James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A130 Appendix I Answers to Odd-Numbered Exercises
29. y 2 22 y42x2 0
y s42x2 2yy2
2s42x 2 2yy2
f sx, y, zd dz dy dx
− y 4 0 ys42y 2s42y
y s42x2 2yy2
2s42x 2 2yy2 f sx, y, zd dz dx dy
− y 1 2 4z2
21
y4
− y 4 0 ys42yy2
2 y 2 4z2
0
y2s4 2 y 2 4z
2s42yy2 ys42y24z2
2s42y24z
2 f sx, y, zd dx dy dz
2 f sx, y, zd dx dz dy
− y 2 22 ys42x2 y2
2s42x 2 y2 y42x 2 24z 2
0
f sx, y, zd dy dz dx
− y 1 2 4z2
21
ys4
2s4 2 4z 2 y4 2 x 2 2 4z 2
0
f sx, y, zd dy dx dz
31. y 2 22 y4 x 2 y22yy2 0
f sx, y, zd dz dy dx
− y 4 0 ysy 2sy
y 22yy2
0
f sx, y, zd dz dx dy
− y 2 0 y422z 0
y sy f sx, y, zd dx dy dz
2sy
− y 4 0 y22yy2 0
y sy
2sy
f sx, y, zd dx dz dy
− y 2 2 22 y22x y2
2 f sx, y, zd dy dz dx
− y 2 0 ys422z
0
y 422z
x
2s422z y422z x
2 f sx, y, zd dy dx dz
33. y 1 0 y1 sx y12y 0
f sx, y, zd dz dy dx − y 1 2
0 yy 0 y12y 0
f sx, y, zd dz dx dy
− y 1 0 y12z 0
y y 2
f sx, y, zd dx dy dz − 0 y1 0 y12y 0
y y 2
0
f sx, y, zd dx dz dy
− y 1 0 y12sx 0
y 12z
f sx, y, zd dy dz dx − sx y1 0 ys12zd2 0
y 12z
sx
f sx, y, zd dy dx dz
35. y 1 0 y1 y yy f sx, y, zd dz dx dy − 0 y1 0 yx 0 yy 0
f sx, y, zd dz dy dx
− y 1 0 y1 z y1 f sx, y, zd dx dy dz − y y1 0 yy 0 y1 y
f sx, y, zd dx dz dy
− y 1 0 yx 0 yx f sx, y, zd dy dz dx − z y1 0 y1 z yx z
f sx, y, zd dy dx dz
37. 64 39. 3 2 , (0, 0, 1 3)
41. a 5 , s7ay12, 7ay12, 7ay12d
43. I x − I y − I z − 2 3 kL5 45. 1 2 kha 4
47. (a) m − y 1 21 y1 x 2 y12y 0 sx 2 1 y 2 dz dy dx
(b) sx, y, zd, where
x − s1ymd y 1 21 y1 x 2 y12y 0
xsx 2 1 y 2 dz dy dx,
y − s1ymd y 1 21 y1 x 2 y12y 0
ysx 2 1 y 2 dz dy dx,
and z − s1ymd y 1 21 y1 x 2 y12y 0
zsx 2 1 y 2 dz dy dx
(c) y 1 21 y1 x 2 y12y 0
sx 2 1 y 2 d 3y2 dz dy dx
49. (a) 3 32 1 11
24
28 30 1 128 45 1 208
(b) S , ,
9 1 44 45 1 220 135 1 660D
1
(c) 240 s68 1 15d
51. (a) 1 8 (b) 1 64 (c) 1
5760 53. L3 y8
55. (a) The region bounded by the ellipsoid x 2 1 2y 2 1 3z 2 − 1
(b) 4s6y45
Exercises 15.7 • page 1043
1. (a) (b)
x
z
0
π
3
4
_2
π
”4, 3, _2’
y
π
”2, _ 2, 1’
x
z
2
π
_ 2 0
(2, 2s3 , 22) s0, 22, 1d
3. (a) (s2 , 3y4, 1) (b) s4, 2y3, 3d
5. Circular cylinder with radius 2 and axis the z-axis
7. Sphere, radius 2, centered at the origin
9. (a) z 2 − 1 1 r cos 2 r 2 (b) z − r 2 cos 2
1
y
11.
x
z
8
z=8-r@
z=r@
y
13. Cylindrical coordinates: 6 < r < 7, 0 < < 2, 0 < z < 20
15.
z 4
x
y
17. 384 19. 8 3 1 128
15 21. 2y5 23. 4 3 (s2 2 1)
25. (a) 512
3 (b) (0, 0, 23 2 )
27. Ka 2 y8, s0, 0, 2ay3d 29. 0
31. (a) yyy C
hsPdtsPd dV, where C is the cone
(b) <3.1 3 10 19 ft-lb
Exercises 15.8 • page 1049
1. (a)
z
(b)
S 3
0
π
3
π
6
6
π π
”6, 3 , 6 ’
y
x
3s3
,
2 2
D , 3s3 S0, D
3s2
2 , 2 3s2
2
3. (a) s2, 3y2, y2d (b) s2, 3y4, 3y4d
5. Half-cone 7. Horizontal plane
9. (a) − 3 (b) 2 ssin 2 cos 2 2 cos 2 d − 1
11.
x
z
1
˙= π 6
y
∏=1
13.
15. 0 < < y4, 0 < < cos
x
∏=2
z
π
2
0
z
3
3π
4
y
π 3π
”3, 2, 4 ’
∏=4
˙= π 3
y
x
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