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5.
Triple Integrals
5A.
Triple integrals in rectangular and cylindrical coordinates
1
Middle:
iy
+
iy2
+
yz]
=
1
+
z

(1)
=
1
+
2z
Outer: z
+
z2]
2
=
6
y=1
0
XY
dz dx dy
Inner: xy2z2]o
=
x3y4
Jd2 Jdfi JdxY
b)
~XY~Z
Middle: ,x
2
1 4
y
4]fi
=
iy6
Outer:
kY7]
=
y.
0
5A2
dz dy dx
(ii) Jdl Jdl' Jdl dxdzdy
(iii)JdlJdlJdl'dydXdz
c) In cylindrical coordinates, with the polar coordinates r and 8
in xzplane, we get
11,
dy dr de
=
Jdr" Jdl Jd2dy dr do
d)
The sphere has equation x2
+
y2
+
z2
=
2, or r2
+
=
2 in cylindrical
coordinates.
The cone has equation z2
=
r2, or z
=
r. The circle in which they intersect has
a radius r found by solving the two equations z
=
r and z2+r2
=
2 simultaneously;
.Y
eliminating z we get r2
=
1, so r
=
1. Putting it all together, we get
\:I
crosssectionview
5A3
By symmetry,
i
=
y
=
2,
so it suffices to calculate just one of these, say
2.
We have
1x
1zy
zmoment
=
z dv
=
z dz dy dx
z=
1Xy
Jdl Jd
Jd
1xy
12
1nner: iz2]
=;(l~y)~
Middle:i(l~y)~]~
=i(l~)~
0
1
1

x4]
=
I

z moment.

,,
0
v=
mass of D
=
volume of D
=
i(base)(height)
=
i

;
.
1
=
i.
Therefore
2
=
&/+=
i;
this is also
Z
and
g,
by symmetry.
5A4
Placing the cone as shown, its equation in cylindrical coordinates is z
=
r
and the density is given by
6
=
r. By the geometry, its projection onto the xyplane
is the interior R of the origincentered circle of radius h.
vertical crosssection
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View Full DocumentTRIPLE INTEGRALS
a) Mass of solid
D
=
/~~~6dV=l~~l~~l~~.rdzdrd6
2~h4
Inner: (h

r)r2;
Middle:



=
.
Outer:

3
hr3
4,
12'
12
r41h
b)
By symmetry, the center of mass is on the zaxis, so we only have to compute its
zcoordinate,
5.
zmoment of D
=
//Lz6dv
=
~2r~hJlhzT~TdzdTd~
h
1
Inner: iz2r2]
=
i(h2r2

r4)
Middle:
g)
=
h5
.

2
2
A
(h2:
2
15

T
,
5A5
Position
S
so that its base is in the xyplane and its diagonal D lies along the xaxis
(the yaxis would do equally well). The octants divide
S into four tetrahedra, which by
symmetry have the same moment of inertia about the xaxis; we calculate the one in the
first octant. (The picture looks like that for 5A3, except the height is 2.)
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 Spring '08
 Auroux
 Integrals

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