Solucionario Larson Cap 13
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C H A P T E R 13 Multiple Integration Section 13.1 Iterated Integrals and Area in the Plane
. . . . . . . . . . . . . 133
Section 13.2 Double Integrals and Volume . . . . . . . . . . . . . . . . . . . 137 Section 13.3 Change of Variables: Polar Coordinates . . . . . . . . . . . . . 143 Section 13.4 Center of Mass and Moments of Inertia . . . . . . . . . . . . . 146 Section 13.5 Surface Area
. . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Section 13.6 Triple Integrals and Applications . . . . . . . . . . . . . . . . . 157 Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates . . . . 162 Section 13.8 Change of Variables: Jacobians . . . . . . . . . . . . . . . . . . 166 Review Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
C H A P T E R 13 Multiple Integration Section 13.1
Iterated Integrals and Area in the Plane
Solutions to Odd-Numbered Exercises
x
1.
0
4x 2
5.
x 2y dy
y y
e
12 x y
4x 2
2 2
y
y ln x 1 dx y ln2 x x 2
x3
9.
0
y x
ye
0
3.
y dx y ln x x
1
2y
y ln 2y 0 y ln 2y
1
4x 2 x 4 2
y 1 y ln2y ln2ey ln y2 y 2 2 2
ey
dy xye
2y
3 x2 2
0
0
7.
x
1 2x y dy 2xy y 2 2
y x
x3
x3
x
0
2
0
x3
ey x dy x 4 ex x 2ey x
x 21 ex x 2ex 2
0
2
u y, du dy, dv ey x dy, v xey x
x y dy dx
1 x2 dy dx
x 2 2y 2 1 dx dy
1
11.
0
1
13.
0
2
15.
1
2
1
0
0
x
4
2
0
1
1y 2
1
x y dx dy
0
0
1
2
0
4y2
0
2
21.
0
sin
2 dx dy 4 y 2
r dr d
2
0
0
1 4
2
0
dx
1 3
4
dy
0
1y 2
1
2 3 2
0
0
3
0
21 23 1 x
1 3 x 2xy 2 x 3
1 2 x xy 2
1
x1 x2 dx
64 4 8y 2 4 dy 3 3
2
19 6y 2 dy
1
3 19y 2y 4
2
3
0
r2 2
2x 4 y 2
sin
0
20 3
2 3
1
dy
0
2
1
x
1 1 1 1 2 1 y 2 y1 y 2 dy y y3 1 y 23 2 2 2 6 2 3
0
19.
2x 2 dx x 2 2x
0
0
0
1
0
0
2
1
dx
y1 x 2
17.
2
1
0
1
1 2 y 2
xy
d
4y2
2
dy
0
2 dy 2y
2
0
0
4
0
0
2
1
1 sin2 d 2
cos 2 d
1 2 1 cos 2 sin 2 4 2 4 2
2
0
2 1 32 8
133
134
Chapter 13
y dy dx
1 dx dy xy
1 x
23.
0
1
1
y2
2 1 x
1
1 2
dx
0
1
25.
Multiple Integration
y ln x 1
1
dy
1
1
1 1 dx x2 2x
1
0
1
1 1 2 2
y 1y 0 dy 1
Diverges
8
27. A
0
8
dy dx
3
dx dy
8
x
0
dx
24
8
0 6
3
dy
8
3 dx 3x
0
0
0
y
8
0
0
8
0
3
y
0
3
A
3
8 dy 8y
3
24
4
0
0
2 x 2
29. A
4x2
2
0
2
dy dx
y
dx
8
y = 4 − x2
4
0
3
2
6
y
4x2
0
0
4
4
x2
dx
2
0
x3 3
4x
0
x
4y
4
dy
0
0
dy dx
33.
x 1
2
y2
4y
3
3
0
2 y 1
1 2
9 2
4y
3
0
3
y = (2 −
0
4
dy 2
dx dy
x
x )2
2
4y
dy
1
0
3
x 1
4
4 y dy
3
3 4 y
2 2y 4 y3 2 3
4 4x x dx
4
0
8 3
4
dx dy
y 2 4 y dy 2
2
2 y 2
y
4y
4
dx
0
Integration steps are similar to those above.
dx dy 2
y2
x
0
0
0
2 x 2
8 x2 4x xx 3 2 4
0
0
1
−1
y
0
2
−2
4
dy dx
4
2 x x 2 dx
3
2
y=x+2
1 1 2x x 2 x3 2 3 A
2 16 8 3 3
1
2
0
2
4 x x 2 dx
2
2 x
4
(1, 3)
3
dx
x2
2
0
4x 2
y
4
y
y = 4 − x2
1
3
2
2 4 y1 21 dy 4 y3 2 3
2 x2 1
4
4 y dy
4x2
1
1
dx dy
0
0
31. A
x
−1
0
4
0
1
16 3
4y
4
A
2
3 0
4
4
3 2
3
9 2
2
3
4
8 3
Section 13.1
3
2x 3
5
0
0
3
3
3
y
0
0
3
3
5
1
2
2
0
A
dx dy
2
3y dy 2
0
0
2
cos2 d
0
ab 2
2
1 cos 2 d
0
2
2 21 sin 2 ab
0
ab 4
a bb2 y2
b
0
dx dy
0
ab 4
y
5y 5 dy 5y y 2 2 4
5
2
a2 x 2 dx ab
dx
0
Therefore, A ab. Integration steps are similar to those above.
5y
2
a
b a
b aa2 x2
y
0
0
A 4
dy
3y 2
0
0
a
dy dx
Therefore, A ab.
5y
x
3y 2
2
5
5y
0
3
2
b aa2 x2
a
135
x a sin , dx a cos d
5 x dx
3
1
dx
0
3 x 5x 2 x
5
2x dx 3
0
5x
dx
y
dy dx
0
5
2x 3
A 37. 4
5x
dy dx
35. A
Iterated Integrals and Area in the Plane
2 0
5
y = ba
b
a2 − x2
y x
a
4 3
y = 23 x
2
y=5−x
1 x 1
2
3
4
5
−1
4
39.
0
2
y
f x, y dx dy, 0 ≤ x ≤ y, 0 ≤ y ≤ 4
0
4
0
41.
4x2
2 0
f x, ydy dx, 0 ≤ y ≤ 4 x2, 2 ≤ x ≤ 2
4y 2
2
4
f x, y dy dx
dx dy
4y 2
0
x
y
y 3
3 1
2 1
−2
10
43.
1
2
3
1
f x, ydx dy, 0 ≤ x ≤ ln y, 1 ≤ y ≤ 10
ln 10
0
2
4
ln y
0
1 −1
x 1
x
−1
45.
1
1 x2
10 x
f x, y dy dx, x 2 ≤ y ≤ 1, 1 ≤ x ≤ 1
1
f x, ydy dx
e
y
0
y
y
f x, y dx dy
y 4
8 3
6 2
4 2 x 1
2
3
−2
−1
x 1
2
136
Chapter 13
1
47.
0
2
Multiple Integration
2
dy dx
0
1
0
1
dx dy 2
49.
0
1y2
1y2
0
y
1
dx dy
1
1x2
dy dx
0
2
y
3
1
2 x
−1
1
1
x
1
2
51.
0
2
x
3
4
dy dx
0
4x
2
2
dy dx
0
0
4y
2
dx dy 4
53.
y
0
1
1
dy dx
x 2
0
2y
dx dy 1
0
y
y
3
2 2
1
1 x 1
2
3
4
x
−1
1
1
55.
0
3 y
x
1
dx dy
y2
0
x= 3 y
5 dy dx 12
x3
2
y
x = y2
2
1
(1, 1) x
1
2
57. The first integral arises using vertical representative rectangles. The second two integrals arise using horizontal representative rectangles.
5
0
50x 2
0
x
5
0
y
x 2y 2 dx dy
0
52
5
50y 2
y
y=
50 − x 2
5
(5, 5) y=x x 5
0
1 2 1 x 50 x 23 2 x5 dx 3 3
15625 24 5
x 2y 2 dx dy
0
(0, 5 2 )
5
x 2y 2 dy dx
1 5 y dy 3
15625 24
52
5
1 15625 15625 15625 50 y23 2 y2 dy 3 18 18 18
Section 13.2
2
59.
0
2
x
1
0
1
y
0
2
0
2
x
y sin x 2
0
1 0
0
dx
0
1 1 1 cos 1 1 1 cos 1 0.2298 2 2 2
4
65.
0
y
2 dx dy ln 52 2.590 x 1y 1
0
y
x2 32
4
x = y3
(8, 2)
2
x
2
1 1 26 27 1 9 9 9
x
1664 15.848 105
x 2y xy 2 dy dx
x 2 −2
(c) Both integrals equal 67520693 97.43
69.
dy
0
13
x232
0
y
2
1
sinx 2 dy dx
x 42y ⇔ x2 32y ⇔ y (b)
3 32
67. (a) x y3 ⇔ y x13
8
x2 2
12 13 23 1 y
1 x sin x 2 dx cos x 2 2
x3 3y 2 dy dx
0
0
0
2x
1 y3
0
x
0
1
0
1 y3 y 2 dy
2
x1 y3 dx dy
2
1 2
1
sin x 2 dx dy
63.
y
0
61.
2
x1 y3 dy dx
Double Integrals and Volume
4
6
8
x = 4 2y
2
4x2
exy dy dx 20.5648
71.
