Calculus 2nd Edition Rogawski

ANSWERS TO ODD-NUMBERED EXERCISES
A113
13.
15.
−1
y
D
y = 4 − x 2
∫ 2 ∫ √ 4−x 2
(x 2 + y 2 )dydx=
−1 0
1
4 17.
1
2 19. 0 21. 18 23.
2
x
√
3
2 + 8π 3 ≈ 9.244
48π − 32
9 ≈ 13.2
25. (a) W : 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 2, r 2 ≤ z ≤ 8 − r 2
(b) 16π
27.
405π
2 ≈ 636.17 29.
2
3 31. 243π
∫
33. 2π ∫ 10 ∫ 40
0 f (r cos θ, rsin θ, z)rdzdrdθ
35. ∫ π ∫ 10 ∫ r 2
0 0 f (r cos θ, rsin θ, z)rdzdrdθ
(
37. z = H R r; V = πR2 H
3 39.
4
3 π a 2 − b 2) 3/2
43.
8π
15 45.
8π
5 47.
5π
8 49. π 51.
4πa 3
3
Section 16.5 Preliminary Questions
1. 5 kg/m 3 2. (a)
41. − π 16
3. The probability that 0 ≤ X ≤ 1 and 0 ≤ Y ≤ 1; the probability
that 0 ≤ X + Y ≤ 1
Section 16.5 Exercises
1.
2
3
(
3. 4 1 − e −100) × 10 −6 C ≈ 4 × 10 −6 C
5. 10,000 − 18,000e −4/5 ≈ 1912
( )
7. 25π 3 × 10 −8 C ≈ 2.356 × 10 −6 C
( ) ( )
9. ≈ 2.593 × 10 10 kg 11. 0, 2 5 13. 4R
3π , 4R
( ) ( )
3π
15. (0.555, 0) 17. 0, 0, 3R 8 19. 0, 0, 9
(
)
8
21.
0, 0,
( 13
2 17−6 √ )
6
23. (1, 2) 25. (0, 0) 27.
16
15π
49.
13
72 51.
1
64
53. C = 15; probability is 5 8 .
55. (a) C = 4 (b)
1
48π + 32 1 ≈ 0.038
Section 16.6 Preliminary Questions
1. (b)
2. (a) G(1, 0) = (2, 0) (b) G(1, 1) = (1, 3)
(c) G(2, 1) = (3, 3)
3. Area (G(R)) = 36 4. Area (G(R)) = 0.06
Section 16.6 Exercises
1. (a) Image of the u-axis is the line y = 1 2 x; image of the v-axis is
the y-axis.
(b) The parallelogram with vertices (0, 0), (10, 5) (10, 2), (0, 7).
(c) The segment joining the points (2, 3) and (10, 8).
(d) The triangle with vertices (0, 1), (2, 1), and (2, 2).
3. G is not one-to-one; G is one-to-one on the domain
{(u, v) : u ≥ 0}, and G is one-to-one on the domain {(u, v) : u ≤ 0}.
(a) The positive x-axis including the origin and the y-axis,
respectively.
(b) The rectangle [0, 1] × [−1, 1].
(c) The curve y = √ x for 0 ≤ x ≤ 1.
(d)
(0, 1)
(0, 0)
y
y =
x
(1, 1)
5. y = 3x − c 7. y = 17 6 x 11. Jac(G) = 1
13. Jac(G) = −10 15. Jac(G) = 1 17. Jac(G) = 4
19. G(u, v) = (4u + 2v, u + 3v)
21.
2329
12 ≈ 194.08
23. (a) Area (G(R)) = 105 (b) Area (G(R)) = 126
x
25. Jac(G) = 2u v ; for R = [1, 4] × [1, 4], area (G(R)) = 15 ln 4
27.
y
(4, 9)
29. (a)
M
4ab
(b) I x = Mb2
3 ; I 0 = M(a2 + b 2 )
3 (c)
31. I 0 = 8000 kg · m 2 ; I x = 4000 kg · m 2
( )
33.
9
2 35.
243
20 37. 24a
35 , 3b 5 39.
a 2 b 4
60
41. I x = MR2
4 ; kinetic energy required is 25MR2
2 J
47. (a) I = 182.5 g· cm 2 (b) ω ≈ 126.92 rad/s
b√
3
(2, 4)
(1, 1)
(3, 6)
P
x
G(u, v) = (1 + 2u + v, 1 + 5u + 3v)