Thomas Calculus 13th [Solutions]

66. average
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2
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Section 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 1131
2 /2 1 2 /2 2 2
3 3 3 sin
cos sin cos sin
0 0 0 8 0 0 8 0 2
3
2
d
3
(2 )
3
16 0 16 8
d d d d d d
0
/2
67.
68.
/2 1 r
/2 1 2 4
/2
2
0 0 0 0 0 3 0 3
1
2 1 3 1
2
M xy 3 3
,
2 0 0 8 0 4 M 4 2 8
M 4 dz r dr d 4 r dr d d ;
M xy
2 1 r
0 0 0
z dz r dr d
r dr d d z and x y 0, by symmetry
/2 2 r
/2 2 2 8
/2
4
/2 2 r
; M
0 0 0 0 0 3 0 3 yz
x dz r dr d
0 0 0
/2 2 3
/2
/2 2 r
/2 2
r cos dr d 4 cos d 4; M 3
0 0 0
xz
y dz r dr d r sin dr d
0 0 0 0 0
/2
/2 2 r
/2 2 3 /2
d
1
M
M
yz 3
0
xy
z dz r dr d r dr d d x
0 0 0 2 0 0 0
M
M xz 3 M
, z
xy
M
M 4
M dz r dr d r dr d d
4 sin 4;
y and
3
2 ,
M
2 /2 2 2 2 /2 2 3
xy
sin cos sin
0 /3 0 0 /3 0
2 /2
2 /2 2 sin 2
1 3 1
2
4 cos sin 4 4
0 /3 0 2 0 2 8 2 0
/3
M
z
xy
( )
3 3
, and x y 0, by symmetry
M 8 8
69.
83 ;
M z d d d d d d
d d d d d
70.
2 /4 a 2
3 2 /4 3 2 2 2
a 2 2
a
a
0 0 0 3 0 0 3 0 2 3
M sin d d d sin d d d
;
xy
2 /4 a 3
2 /4 2
sin cos
a
sin
a a
0 0 0 4 0 0 16 0 8
4 4 4
M d d d d d d
M
xy
a
4
3 3a
2 2
2 2
3 2 2 a
z , and x y 0, by symmetry
M
8 3
8 2 16
a
3
71.
72.
2 4 r
2 4 3/2 64
2
128
0 0 0 0 0 5 0 5
1
2 4 2 32
2
64
M xy 5
,
2 0 0 3 0 3 M 6
M dz r dr d r dr d d
;
M xy
2 4 r
0 0 0
r dr d d z and x y 0, by symmetry
2
/3 1 1 r
/3 1 2 /3
2 2
2
2 1 1
/3 0 1 r
/3 0 /3 3
M dz r dr d r r dr d r d
2
3/2
1
0
z dz r dr d
2
/3
2 2 4
/3 1 1 r 2 /3 1 2 2
d
; M
3 /3 3 3 9 yz
2 r cos dz dr d 2 r 1 r cos dr d
/3 0 1 r
/3 0
/3
1
2 1 1 1 2 2 /3 /3
3 3
8 sin r r
8 1 r 1 2 r cos d
/3 8 cos d
/3
8 sin /3 8 2 2 8
0
x
M yz 9 3
M 32 , and y z 0, by symmetry
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