0
1cos
0
6r 2 cos dr d
0
15 2
73. An iterated integral is a double integral of a function of two variables. First integrate with respect to one variable while holding the other variable constant. Then integrate with respect to the second variable. 75. The region is a rectangle.
Section 13.2
77. True
Double Integrals and Volume
For Exercise 1–3, xi yi 1 and the midpoints of the squares are
12, 21 , 32, 12 , 52, 12 , 72, 12 , 12, 32 , 32, 32 , 52, 32 , 72, 32 .
y 4 3 2
1 x 1
2
1. f x, y x y 8
f x , y x y 1 2 3 4 2 3 4 5 24 i
i
i
i
i1
4
0
2
0
4
x y dy dx
0
xy
y2 2
2 0
4
dx
0
2x 2 dx x 2 2x
4 0
24
3
4
137
138
Chapter 13
Multiple Integration
3. f x, y x 2 y 2 8
2
f x , y x y 4 i
i
i
i
i1
4
0
4
5.
0
10 26 50 10 18 34 58 52 4 4 4 4 4 4 4
2
4
x 2 y 2 dy dx
0
y3 3
x2y
0
2 0
4
dx
2x 2
0
8 2x3 8x dx 3 3 3
4 0
160 3
4
f x, ydy dx 32 31 28 23 31 30 27 22 28 27 24 19 23 22 19 14
0
400 Using the corner of the ith square furthest from the origin, you obtain 272.
2
7.
0
1
y
2
1 2x 2y dy dx
0
y 2xy y
1
2
dx
0
0
3
2
2
2 2x dx
0
1
2x x 2
2 x
1
0
2
3
8
6
9.
0
3
6
x y dx dy
y2
0
6
0
1 2 x xy 2
3
dy
y
y2
(3, 6) 6
9 5 3y y 2 dy 2 8
5 9 3 y y 2 y3 2 2 24
4
2
6 0
x 2
36
a2 x2
a
11.
x y dy dx
a a2 x2
a
a
1 xy y 2 2
a2 x2
a2 x2
4
6
y
dx
a
a
a
2xa2 x 2 dx
2 a2 x232 3
5
13.
0
3
3
xy dx dy
0
0
25 2
x y dy dx
y 5
1 2 xy 2
5
dx
0
4 3 2
3
x dx
1
0
x 1
254 x
3
2
0
225 4
a
−a
0
0
0
a
5
3
a
−a
2
3
4
5
x
Section 13.2
2
15.
0
y
4
y 2 2 dx dy y2 x y
2
y 2 2 dx dy y2 x y
2
2
0
3
4y
4
17.
0
4y
dx
2
x
1
2
ln
5x 2
1
0
x 1
0
dx
12 ln 2x
2
5
0
ln
5 2 y
2y ln x dy dx
4
ln x y2
(1, 3)
3
4x2
dx
2
4x
1
ln x4 x22 4 x2 dx
x
4
0
3x4
5
x dy dx
0
4
1
25x 2
0
0
4
0
2
0
y dy dx 2
4
0
y2 4
25y2
3
dy
4y3
25 18
x= 4y 3
9y y 25 18 3 1
3
3
0
x 1
2
25
y
dx
4
0
3
dx 4
2
0
1 x 1
2
23.
0
y
2
4 x ydx dy
0
4x
0
2
4y
0
2y2
x2 xy 2
y3 y3 6 3
8 8 8 4 6 3
2
2
3
4
y
dy
y
0
y2 y2 dy 2
(4, 3)
1
9 y 2 dy
0
2
25 − y 2
x=
2
3
4
5 4
1 2 x 2
0
x dx dy
4y3
3
4
y
25y 2
3
x dy dx
21.
3
26 25
3
0
0
19.
2
2
1 5 ln 2 2
1
y=x
dx
ln
2x 2
4x
x=2
3
2x
lnx 2 y 2
0
y = 2x
4
4x2
1
2y ln x dx dy
x
y
y dy dx x2 y 2
2
1 2
1 2
2x
Double Integrals and Volume
2
1
y=x
0
x 1
2
3
4
5
4
139
140
Chapter 13
23x4
6
25.
0
0
Multiple Integration
6
12 2x 3y dy dx 4
0
6
0
3 1 12y 2xy y 2 4 2
dx
5
0
3
y = − 23 x + 4
4
1 2 x 2x 6 dx 6
181 x
y
23x4
3 2 1
6
x 2 6x
x
0
1
−1
2
3
4
12
1
27.
y
0
1
1 xy dx dy
0
0
1
0
y
x 2y x 2
y
dy
0 1
y3 y dy 2
y=x x
y2 y4 2 8
29.
0
0
1
0
3 8
1 dy dx x 12 y 12
0
x 11 y 1 2
0
dx
0
1 1 dx x 12 x 1
0
4x2
2
31. 4
1
0
4 x 2 y 2 dy dx 8
0
1
33. V
0
1
0
x
2
35. V
xy dy dx
0
0
x
1 2 xy 2
18 x
1
0
0
1 2
1
x 2 dy dx
0
2
x3 dx
0
dx
32 3
4 0
4x3
3 2 0
y
2
x 2y
0
1 8
4
dx
4
0
y
4
1
y=x 3 2 1
x 1
x
−1
37. Divide the solid into two equal parts.
1
0
y=x
1 x 2 dy dx
0
1
2
dx
0
x 1
1
x1 x 2 dx
0
1
x
y1 x 2
0
2
y
x
V2
1
2 1 x 232 3
1 0
2 3
2
3
4x 2 dx
1
5
6
Section 13.2
0
xy
0
2
1 2 y 2
4x 2
0
0
0
1 1 4 x 232 2x x3 3 6
2 0
2
4
1 x4 x 2 2 x 2 dx 2
x 2 y 2 dy dx
1 x 24 x 2 4 x 232 dx, 3
4
dx
16 cos2
0
x 2 sin
32 cos4 d 3
32 3
4 3 16
4 16
16 3
8
y
y 2
x 2 + y2 = 4
4 − x2
y=
1
1 x
−1
1 −1
x 1
2
4x2
2
43. V 4
0
0.5x1
2
4 x 2 y 2 dy dx 8
45. V
0
0
0
2 dy dx 1.2315 1 x2 y 2
47. f is a continuous function such that 0 ≤ f x, y ≤ 1 over a region R of area 1. Let f m, n the minimum value of f over R and f M, N the maximum value of f over R. Then
f m, n
dA ≤
R
Since
f x, y dA ≤ f M, N
R
dA.
R
dA 1 and 0 ≤ f m, n ≤ f M, N ≤ 1, we have 0 ≤ f m, n1 ≤
R
Therefore, 0 ≤
f x, y dA ≤ f M, N1 ≤ 1.
R
f x, y dA ≤ 1.
R
1
49.
12
0
12
ex 2 dx dy
y2
0
2x
1
ex 2 dy dx
0
51.
arccos y
0
sin x1 sin2 x dx dy
0
12
2xex 2 dx
ex 2
12
0
2
e14
cos x
sin x1 sin2 x dy dx
0
2
1 sin2 x12 sin x cos x dx
0
e14 1 1
0
0
0.221
12 23 1 sin
2
y
y
y = 2x 1
2
1 2
1
y = cos x
x 1 2
1
141
0
0
0
2
4x 2
2
41. V 4
x y dy dx
2
4x 2
2
39. V
Double Integrals and Volume
π 2
π
x
2
x32
0
1 22 1 3
142
Chapter 13 1 8
53. Average
Multiple Integration
4
0
2
x dy dx
0
1 8
4
2x dx
0
8 x2
4 0
1 4
55. Average
2
57. See the definition on page 946.
1 4
2
0
2
x 2 y 2 dx dy
0
2
0
x3 xy 2 3
2 0
14 83 y 32 y
2
3
0
59. The value of
dy
1 4
2
0
8 3
f x, y dA would be kB.
R
1 1250
61. Average
1 1250
325
300
250
100x 0.6y 0.4 dx dy
200
325
100y 0.4
300
x1.6 1.6
63. f x, y ≥ 0 for all x, y and
250 200
5
f x, y dA
0
0
2
P0 ≤ x ≤ 2, 1 ≤ y ≤ 2
2
0
2
1
dy
325
128,844.1 1250
1 dy dx 10 1 dy dx 10
5
0 2
0
y 0.4 dy 103.0753
300
y1.4
1.4 325 300
25,645.24
1 dx 1 5 1 1 dx . 10 5
65. f x, y ≥ 0 for all x, y and
3
f x, y dA
0
3
0
1
P0 ≤ x ≤ 1, 4 ≤ y ≤ 6
0
6
3
1 9 x y dy dx 27
1 y2 9y xy 27 2 6
4
6 3
3
dx
1 9 x y dy dx 27
0
1
0
1 1 x x2 x dx 2 9 2 18
3 0
1
2 7 4 x dx . 27 27
67. Divide the base into six squares, and assume the height at the center of each square is the height of the entire square. Thus,
z
V 4 3 6 7 3 2100 2500m3.
7
(15, 15, 7) (5, 5, 3)
(15, 5, 6)
(5, 15, 2)
(25, 5, 4)
20 y
30
(25, 15, 3)
x
1
69.
0
2
6
sin x y dy dx
0
m 4, n 8
71.
