Thomas Calculus 13th [Solutions]

Section 15.8 Substitutions in Multiple Integrals 1139
1 2 1 x 2 2 1 u 2 2
2
2 2 2 1 u 2 2
2
(2 ) 4 4
0 0 0 0 0 0
4
x y dx dy u v uv u v dv du u v dv du
2 4 2 3 5
u
2 5 6
2
u v
2
u v
1
v du
112
u du
112 1
u
56
1 3 5 0 15 1 15 6 1 45
17. (a) x u cosv and
(b)
x u sin v and
( x, y) cos v u sin v 2 2
y u sin v u cos v u sin v u
( u, v)
sin v u cosv
( x, y) sin v u cosv
2 2
y u cos v u sin v u cos v u
( u, v)
cos v u sin v
18. (a)
(b)
cos v u sin v 0
( x, y, z) 2 2
x u cos v, y u sin v, z w sin v u cos v 0 u cos v u sin v u
( u, v, w)
0 0 1
2 0 0
1 ( x, y, z)
x 2u 1, y 3v 4, z ( w 4) 0 3 0 (2)(3) 1 3
2 ( u, v, w) 2
0 0 1
2
19.
sin cos cos cos sin sin
sin sin cos sin sin cos
cos sin 0
cos cos sin sin sin cos sin sin
(cos ) ( sin )
cos sin sin cos sin sin sin cos
2 2 2 2 2 2 2 2
cos sin cos cos sin cos sin sin sin cos sin sin
2 2 2 3 2 2 2 2
sin cos sin sin cos sin sin
b
g( b)
20. Let u g( x) J ( x) du g ( x) f ( u) du f g( x) g ( x)
dx in accordance with Theorem 7 in
dx a g( a)
Section 5.6. Note that g ( x ) represents the Jacobian of the transformation u g( x ) or
x g
1 ( u).
21.
3 4 1 ( y/2) 3 4 2
1 ( y/2)
2x y z
xy 3 4
1 y
dx dy dz x xz dy dz ( y 1) z dy dz
0 0 y/2 2 3 0 0 2 2 3
y/2
0 0 2 2 3
2 2 4
3 3 3
2 3
( y 1) y yz
dz 9 4z 1 dz 2 4z dz 2z
2z
12
0 4 4 3 0 4 3 4 0 3 3
0
0
22.
a 0 0
J ( u, v, w) 0 b 0 abc;
the transformation takes the ellipsoid region x
a b c
0 0 c
2 2 2
the spherical region u v w 1 in uvw-space
which has volume V 4
3
abc du dv dw
G
4
abc
3
2 2
y 2
z 1
2 2 2
V
R
in
xyz -space into
dx dy dz
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