Thomas Calculus 13th [Solutions]

1118 Chapter 15 Multiple Integrals
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topolar {x r Cos[t], y r Sin[t]};
fp f/.topolar //Simplify
Integrate[r fp, {t, 0, 2 }, {r, 0, 1},{z, 0, 1}]
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15.6 MOMENTS AND CENTERS OF MASS
1.
2
1 2 x
1 2
M 7
0 3 dy dx
x
3 0
2 x x dx
2
;
3 1 3 2 5
0
2 x x x dx
4
;
x 5 and y 38
14
35
2 2
M 1 2 x
1 2 x
y 3 3
0 x
x dy dx 0
xy x
dx
2 M 1 2 x
1 2
2 x
3 3
1 2 4 19
x 3 4 5
0 x
y dy dx 2 0 y dx x 2 0
x x dx 5
2
2.
M 3 3 3
0 0 dy dx 0 3 dx 9 ;
3 3
3 3 2
3 y
Ix
y dy dx dx
0 0 0 3
0
3 3 2 3 2
3 3 2
I y x dy dx x y dx 3x dx 27
0 0 0 0 0
27 ;
3.
2 4 y 2
2
14
2
0 /2 0 4 y
M dx dy y dy
y
2 3
; M 2 4 y
2 2
4 y
1
y
2
0 y /2 x dx dy 2 0 x dy 2
y /2
2 4
2 y 128
1
2 0 16 8 y y dy
4 3
;
2 4 y
2 2
3
y 10 64
2
M x y dx dy
0 y /2 0 4 y y dy x and y 5
2 3 35
7
4.
3 3 x 3
M dy dx 9
0 0 0 (3 x ) dx
2
;
x 1 and y 1, by symmetry
3 3 x
3 3 x 3 2
M 9
y x dy dx xy dx
0 0 0 0 0 3 x x dx
2
5.
M
2 2
a a x
2
dy dx a ;
0 0
4
x y 4a
3 , by symmetry
2 2 2 2 3
a a x a a x a 2 2
M a
y
x dy dx xy dx x a x dx
0 0 0 0 0
3
6.
sin x
M dy dx sin x dx 2;
0 0 0
1 (1 cos 2
4 0 x ) dx 4 x 2
and y
8
sin x
2
sin x
1 1 2
M x y dy dx y dx sin x dx
0 0 2 0 0 2 0
7.
x
2 3 4 x
2 4 x 2 2 y
2 2
3/2
2
2
4 4 ;
2 4 x
2 3 2 3 2
4 x
2
I y dy dx dx x dx I 4 , by symmetry;
Io Ix I y
8
y
8.
2 2
2 sin x / x 2 2 2 1
2
sin 0 (1 cos 2 )
0
2 2
I x dy dx x dx x dx
y
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