4
2
y cos x dx dy
0
(a) 1.78435
(a) 11.0571
(b) 1.7879
(b) 11.0414
m 4, n 8
8 2y 2 dy 3
Section 13.3
73. V 125
75. False
z
(4, 0, 16) 16
Matches d.
Change of Variables: Polar Coordinates
1
(4, 4, 16)
V8
0
5
(4, 4, 0)
1
0
1
f x dx
x
0
1
0
1
e t dt dx 2
1
0
t
t
1 2
et dt dx
x 1
1
e t dx dt 2
0
2
te t dt
0
1 2 et 2
Section 13.3
1 x 2 y 2 dx dy
0
y
5
x
1y 2
(0, 4, 0)
(4, 0, 0)
77. Average
143
1 0
x
1 1 e 1 1 e 2 2
1
Change of Variables: Polar Coordinates
1. Rectangular coordinates
3. Polar coordinates
5. R r, : 0 ≤ r ≤ 8, 0 ≤ ≤
7. R r, : 0 ≤ r ≤ 3 3 sin , 0 ≤ ≤ 2 Cardioid
2
9.
0
6
3r 2 sin dr d
0
2
0
2
r 3 sin
6 0
π 2
d
216 sin d
0
0
4
216 cos
2
11.
0
3
9 r 2 r dr d
2
2
0
0
31 9 r
3
2 3 2
2
0
2
π 2
d
2
5 3 5
0
55 6
0 1
2
13.
0
1sin
r dr d
2
2 1sin 0
0
0
2r
2
0
8
3 2 9 32 8
2
3
d
π 2
1 1 sin 2 d 2
1
2
1 1 sin cos cos 2 2
2
sin 2 8 sin2 0 1
1
0 1
2
144
Chapter 13
y dx dy
x 2 y 23 2 dy dx
a2 y 2
a
15.
0
0
a
r 2 sin dr d
0
0
2xx 2
xy dy dx
2
0
2
2
x 2 y 2 dy dx
0
2
r 4 dr d
243 5
2
r3 cos sin dr d 4
d
2
8x 2
cos5 sin d
4
x 2 y 2 dy dx
4x 2
0
x y dy dx
2
2
0
0
4y 2
1 2
25.
1y 2
0
8 3
r cos r sin r dr d
0
2
0
y arctan dx dy x
2
1 2
4
0
2
29. V
1
r3 sin 2 dr d
0
2
5
r 2 dr d
0
0
2
0
2
2
cos sin r2 dr d
0
2
83 sin cos
0
π 2
y arctan dx dy x
16 3
(
1 , 2
1 2
2
( ( 2, 2)
r dr d
3 3 2 d 2 4
4
0
3 2 64
0 1
2
r cos r sin r dr d
0
31. V 2
0
2
1 2
0 3
1
0
y
2
1
4
0
27. V
4y 2
1
2 3
162 d 3
0
cos sin d
0
π 2
r 2 dr d
42 3
2
0
4
0
2
a3 3
4 cos6 6
22
0
0
243 10
0
0
2
0
0
23.
3
0
3
2
a3 cos
sin d
2 cos
22
x
a3 3
0
0
0
0
0
21.
2
9x 2
2
19.
0
0
3
17.
Multiple Integration
128 3
4 cos
0
2
0
1 8
2
sin 2 d
0
1 cos 2 16
2
0
1 8
250 3
16 r2 r dr d 2
2
0
1 sin 1 cos2 d
4 cos
3 16 r 1
2 3
0
128 cos3 cos 3 3
d
2
0
2 3
2
64 sin3 64 d
0
64 3 4 9
Section 13.3
33. V
2
4
16 r 2 r dr d
a
0
2
3 16 r 1
2 3
4 a
0
d
Change of Variables: Polar Coordinates
1 16 a2 32 3
One-half the volume of the hemisphere is 64 3. 2 64 16 a23 2 3 3
16 a23 2 32 16 a2 322 3 3 a2 16 322 3 16 8 2 3 3 2 24 2 2 2.4332 a 4 4 2
35. Total Volume V
2
4
50e r2 4
4 0
0
2
0
0
2
25er 4 r dr d
2
d
50e4 1 d
0
1 e4 100 308.40524 Let c be the radius of the hole that is removed. 1 V 10
2
c
25er
r dr d
4
2
0
0
2
2
0
c
50e
r 2 4
0
d
50ec 4 1 d ⇒ 30.84052 1001 ec 2
4
2
0
⇒ ec
4
2
0.90183
c2 0.10333 4
c2 0.41331 c 0.6429 ⇒ diameter 2c 1.2858
37. A
6 cos
0
0
2
39.
0
r dr d
1cos
r dr d
0
3
0
2 sin 3
0
1 2 1 2
r dr d
18 cos d 9 2
0
41. 3
1 cos 2 d 9
0
2
1 sin 2 2
0
9
1 2 cos cos2 d
0 2
2
2 1 2 cos 1 cos
d 21 2 sin 21 21 sin 2 2
0
0
3 2
3
0
4 sin2 3 d 3
3
1 cos 6 d 3
0
43. Let R be a region bounded by the graphs of r g1 and r g2, and the lines a and b. When using polar coordinates to evaluate a double integral over R, R can be partitioned into small polar sectors.
1 sin 6 6
3
0
3 2
45. r-simple regions have fixed bounds for .
-simple regions have fixed bounds for r.
145
146
Chapter 13
Multiple Integration
47. You would need to insert a factor of r because of the r dr d nature of polar coordinate integrals. The plane regions would be sectors of circles.
2
49.
5
4
r1 r 3 sin dr d 56.051
0
Note: This integral equals
2
5
sin d
4
r1 r3 dr
0
51. Volume base height
53. False
z 16
8 12 300
Let f r, r 1 where R is the circular sector 0 ≤ r ≤ 6 and 0 ≤ ≤ . Then,
Answer (c)
r 1 dA > 0
but
R
6
4
4 6
x
55. (a) I 2
e x
2 dA
2 y2
2
4
0
y
er
22
r dr d 4
2
0
0
e r 22
0
2
d 4
d 2
0
(b) Therefore, I 2.
49x 2
7
57.
4000e0.01 x
2 y2
7 49x 2
2
dy dx
7
4000e0.01r r dr d 2
0
0
2
0
200,000e
7
0.01r 2
0
d
2 200,000 e0.49 1 400,000 1 e0.49 486,788
4
59. (a)
3x
2
23
4
61. A
43
4 csc
2 csc
3x
x
f dy dx
(2, 2) 2 ,2 3
(
1
1
2
0
0
2
0
4
xy dy dx
0
Center of Mass and Moments of Inertia
3
xy2 2
3 0
4
dx
0
2
r cos r sin r dr d
0
9 9x2 x dx 2 4
2
0
4
36
0
2
cos sin
0
2
r 3 dr d
4 cos sin d
0
4
sin2 2
2
0
(4, 4) 4 ,4 3
(
3
43 x
f r dr d
y=x
3x
4
4
f dy dx
y=
r r2
r22 r12 1
r2 r1 r r 2 2 2
4
3. m
f dy dx
2
Section 13.4 1. m
5
2
3
(c)
f dx dy
y3
2
(b)
y
y
2
3
(
( x
4
5
r 1 0 for all r.
Section 13.4
a
5. (a)
m
b
0
ky dy dx
0
a
My
k dy dx kab
m
(b)
0
0
a
b
0
a
b
0
Mx
Center of Mass and Moments of Inertia
kx dy dx
0
Mx
0
My
0
b
ky 2 dy dx
kab3 3
kxy dy dx
ka2b2 4
b
0
x
My b2 a m kab 2
My ka2b24 a x m kab22 2
y
Mx kab22 b m kab 2
y
ka2
a2, b2
x, y
a
(c) m
x, y
b
0
a
kx dy dx k
0
a
0
b
Mx
0
ka2b2 4
kx 2 dy dx
ka3b 3
b
My
0
0
Mx kab33 2 b m kab22 3
a2, 23b
1 xb dx ka2b 2
kxy dy dx
0
a
My ka3b3 2 2 a x m ka b2 3 y
Mx ka2b24 b 2 m ka b2 2
x, y
7. (a)
kab2 2
0
a
ka2b 2
ky dy dx
0
a
kab2 2
b
23 a, b2
k m bh 2
y
y=
b x by symmetry 2
b2
Mx
0
2hxb
h
b
ky dy dx
0
2hx b y=−
2h xb b
ky dy dx
b2 0
x b
kbh2 kbh2 kbh2 12 12 6 y
x, y
Mx kbh26 h m kbh2 3
b2, h3
—CONTINUED—
2 h (x − b ) b
147
148
Chapter 13
Multiple Integration
7. —CONTINUED—
b2
(b) m
0
2hxb
0
b2
Mx
0
0
0 2hxb
0
0
2hxb
kxy dy dx
kb2h2 12
2h xb b
b
kx dy dx
0
kx dy dx
b2 0
1 2 1 1 kb h kb2h kb2h 12 6 4
b2
0
2hxb
2h xb b
b
kxy dy dx
0
kxy dy dx
b2 0
1 2 2 5 1 kh b kh2b2 kh2b2 32 96 12
b2
0
kbh3 12
kb2 2
b2
My
ky 2 dy dx
b2 0
Mx kbh312 h m kbh26 2
2h xb b
b
kxy dy dx
y
Mx
2hxb
b
2
kx dy dx
0
2h xb b
kx2 dy dx
b2 0
11 7 1 3 kb h kb3h kb3h 32 96 48
x
My 7kb3h48 7 b m kb2h4 12
y
Mx kh2b212 h m kb2h4 3
9. (a) The x-coordinate changes by 5: x, y
a2 5, 2b
a 2b (b) The x-coordinate changes by 5: x, y 5, 2 3
a5
(c) m
5
a5
Mx
5
a5
My
5
x
kbh2 6
b2 0
My h 12 b m kbh26 2
ky dy dx
2h xb b
b
ky 2 dy dx
x
(c) m
2h xb b
b2 0
2hxb
b2
My
b
ky dy dx
b
0
1 25 2 kx dy dx k a 5 b kb 2 2
b
0
1 25 2 2 2 kxy dy dx k a 5 b kb 4 4
b
kx 2 dy dx
0
1 125 3 k a 5 b kb 3 3
My 2 a 15a 75 m 3 a 10
M b y x m 2
11. (a)
x 0 by symmetry m
a2k 2
yk dy dx
Mx 2a3k m 3
a2k 3
a
Mx y
a 0
a
(b) m Mx
a2 x 2
a 0 a
My
a2 x 2
a 0 a
2
a2 x 2
a2 x 2
a 0
2
4a
k a y y dy dx
a4k
16 3 24
k a y y 2 dy dx
a5k
15 32 120
kx a y y dy dx 0
x
My 0 m
y
Mx a 15 32 m 5 16 3
2a3k 3
Section 13.4
x
4
13. m
0
32k 3
kxy dy dx
0
Mx
0
0
1 1x 2
kx
Mx
dy dx 32k
2y
0
x
My 32k m 1
y
Mx 256k m 21
k dy dx
1 0 1 1x 2
1
x
My
m
256k kxy 2 dy dx 21
0
4
x 0 by symmetry 1
x
4
15.
