape
this is?
625
00:45:18 --> 00:45:20
Well, it's an ellipse,
but it's even more than that.
626
00:45:20 --> 00:45:23
It's also a circle.
If you slice a sphere by a
627
00:45:23 --> 00:45:25
plane, you always get a circle.
But, of course,
628
00:45:25 --> 00:45:28
it's a slanted circle.
So, if you look at it in the xy
629
00:45:28 --> 00:45:31
plane, if you project it to the
xy plane, that you will get an
630
00:45:31 --> 00:45:35
ellipse.
OK, so we want to look at this
631
00:45:35 --> 00:45:38
guy in here.
So, how do we do that?
632
00:45:38 --> 00:45:42
Well, so maybe I should
actually draw quickly a picture.
633
00:45:42 --> 00:45:47
So, in the yz plane,
it looks just like this,
634
00:45:47 --> 00:45:51
OK?
But, if I look at it from above
635
00:45:51 --> 00:45:54
in the xy plane,
then its shadow,
636
00:45:54 --> 00:45:59
well, see, it will sit entirely
where y is positive.
637
00:45:59 --> 00:46:02
So, it sits entirely above
here, and it goes through here
638
00:46:02 --> 00:46:04
and here.
And, in fact,
639
00:46:04 --> 00:46:08
when you project that slanted
circle, now you will get an
640
00:46:08 --> 00:46:12
ellipse.
And, well, I don't really know
641
00:46:12 --> 00:46:20
how to draw it well,
but it should be something like
642
00:46:20 --> 00:46:24
this.
OK, so now if you want to try
643
00:46:24 --> 00:46:29
to set up that double integral,
sorry, the triple integral,
644
00:46:29 --> 00:46:37
well, so let's say we do it in
rectangular coordinates because
645
00:46:37 --> 00:46:41
we are really evil.
[LAUGHTER]
646
00:46:41 --> 00:46:43
So then, the bottom surface,
OK, so we do it with z first.
647
00:46:43 --> 00:46:46
So, the bottom surface is the
slanted plane.
648
00:46:46 --> 00:46:51
So, the bottom value would be z
equals one minus y.
649
00:46:51 --> 00:46:56
The top value is on the sphere.
So, the sphere corresponds to z
650
00:46:56 --> 00:47:01
equals square root of one minus
x squared minus y squared.
651
00:47:01 --> 00:47:05
So, you'd go from the plane to
the sphere.
652
00:47:05 --> 00:47:09
And then, to find the bounds
for x and y, you have to figure
653
00:47:09 --> 00:47:13
out what exactly,
what the heck is this region
654
00:47:13 --> 00:47:15
here?
So, what is this region?
655
00:47:15 --> 00:47:19
Well, we have to figure out,
for what values of x and y the
656
00:47:19 --> 00:47:23
plane is below the ellipse.
So, the condition is that,
657
00:47:23 --> 00:47:25
sorry, the plane is below the
sphere.
658
00:47:25 --> 00:47:31
OK, so, that's when the plane
is below the sphere.
659
00:47:31 --> 00:47:37
That means one minus y is less
than square root of one minus x
660
00:47:37 --> 00:47:41
squared minus y squared.
So, you have to somehow
661
00:47:41 --> 00:47:43
manipulate this to extract
something simpler.
662
00:47:43 --> 00:47:47
Well, probably the only way to
do it is to square both sides,
663
00:47:47 --> 00:47:51
one minus y squared should be
less than one minus x squared
664
00:47:51 --> 00:47:55
minus y squared.
And, if you work hard enough,
665
00:47:55 --> 00:47:57
you'll find quite an ugly
equation.
666
00:47:57 --> 00:48:00
But, you can figure out what
are, then, the bounds for x
667
00:48:00 --> 00:48:03
given y, and then set up the
integral?
668
00:48:03 --> 00:48:06
So, just to give you a hint,
the bounds on y will be zero to
669
00:48:06 --> 00:48:09
one.
The bounds on x,
670
00:48:09 --> 00:48:10
well, I'm not sure you want to
see them,
671
00:48:10 --> 00:48:14
but in case you do,
it will be from negative square
672
00:48:14 --> 00:48:18
root of 2y minus 2y squared to
square root of 2y minus 2y
673
00:48:18 --> 00:48:20
squared.
So, exercise,
674
00:48:20 --> 00:48:25
figure out how I got these by
starting from that.
675
00:48:25 --> 00:48:27
Now, of course,
if we just wanted the volume of
676
00:48:27 --> 00:48:28
this guy, we wouldn't do it this
way.
677
00:48:28 --> 00:48:31
We do symmetry,
and actually we'd rotate the
678
00:48:31 --> 00:48:34
thing so that our spherical cap
was actually centered on the z
679
00:48:34 --> 00:48:37
axis because that would be a
much easier way to set it up.
680
00:48:37 --> 00:48:39
But, depending on what function
we are integrating,
681
00:48:39 --> 00:48:42
we can't always do that.
682
00:48:42 --> 00:48:47