Center of Mass and Moments of Inertia
1 0
k 2
k ky dy dx 2 8
M 2 k 2 y x 2 m 8 k 4
3
32k 3
y
3 8 32k 7
2
1 1 + x2
y=
y 3
y=
x
2
x
−1
1
1 x 1
2
3
4
−1
17. y 0 by symmetry
16y 2
4
m
4 0
x
kx dx dy
8192k 15
4 0
L by symmetry 2
sin xL
L
m
0
16y 2
4
My
19. x
kx 2 dx dy
My 524,288k m 105
524,288k 105
15
8192k
sin xL
L
Mx
0
64 7
y
y
ky dy dx
0
ky 2 dy dx
0
Mx 4kL m 9
4
16
kL 9
y
x = 16 − y 2
8
2
4 x 4
y = sin π x L
1
8
−4
x
−8
21. m Mx
a2k 8
ky dA
π 2
4
0
R
My
L
L 2
kx dA
R
a
kr 2 sin dr d
0
4
0
ka3 2 2 6
a
kr 2 cos dr d
0
x
My ka32 m 6
y
Mx ka3 2 2 m 6
8
a2k
ka32 6
4a2 3 8
a2k
4a 2 2 3
kL 4
y=x r=a
a
0
4kL 9
149
150
Chapter 13
ex
2
23. m
0
0
2
k
1 e4 4
y 0 by symmetry
k ky 2 dy dx 1 e6 9
m
k 1 5e4 8
My
x
e
0
25.
x
e
0
My
ky dy dx
0
2
Mx
Multiple Integration
kxy dy dx
0
2 cos 3
6 0
kr dr d
k 3
kx dA
R
My k e4 5 m 8e4
k e4 1 2 e4 1 0.46
y
Mx k e6 1 m 9e6
k e4 1 9 e6 e2 0.45
e4 5
4 e6 1
4e4
6
k dA
R
x
4e4
6
2 cos 3
6 0
kr 2 cos dr d 1.17k
My 3 1.17k 1.12 m k
x π 2
y
π θ= 6
2
r = 2 cos 3θ y = e −x
1
0 1
π θ =−6
x 1
2
29. m a2
27. m bh
b
Ix
0
h
0
b
Iy
0
h
3 I bh y m 3 Iy m
b3h 3
x
Ix
a2 4
y2
dA
y 2 dA
R
Iy
3 1 h bh 3 1
2
3
2
x 2 dA
b b 3 3
I0 Ix Iy
3 h h 3 3
xy
4
a4 4
r3 cos2 dr d
a4 4
0
2
a
0
0
a4
r3 sin2 dr d
a
0
R
b2
bh
2
0
x2
dA
2
0
R
I0 Ix Iy xy
a 4 a 4 4 2
mI a4 1a 4
x
2
a 2
33. ky
R
Iy
Ix
b3h 3
x 2dy dx
0
x
31. m
bh3 3
y 2 dy dx
r3
sin2
0
a4 dr d 16
a
r3
cos2
0
a4 dr d 16
mI 16a 4a 4
2
b
mk
0
a
0
a 2
kab2 2
b
kab4 4
y3 dy dx
0 b
Iy k
0
y dy dx
0
Ix k
a
a4 a4 a4 16 16 8 x
a
a
x 2y ydy dx
0
I0 Ix Iy
ka3b2 6
3kab4 2kb2a3 12
m kakabb 26 a3 a3 33 a I b kab 4 b 2 y b m kab 2 2 2 2 x
Iy
3 2
x
4 2
2
2
2
Section 13.4 35. kx
37. kxy
2
4x
2
mk
0
0
4x 2
Ix k
0
xy 2 dy dx
0 4x 2
2
Iy k
0
x3 dy dx
0
32k 3
Ix
0
Iy
0
2 23 3 3
8 mI 32k3 4k 3
6
Iy
x
kxy3 dy dx 16k
x
kx3 y dy dx
0
I0 Ix Iy
4 m 16k3 4k 3
4
x
26 3
32k 3
kxy dy dx
0
4
16k 3
x
0
4
I0 Ix Iy 16k
y
4
m
x dy dx 4k
0
2
x
Center of Mass and Moments of Inertia
512k 5
592k 5
x
3 48 4 15 32k 5 5 m 512k 5
y
mI 16k1 32k3 32
Iy
x
39. kx
x
1
m
0
x
0
kxy 2 dy dx
2
x
x
1
Iy
3k 20
x
1
Ix
kx dy dx
2
0
x
kx3 dy dx 2
I0 Ix Iy
3k 56
k 18
55k 504
x
m 18k 203k
30
y
mI 563k 203k
70
Iy
x
b
b
x a 2 dy dx 2k
0
b
2k
2k
b
0
b
x a 2b2 x2 dx
b
x 2b2 x 2 dx 2a
b
b
xb2 x2 dx a2
b
b2 x 2 dx
b4 a2b2 kb2 2 0
b 4a2 8 2 4
4
43. I
14
b2 x 2
b
41. I 2k
9
x
0
4
kx x 6 2 dy dx
0
9 x
kxx x 2 12x 36 dx k
2
92
24 72 72 52 x x 7 5
4 0
42,752k 315
6
2
151
152
Chapter 13
a 2x 2
a
45. I
0
Multiple Integration
0
0
a
k 4
a
a
0
3
0
a2 x2
k 4 a y 4
a2 x 2
dx
0
dx
0
a4 4a3a2 x2 6a2 a2 x2 4a a2 x2 a2 x2 a4 2a2x2 x4 a4 dx
0
k 4
a
k a y dy dx
0
a4 4a3y 6a2 y2 4ay3 y4
0
k 4
a2 x 2
a
k a y y a 2 dy dx
7a4 8a 2x 2 x 4 8a3a2 x 2 4ax 2a 2 x 2 dx
k 8a2 3 x5 x a x x 4a3 xa2 x 2 a2 arcsin 7a4x x 2x 2 a2 a2 x 2 a4 arcsin 4 3 5 a 2 a
k 8 1 1 7 17 7a5 a5 a5 2a5 a5 a5k 4 3 5 4 16 15
a 0
49. x, y kxy.
47. x, y ky. y will increase
Both x and y will increase 51. Let x, y be a continuous density function on the planar lamina R. The movements of mass with respect to the x- and y-axes are
y x, y dA and My
Mx
R
x x, y dA.
R
If m is the mass of the lamina, then the center of mass is
x, y
m , m . My Mx
53. See the definition on page 968
L L 55. y , A bL, h 2 2
b
Iy
0
b
0
ya y
L
0
y
L 2
57. y
2L bL L .A ,h 3 2 3
b2
2
y L2 3 3
L
Iy 2
dy dx
0
L 0
dx
L3b 12
Iy L L3b12 L hA 2
L2 bL 3
ya
2 3
2Lxb
y 2L3
b2
y
0
b2
2 3
0
2L 3
2
dy dx
3 L
dx
2L xb
L 2Lx 2L 27 b 3
b 2Lx 2L 2 L3x 3 27 8L b 3 L3b36
2L L 2 3 L b6 2
dx 3
4 b2 0
L3b 36
Section 13.5
Section 13.5
Surface Area
1. f x, y 2x 2y
3. f x, y 8 2x 2y
R triangle with vertices 0, 0, 2, 0, 0, 2
R x, y: x 2 y 2 ≤ 4
fx 2, fy 2
fx 2, fy 2
1 fx fy 3 2
2
S
0
1 fx 2 fy 2 3
2
2x
2
3 dy dx 3
0
x2 2
2 0
4x 2
2
2 x dx
S
2 4x 2
0
3 2x
6
2
2
3r dr d 12
0
0
y = 4 − x2
R
1
x
−1
y = −x + 2
1 −1
R
1
3 dy dx
y
y
2
Surface Area
y = − 4 − x2 x 1
2
5. f x, y 9 x 2
y
R square with vertices, 0, 0, 3, 0, 0, 3, 3, 3
3
fx 2x, fy 0
2
R
1 fx 2 fy 2 1 4x 2
3
S
0
3
1
3
1 4x 2 dy dx
0
3 1 4x 2 dx
x
1
34 2x 1 4x
3
3
y
R rectangle with vertices 0, 0, 0, 4, 3, 4, 3, 0
4
3 fx x1 2, fy 0 2
3
3
0
4
4 9x
2
0
4 9x
3
dx
dy dx
27 4 9x
3
3 2
0
1 94 x
0
4
R
2
1 fx 2 fy 2
4
4 9x
2
1
2
x 1
2
3
4
4 31 31 8 27
9. f x, y ln sec x
R x, y: 0 ≤ x ≤
y
, 0 ≤ y ≤ tan x 4
2
y = tan x
fx tan x, fy 0
1
R
1 fx 2 fy 2 1 tan2 x sec x
S
4
0
3
0 4 6 37 ln 6 37
ln 2x 1 4x 2
2
7. f x, y 2 x3 2
S
2
0
tan x
0
sec x dy dx
4
0
4
sec x tan x dx sec x
0
2 1
π 4
π 2
x
153
154
Chapter 13
Multiple Integration
11. f x, y x 2 y 2
y
R x, y: 0 ≤ f x, y ≤ 1
x 2 + y2 = 1
1
0 ≤ x 2 y 2 ≤ 1, x 2 y 2 ≤ 1 x
1
x y fx , fy x 2 y 2 x 2 y 2
1 x
1 fx 2 fy 2
1x 2
1
S
1 1x 2
2 dy dx
2
x2 y2 2 2 2 y x y2
2
0
1
2 r dr d 2
0
13. f x, y a2 x 2 y 2
y
R x, y: x 2 y 2 ≤ b2, b < a
a
x y fx , fy a2 x 2 y 2 a2 x 2 y 2
b x
b
S
2
2
2
b b2 x 2
a2
a dy dx x2 y 2
2
0
b
0
a a2 r 2
15. z 24 3x 2y 16
3 2x12
12
8
0
b
x
a
−b
r dr d 2a a a2 b2
y
1 fx 2 fy 2 14
S
−b
x2 y2 a 1 2 a x 2 y 2 a2 x 2 y 2 a2 x 2 y 2
1 fx fy 2
x 2 + y 2 ≤ b2
b
14 dy dx 48 14
8
0 4 x 4
8
12
16
17. z 25 x 2 y 2 1 fx 2 fy 2
9x
3
S2
1 25 x
x2 2
y2
y2 25 x 2 y 2
5
2
2
3
0
0
5 25 r 2
y
R triangle with vertices 0, 0, 1, 0, 1, 1
1
S
0
x
0
5 4x 2 dy dx
1 27 5 5 12
x
−1 −2
19. f x, y 2y x 2 1 fx 2 fy 2 5 4x 2
1
−2 −1
r dr d 20
y=x
1
R
x 1
x 2 + y2 = 9
2
25 x 2 y 2
5 dy dx 2 y 2 2 25 x 9x
3
2
y
1
2
Section 13.5 21. f x, y 4 x 2 y 2
Surface Area
23. f x, y 4 x 2 y 2
R x, y: 0 ≤ f x, y
R x, y: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
0 ≤ 4 x 2 y 2, x 2 y 2 ≤ 4
fx 2x, fy 2y
fx 2x, fy 2y
1 fx 2 fy 2 1 4x 2 4y 2
1 fx 2 fy 2 1 4x 2 4y 2
4x 2
2
S
2 4x 2 2
1 4x 2 4y 2 dy dx
2
1 4r 2 r dr d
1
0
1 4x2 4y2 dy dx 1.8616
0
17 17 1
0
0
1
S
6
y
x 2 + y2 = 4
1
x
−1
1 −1
25. Surface area > 4 6 24.
27. f x, y ex R x, y: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
Matches (e)
fx ex, fy 0
z
1 fx 2 fy 2 1 e2x
10
1
S
1
0
1 e2x dy dx
0
1
5
5
y
1 e2x 2.0035
0
x
29. f x, y x3 3xy y3
31. f x, y ex sin y
R square with vertices 1, 1, 1, 1, 1, 1, 1, 1
fx ex sin y, fy ex cos y
fx 3x 2 3y 3x 2 y, fy 3x 3y 2 3 y 2 x
1 fx2 fy2 1 e2x sin2 y e2x cos2 y
S
1
1
1
1
1 9x 2 y2 9 y 2 x2 dy dx
S
33. f x, y exy R x, y: 0 ≤ x ≤ 4, 0 ≤ y ≤ 10 fx yexy, fy xexy 1 fx 2 fy 2 1 y 2e2xy x 2e2xy 1 e2xy x 2 y 2
4
S
0
10
0
1 e2xyx 2 y 2 dy dx
1 e2x
2
4x2
2
4x2
1 e2x dy dx
155
156
Chapter 13
Multiple Integration 37. f x, y 1 x 2; fx
35. See the definition on page 972.
S
x 12 x 2
, fy 0
1 fx2 fy2 dA
R
1
x
16
0
0
1 1 x 2
1
16
0
502 x 2
50
39. (a) V
0
20 50
2
0
dx 161 x 21 2
1 0
xy xy dy dx 20 100 5
0
50
x
1 x 2
dy dx
x2
x x 502 x 2 502 x 2 502 x 2 dy 200 5 10
10 x 50 x 2 502 arcsin
1 x 25 x4 x3 502 x 23 2 250x x2 50 4 800 15 30
50 0
30,415.74 ft3 xy 100
(b) z 20
1 fx 2 fy 2
S
41. (a) V
1 100 1 100
502 x 2
50
0
1 100y 2
2
1002 x 2 y 2 x2 2 100 100
1002 x 2 y 2 dy dx
0
2
0
50
1002 r 2 r dr d 2081.53 ft2
0
y
f x, y
24
R
8
20
625
x2
y2
dA
where R is the region in the first quadrant
R
8
2
4
625 r 2 r dr d
x
4
2
0
R 8
25
0
4
4
25
2 625 r 23 2 3
4
d
8 0 609 609 2 3
812 609 cm3 (b) A
16 12
1 fx 2 fy 2 dA 8
R
8
R
R
25 dA 8 625 x 2 y 2
lim 200 625 r 2 b→25
1
b 4
2
0
25
4
x2 y2 dA 625 x 2 y 2 625 x 2 y 2 25
625 r 2
100 609 cm2 2
r dr d
8
12
16
20
24
16
Section 13.6
Section 13.6
3
1.
2
0
Triple Integrals and Applications
1
0
3
x y z dx dy dx
0
2
0
2
0
1
x
0
xy
0
1
x dz dy dx
0
4
1
1
0
0
1
1
2
0
zex
1
2
0
1
2
4
0
0
2zex y
4
7.
1
4
0
1x
x cos y dz dy dx
0
2
4
0
4x
0
0
4x 2
0
0
4
1
0
13.
0
0
dz
2 0
0
a
a2 x 2
a2 x 2y 2
19. 8
0
0
4x 2
2
0
0
0
z2 2
dy dx
2
2
4
0
0
4
1
15 1 1 2 e
x1 xcos y dy dx
0
4
dx
x1 xdx
x2 x3
3 4
2
8
0
0
64 40 3 3
128 15
x sin y ln z
4
2
dy dx
1
0
x 2 ln4cos y
4x2
0
2
dx
0
x 2 ln 4 1 cos 4 x 2 dx 2.44167
3
9x2
3
9x2
2
2
9x y
dz dy dx
0
4y 2
x dx dy
2
2
18
1 10
x3 dy dx 2
1
z1 e1 dz 1 e1
1x
2 0
1 2
0
0
15.
2
dz dx dy
0
3
2
1
dz dy dx
x
dz 3z z2
2zxex dx dz
0
4y 2
2
1
4xy
2
17.
1
x4 x5 dx 2 10
4
0
4x
4
2
4
0
4x
0
x cos yz
4x 2
x 2 sin y dz dy dx z
dx
dx dz
x1 xsin y
2
x dz dy dx
0
2
2
11.
x2
4x 2
2
0
1
x
0
0
4
x
x 2y 2 2
0
9.
1 1 y y 2 yz 2 2
dy dx
2
3
0
0
2zex dy dx dz
x 2y dy dx
x
dy dx
0
1 y z dy dz 2
x
0
5.
xz
0
1
0
1
xy
x
0
1 2 x xy xz 2
0
3
3.
Triple Integrals and Applications
2
4 y 22 dy
0
a
dz dy dx 8
0
a2 x 2
a y x
256 15
a2 x2
y a2 x 2 y 2 a2 x 2 arcsin
2
a2 x 2 y 2 dy dx
2
0
4
0
0
a
4
2
8 1 16 8y 2 y 4 dy 16y y3 y5 3 5
a
0
1 a2 x 2 dx 2 a2x x3 3
a 0
4 a3 3
2
0
dx
157
158
Chapter 13
4x 2
2
21.
0
0
Multiple Integration
4x 2
2
dz dy dx
2
4 x 22 dx
0
0
0
8 1 16 8x 2 x 4 dx 16x x3 x5 3 5
23. Plane: 3x 6y 4z 12
2
0
256 15
25. Top cylinder: y 2 z2 1 Side plane: x y
z 3
z
1 2
y
3
4 x
1
x
3
0
(124z) 3
0
1
y
(124z3x) 6
dy dx dz
1
0
0
x
0
1y 2
dz dy dx
0
27. Q x, y, z: 0 ≤ x ≤ 1, 0 ≤ y ≤ x, 0 ≤ z ≤ 3
3
xyz dV
0
Q
1
0
1
3
xyz dx dy dz
y
0
0
1
0
1
0
1
0
1
1
0
y
x
xyz dy dx dz
3
0
3
0
1
y
x
0
1
R
xyz dx dz dy
x
y
1
x
xyz dy dz dx
0 3
xyz dz dx dy
0 3
0
xyz dz dy dx
9 16
29. Q x, y, z: x2 y2 ≤ 9, 0 ≤ z ≤ 4
4
xyz dV
0
Q
4
0
3
3
xyz dy dx dz
9y2
xyz dx dy dz
3 9y2 4
4
9y2
x
xyz dx dz dy
9y2
9y2
4
3 0 3
5
9x2
4
xyz dz dx dy
3 9y2 0 3
z
3 9x2
3 0 3
3
y=x
1
0
9x2
xyz dy dz dx
9x2
9x2
4
3 9x2 0
xyz dz dy dx 0
3
3
4
y
Section 13.6
dz dy dx
0
x dz dy dx
6
31.
4(2x 3)
mk
0
4(2x 3)
Myz k
0
2(y 2)(x 3)
0
8k
6
Triple Integrals and Applications
0
2(y 2)(x 3)
0
12k x
Myz 12k 3 m 8k 2
4
33.
4
0
0
4
4k
0
4
0
0
2k
0
b
x4 x dy dx
0
4
xz dz dy dx k
0
0
0
4 x2 x dy dx 2
128k 16x 8x 2 x3 dx 3
39. y will be greater than 0, whereas x and z will be unchanged. 1 m kr 2h 3 x y 0 by symmetry
r
0
z
3kh2 r2 4kh2 3r 2
r 2x 2
h
z dz dy dx
h x 2y 2 r
0
r
0
0
b
b
0
b
0
b
0
b
0
x 2y dz dy dx
kb6 6
xy 2 dz dy dx
kb6 6
xyz dz dy dx
kb6 8
0 b
0 b
Mxy k
0
r 2x 2
r 2 x 2 y 2 dy dx
0
r
r 2 x 23 2 dx
0
kr 2h2 4 Mxy kr 2h2 4 3h m kr 2h 3 4
kb5 4
b
Mxz k
0
xy dz dy dx
0
Myz kb6 6 2b 5 x m kb 4 3
37. x will be greater than 2, whereas y and z will be unchanged.
Mxy 4k
b
Myz k
Mxy z 1 m
41.
b
mk
b
4
0
35.
0
4x
4
128k 4x x 2 dx 3
Mxy k
4
4
x dz dy dx k
0
4
0
4x
mk
y
Mxz kb6 6 2b 5 m kb 4 3
z
Mxy kb6 8 b 5 m kb 4 2
159
160
43.
Chapter 13
m
Multiple Integration
128k 3
x y 0 by symmetry z 42 x 2 y 2
42 x 2
4
Mxy 4k
0
0
2k
0
2
1024k 3
4
0
42 x2
0
dx
4k 3
4
4 x 23 2 dx 2
0
let x 4 sin
cos4 d
by Wallis’s Formula
5 y 12
3
y
mk
0
0
Myz k
0
0
0
0
4 x 4
(5 12)y
8
12
16
20
y dz dy dx 1200k
(5 12)y
z dz dy dx 250k
0
x
Myz 1000k 5 m 200k
y
Mxz 1200k 6 m 200k
z
Mxy 250k 5 m 200k 4
a
a
0
a
a
0
0
Ix Iy Iz
a
a
a 0
a
dz ka
0
0
2ka5 by symmetry 3
y 2 z2xyz dx dy dz
0
0
0
ka2 2
a
y 4z y2z3 4 2
Ix Iy Iz
a 0
dz
ka4 8
ka8 by symmetry 8
y 2 z2 dy dz
0
1 1 3 1 2 a az dz ka a3z az3 3 3 3
a
(b) Ix k
0
a
y 2 z2 dx dy dz ka
1 3 y z2y 3
ka
a
47. (a) Ix k
0
x dz dy dx 1000k
0
(3 5)x12
20
y = − 35 x + 12
8
0
0
Mxy k
16 12
(5 12)y
(3 5)x12
Mxz k
dz dy dx 200k
0
0
20
20
(5 12)y
(3 5)x12
20
3
128k 2
(3 5)x12
20
1 16y x 2y y3 3
0
Mxy 64k m 1
45. f x, y
42 x 2 y 2 dy dx 2k
0
64k z
z dz dy dx
0
42 x 2
4
42 x 2y 2
ka2 2
a
0
0
2ka 3
a
y3z yz3 dy dz
0
a
0
5
a
a2z 2z3 dz
ka8 a2z 4
2 2
2z4 4
a 0
ka8 8
Section 13.6
4
4
0
0
4
4
0
k
0
0
4
0
4
4
0
4
0
4
0
y3 4 x 3
4 0
4
4
0
4
0
4
0
0
4
0
4x
4
0
0
4 0
a
a2 x 2
2k 3
L 2
k
L 2
L 2
2
L 2
4 0
1024k 3
x 2y y34 x dx
0
4
8x 2 644 x dx
dx k
0
L 2 a a2 x 2
4
yx 2 y 2 dz dy dx k
2 3
0
1 x 2y4 x y4 x3 dy dx 3
4 1 32 8x 4x 2 x3 dx 8k 32x 4x 2 x3 x4 3 4 0
L 2
8 644 x 4 x3 dx 3
4
4
0
2048k 3 4
x 2y 2 y 4 4 x 2 4
0
51. Ixy k
0
0
4
k
8k
4
yx 2 z2 dz dy dx k
Iz k
0
4
64 4 x dx 256k 3
1 4 1 1 4x 2 x3 4 x3 dx 8k x3 x4 4 x4 3 3 4 12
8k
4
0
512k 3
1 y34 x y4 x3 dy dx 3
0
dx k
4x
Iy k
0
4
y y 2 z2 dz dy dx k
2 k 324 x2 4 x4 3 4
4x 2
0
0
0
0
dx k
y4 y2 4 x 4 x3 4 6
4
4
x 2 y 24 x dy dx
0
4x
(b) Ix k
0
4
x 2 y 2 dz dy dx k
x 2y
4
1 x 24 x 4 x3 dy dx 3
4x
4
k
0
0
k
0
4
0
Iz k
4
4
x 2 z2 dz dy dx k
1 4 1 1 4x2 x3 4 x3 dx 4k x3 x 4 4 x4 3 3 4 12
0
4
64 4 4 x 4 x3 dx 3 3
256k
0
0
4k
0
4
4x
Iy k
0
0
4
dx k
0
1 3 y 24 x 4 x dy dx 3
0
4
32 1 4 x2 4 x4 3 3
4
4
y 2 z2 dz dy dx k
y3 y 4 x 4 x3 3 3
k
4x
49. (a) Ix k
Triple Integrals and Applications
L 2
z2 dz dx dy k
a
L 2 a
4 0
2048k 3
2 2 a x2 a2 x 2 dx dy 3
x 1 x a2 x a2 x 2 a2 arcsin x2x 2 a2 x 2 a2 a4 arcsin 2 a 8 a a4 a4 a4Lk dy 4 16 4
Since m a2Lk, Ixy ma2 4. —CONTINUED—
a a
dy
161
162
Chapter 13
Multiple Integration
51. —CONTINUED—
L2
Ixz k
a2 x 2
a
L2 a a2 x 2 L2
2k
L2
a
a2 x 2
L2 a a2 x 2
a
L2 a
y2 x xa2 x 2 a2 arcsin 2 a
L2
L2
Iyz k
y 2 dz dx dy 2k
a a
L2
L2 a
2
Iy Ixy Iyz
ma2 ma2 ma2 4 4 2
Iz Ixz Iyz
mL2 ma2 m 3a2 L2 12 4 12
1
L2
a a
dy
ka4 4
L2
dy
L2
ka4L ma2 4 4
2
ma mL m 3a2 L2 4 12 12
2ka2 L3 1 mL2 3 8 12
y 2 dy
x 2a2 x 2 dx dy
1 x x2x 2 a2a2 x 2 a4 arcsin a L2 8
Ix Ixy Ixz
1
dy ka2
a
L2
2k
53.
L2
x 2 dz dx dy 2k
y 2a2 x 2 dx dy
1x
x 2 y 2x 2 y 2 z2 dz dy dx
1 1 0
55. See the definition, page 978. See Theorem 13.4, page 979.
57. (a) The annular solid on the right has the greater density. (b) The annular solid on the right has the greater movement of inertia. (c) The solid on the left will reach the bottom first. The solid on the right has a greater resistance to rotational motion.
Section 13.7
4
1.
0
2
0
Triple Integrals in Cylindrical and Spherical Coordinates
2
4
r cos dr d dz
0
0
4
0
2
3.
2 cos2
0
0
2
0
2 cos r2
2
d dz
0
2
4
2 cos d dz
0
0
4r2
r sin dz dr d
2
0
0
2 cos2
2 sin
2
r4 r 2sin dr d
2
4
0
0
0
z
0
2 sin d d d
0
4
7.
cos
2
0
rer d dr dz e4 3
1 3
2
0
0
2
8 cos4 4 cos8 sin d
2
0
4
0
cos3 sin d d
2dz 8
0
0
0
5.
4
dz
1 12
2r
2 cos2
r4 sin 4
2
cos4
4
0
d
d
0
8 cos5 4 cos9 5 9
0
2
2
0
8
52 45
Section 13.7
2
9.
0
3
2
r
e
0
r dz dr d
2
3
0
0
2
2
11.
2
2
2 sin d d d
0
2
2
4 sec
0
0
2
0
2
0
2
2
6
sin d d
cos
0
323 3
2
z 4
2
6
d
d
4
4
x
y
0
643 3
2
a cos
2
2
cot csc
arctan12 0
3 sin2 cos d d d 0
3 sin2 cos d d d 0
2
0
0
0
a cos
ra2 r 2 dr d
0
a2 r2
r dz dr d
2
0
4 2 2a 3 a3 3 4 3 2 3 9
ra2 r 2 dr d
a cos
0
2a3 3 4 9
2
2
krr dz dr d
0
2
kr 29 r cos 2r sin dr d
0
2
0
9r cos 2r sin
k 3r 3
r4 r4 cos sin 4 2
0
k 24 4 cos 8 sin d
0
0
1 sin3 d
2a3 cos3 cos 3 3
d
2
0
0
1 a2 r 232 3
0
0
0
2
21. m
0
a cos
0
2
4 1 1 sin3 d a3 cos sin2 2 3 3
a cos
0
r dz dr d 4
0
a2 r2
0
4 a3 3
2a3 3
r 2 cos dz dr d 0
a sec
0
3 sin2 cos d d d
x
0
2a cos
0
2
64 3
y
3
a
4
2
64 3
3
1 2
a a2 r2
a
0
19. V 2
1
0
2
17. V 4
2
r2
arctan12
0
(b)
0
r 2 cos dz dr d 0
2
15. (a)
d
4
0
0
(b)
3
3
1 e9 4
13. (a)
z
1 1 e9d 2
4
6
0
2
1 2 er 2
0
rer dr d
0
0
Triple Integrals in Cylindrical and Spherical Coordinates
k 24 4 sin 8 cos k 48 8 8 48k
2
0
2 0
d
163
164
Chapter 13
23. z h
h h x2 y2 r0 r r0 r0
2
V4
0
Multiple Integration
4h r0
0
2
r0
r0r r 2 dr d
1 r02h 2 3
2
0
0
r3 dz dr d
r0
0
r0r3 r 4 dr d
2
0
2
b
r3 dz dr d
0
2
b
r3 dr d
0
a
2
kh
h
a
4kh
1 k r04h 10
b4 a4 d
0
Since the mass of the core is m kV k3 r02h from Exercise 23, we have k 3m r02h. Thus,
kb4 a4h 2
1 Iz k r04h 10
kb2 a2b2 a2h 2
1
1 3m r04h 10 r02h
3 mr 2 10 0
31. V
2
0
0
r 2 z dz dr d
0
Mxy k r03h230 h m k r03h6 5
Iz 4k
0
29. m kb2h a2h khb2 a2
0
4kh r05 r0 20
h(r0 r)r0
r0
1 k r03 h2 30
0
2
4kh r0
h(r0 r)r0
r0
2
0
z
r 2 dz dr d
0
Mxy 4k
27. Iz 4k
0
1 k r03h 6
r0 d 6
4h r03 r0 6
h(r0 r)r0
r0
0
3
0
2
m 4k
0
2
4h r0
r dz dr d
0
kx 2 y 2 kr x y 0 by symmetry
hr0 rr0
r0
0
25.
4 sin
2 sin d d d 16 2
1 ma2 b2 2
33. m 8k
0
2
0
2ka4
2
0
0
2
2
0
ka4
a
3 sin d d d sin d d
0
2
sin d
0
ka4
2
ka4cos
0
Section 13.7
35.
Triple Integrals in Cylindrical and Spherical Coordinates
2 m kr 3 3
37. Iz 4k
x y 0 by symmetry
2
Mxy 4k
2
0
1 kr 4 2
kr 4 4
0
2
2
0
3 cos sin d d d sin 2 d d
1 k r 4 cos 2 8 z
2
cos5 sin3 d d
0
2
4
cos5 1 cos2 sin d
k 192
2
1
6
1 cos8 8
41.
1
g2
g1
h2r cos , r sin
h1r cos , r sin
zz (b) 0: sphere of radius 0
43. (a) r r0: right circular cylinder about z-axis
0: plane parallel to z-axis
0: plane parallel to z-axis
z z0: plane parallel to xy-plane
0: cone
0
4
f r cos , r sin , zr dz dr d
y x
tan
zz
2
x2 y 2 r 2
y r sin
a
2
Mxy kr 44 3r m 2kr33 8
39. x r cos
45. 16
2
5 k 6 cos
1 k r 4 4
0
4 sin3 d d d
0
0
4
cos
sin 2 d
0
2
2
2 k 5
0
2
4
4 k 5
r
0
2
a2 x 2
0
a2 x 2y 2
0
a2 x2 y 2z2
0
dw dz dy dx
a2 x 2
a
16
0
0
a
0
0
2
16
0
2
0
4
a
0
2
0
a4
2
0
a2 r 2
a2 r 2 z2 dzr dr d
0
a
0
8
a2 x 2 y 2 z2 dz dy dx
0
2
16
a2 x 2y 2
1 z za2 r 2 z2 a2 r 2 arcsin 2 a2 r 2
2 a r 2r dr d 2 a2r 2 r 4 2 4
d
a4 2 2
a 0
d
a2 r2
0
r dr d
165
166
Chapter 13
Multiple Integration
Section 13.8
Change of Variables: Jacobians
1 1. x u v 2
3. x u v 2 yuv
1 y u v 2
x y y x 11 12v 1 2v u v u v
y x 1 x y u v u v 2
12 1212
1 2
5. x u cos v sin y u sin v cos x y y x cos2 sin2 1 u v u v 7. x eu sin v y eu cos v y x x y eu sin veu sin v eu cos veu cos v e2u u v u v 9. x 3u 2v y 3v v
y 3
u
x 2v x 2 y3 3 3
x, y
u, v
0, 0
0, 0
3, 0
1, 0
2, 3
0, 1
v
(0, 1)
1
(1, 0) u
1
x 2y 3 9
1 11. x u v 2 1 y u v 2 y x 1 x y u v u v 2
21 1212 21
1
4x 2 y 2 dA
1
1 1
R
1
12 dv du
14 u v
4
1 u v2 4
2
1
1 1
1
u2 v 2 dv du
1
2 u2
u3 u 1 du 2 3 3 3
13. x u v
R
1
8 3
4
y x x y 10 11 1 u v u v
3
yx y dA
1
v
yu
0
4
0
3
2
3
uv1 dv du
0
8u du 36
1 u 1
2
3
4
Section 13.8
15.
xy2
e
Change of Variables: Jacobians
dA
y = x1
y
R
4
1 4 x R: y , y 2x, y , y 4 x x
y = 2x 3
y = 4x
x u x, y y u, v u
1 v12 x v 2 u32 y 1 v12 v 2 u12
R 1 x
1 1 2 u12v12 1 1 1 1 4 u u 2u 1 u12 2 v12
Transformed Region: y
y = 41 x
2
y x vu, y uv ⇒ u , v xy x
1
2
3
4
3
4
v
1 ⇒ yx 1 ⇒ v 1 x
3
S 2
4 y ⇒ yx 4 ⇒ v 4 x y 2x ⇒ y
u
y 2 ⇒ u2 x
1
y 1 1 x ⇒ ⇒ u 4 x 4 4
2
exy2 dA
4
14 1
R
2u1 dv du e u 2
ev2
vxy0
u x y 8,
vxy4
1 x u v 2
1 y u v 2
1 e2 e12 du u 14
17. u x y 4,
2
du
1
14
e2 e12ln u
2
e2 e12 ln 2 ln
14
1 e12 e2ln 8 0.9798 4
y
6
x−y=0
x+y=8
4
2
1 x, y u, v 2
v2 4
x−y=4
x+y=4
x
8
x yexy dA
4
R
1 2
2
4
uev
0
4
6
12 dv du
8
ue4 1 du
4
19. u x 4y 0,
vxy0
u x 4y 5,
vxy5
1 x u 4v, 5
1 y u v 5
14 u e 2
4
1
8 4
12e4 1
y
x−y=0 2
x + 4y = 5
y x 1 x y u v u v 5
1
51 1545 15
5
x yx 4y dA
uv
0
R
5
5
0
0
x
−1
3 −1
x + 4y = 0 −2
1 du dv 5
1 2 32 u v 5 3
5 0
dv
2 3 5 23v
5
32
0
100 9
4
x−y=5
167
168
Chapter 13
Multiple Integration
1 1 21. u x y, v x y, x u v, y u v 2 2 x y y x 1 u v u v 2
a
x y dA
u
u
0
R
12 dv du
a
u
u u du
0
y
5 u 2
a
52
0
2 a52 5
y
v=u a a
x+y=a x 2a
a
23.
x −a
v = −u
x2 y 2 2 1, x au, y bv a2 b
(a)
x2 y 2 21 a2 b
u2 v 2 1 v
au2 bv2 2 1 a2 b
y
1
u2 v 2 1
b
S R
u x
a
(b)
x, y x y y x u, v u v u v ab 00 ab
(c) A
ab dS
S
ab12 ab 25. Jacobian
x y y x x, y u, v u v u v
27. x u1 v, y uv1 w, z uvw
1v x, y, z v1 w u, v, w vw
u 0 u1 w uv uw uv
1 v u2v1 w u2vw u uv21 w uv2w 1 vu2v uuv 2 u2v
29. x sin cos , y sin sin , z cos
sin cos x, y, z sin sin , , cos
sin sin sin cos 0
cos cos cos sin sin
cos
sin cos sin sin cos cos sin
sin2 cos2 sin2 sin2 2
2
2
2
cos
2 sin cos sin2 cos2 sin
sin2 cos2 sin2 2 sin cos2 2 sin3
2 sin cos2 sin2 2 sin
1
. . . . . . . . . . . . . 133
Section 13.2 Double Integrals and Volume . . . . . . . . . . . . . . . . . . . 137 Section 13.3 Change of Variables: Polar Coordinates . . . . . . . . . . . . . 143 Section 13.4 Center of Mass and Moments of Inertia . . . . . . . . . . . . . 146 Section 13.5 Surface Area
. . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Section 13.6 Triple Integrals and Applications . . . . . . . . . . . . . . . . . 157 Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates . . . . 162 Section 13.8 Change of Variables: Jacobians . . . . . . . . . . . . . . . . . . 166 Review Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
C H A P T E R 13 Multiple Integration Section 13.1
Iterated Integrals and Area in the Plane
Solutions to Odd-Numbered Exercises
x
1.
0
4x 2
5.
x 2y dy
y y
e
12 x y
4x 2
2 2
y
y ln x 1 dx y ln2 x x 2
x3
9.
0
y x
ye
0
3.
y dx y ln x x
1
2y
y ln 2y 0 y ln 2y
1
4x 2 x 4 2
y 1 y ln2y ln2ey ln y2 y 2 2 2
ey
dy xye
2y
3 x2 2
0
0
7.
x
1 2x y dy 2xy y 2 2
y x
x3
x3
x
0
2
0
x3
ey x dy x 4 ex x 2ey x
x 21 ex x 2ex 2
0
2
u y, du dy, dv ey x dy, v xey x
x y dy dx
1 x2 dy dx
x 2 2y 2 1 dx dy
1
11.
0
1
13.
0
2
15.
1
2
1
0
0
x
4
2
0
1
1y 2
1
x y dx dy
0
0
1
2
0
4y2
0
2
21.
0
sin
2 dx dy 4 y 2
r dr d
2
0
0
1 4
2
0
dx
1 3
4
dy
0
1y 2
1
2 3 2
0
0
3
0
21 23 1 x
1 3 x 2xy 2 x 3
1 2 x xy 2
1
x1 x2 dx
64 4 8y 2 4 dy 3 3
2
19 6y 2 dy
1
3 19y 2y 4
2
3
0
r2 2
2x 4 y 2
sin
0
20 3
2 3
1
dy
0
2
1
x
1 1 1 1 2 1 y 2 y1 y 2 dy y y3 1 y 23 2 2 2 6 2 3
0
19.
2x 2 dx x 2 2x
0
0
0
1
0
0
2
1
dx
y1 x 2
17.
2
1
0
1
1 2 y 2
xy
d
4y2
2
dy
0
2 dy 2y
2
0
0
4
0
0
2
1
1 sin2 d 2
cos 2 d
1 2 1 cos 2 sin 2 4 2 4 2
2
0
2 1 32 8
133
134
Chapter 13
y dy dx
1 dx dy xy
1 x
23.
0
1
1
y2
2 1 x
1
1 2
dx
0
1
25.
Multiple Integration
y ln x 1
1
dy
1
1
1 1 dx x2 2x
1
0
1
1 1 2 2
y 1y 0 dy 1
Diverges
8
27. A
0
8
dy dx
3
dx dy
8
x
0
dx
24
8
0 6
3
dy
8
3 dx 3x
0
0
0
y
8
0
0
8
0
3
y
0
3
A
3
8 dy 8y
3
24
4
0
0
2 x 2
29. A
4x2
2
0
2
dy dx
y
dx
8
y = 4 − x2
4
0
3
2
6
y
4x2
0
0
4
4
x2
dx
2
0
x3 3
4x
0
x
4y
4
dy
0
0
dy dx
33.
x 1
2
y2
4y
3
3
0
2 y 1
1 2
9 2
4y
3
0
3
y = (2 −
0
4
dy 2
dx dy
x
x )2
2
4y
dy
1
0
3
x 1
4
4 y dy
3
3 4 y
2 2y 4 y3 2 3
4 4x x dx
4
0
8 3
4
dx dy
y 2 4 y dy 2
2
2 y 2
y
4y
4
dx
0
Integration steps are similar to those above.
dx dy 2
y2
x
0
0
0
2 x 2
8 x2 4x xx 3 2 4
0
0
1
−1
y
0
2
−2
4
dy dx
4
2 x x 2 dx
3
2
y=x+2
1 1 2x x 2 x3 2 3 A
2 16 8 3 3
1
2
0
2
4 x x 2 dx
2
2 x
4
(1, 3)
3
dx
x2
2
0
4x 2
y
4
y
y = 4 − x2
1
3
2
2 4 y1 21 dy 4 y3 2 3
2 x2 1
4
4 y dy
4x2
1
1
dx dy
0
0
31. A
x
−1
0
4
0
1
16 3
4y
4
A
2
3 0
4
4
3 2
3
9 2
2
3
4
8 3
Section 13.1
3
2x 3
5
0
0
3
3
3
y
0
0
3
3
5
1
2
2
0
A
dx dy
2
3y dy 2
0
0
2
cos2 d
0
ab 2
2
1 cos 2 d
0
2
2 21 sin 2 ab
0
ab 4
a bb2 y2
b
0
dx dy
0
ab 4
y
5y 5 dy 5y y 2 2 4
5
2
a2 x 2 dx ab
dx
0
Therefore, A ab. Integration steps are similar to those above.
5y
2
a
b a
b aa2 x2
y
0
0
A 4
dy
3y 2
0
0
a
dy dx
Therefore, A ab.
5y
x
3y 2
2
5
5y
0
3
2
b aa2 x2
a
135
x a sin , dx a cos d
5 x dx
3
1
dx
0
3 x 5x 2 x
5
2x dx 3
0
5x
dx
y
dy dx
0
5
2x 3
A 37. 4
5x
dy dx
35. A
Iterated Integrals and Area in the Plane
2 0
5
y = ba
b
a2 − x2
y x
a
4 3
y = 23 x
2
y=5−x
1 x 1
2
3
4
5
−1
4
39.
0
2
y
f x, y dx dy, 0 ≤ x ≤ y, 0 ≤ y ≤ 4
0
4
0
41.
4x2
2 0
f x, ydy dx, 0 ≤ y ≤ 4 x2, 2 ≤ x ≤ 2
4y 2
2
4
f x, y dy dx
dx dy
4y 2
0
x
y
y 3
3 1
2 1
−2
10
43.
1
2
3
1
f x, ydx dy, 0 ≤ x ≤ ln y, 1 ≤ y ≤ 10
ln 10
0
2
4
ln y
0
1 −1
x 1
x
−1
45.
1
1 x2
10 x
f x, y dy dx, x 2 ≤ y ≤ 1, 1 ≤ x ≤ 1
1
f x, ydy dx
e
y
0
y
y
f x, y dx dy
y 4
8 3
6 2
4 2 x 1
2
3
−2
−1
x 1
2
136
Chapter 13
1
47.
0
2
Multiple Integration
2
dy dx
0
1
0
1
dx dy 2
49.
0
1y2
1y2
0
y
1
dx dy
1
1x2
dy dx
0
2
y
3
1
2 x
−1
1
1
x
1
2
51.
0
2
x
3
4
dy dx
0
4x
2
2
dy dx
0
0
4y
2
dx dy 4
53.
y
0
1
1
dy dx
x 2
0
2y
dx dy 1
0
y
y
3
2 2
1
1 x 1
2
3
4
x
−1
1
1
55.
0
3 y
x
1
dx dy
y2
0
x= 3 y
5 dy dx 12
x3
2
y
x = y2
2
1
(1, 1) x
1
2
57. The first integral arises using vertical representative rectangles. The second two integrals arise using horizontal representative rectangles.
5
0
50x 2
0
x
5
0
y
x 2y 2 dx dy
0
52
5
50y 2
y
y=
50 − x 2
5
(5, 5) y=x x 5
0
1 2 1 x 50 x 23 2 x5 dx 3 3
15625 24 5
x 2y 2 dx dy
0
(0, 5 2 )
5
x 2y 2 dy dx
1 5 y dy 3
15625 24
52
5
1 15625 15625 15625 50 y23 2 y2 dy 3 18 18 18
Section 13.2
2
59.
0
2
x
1
0
1
y
0
2
0
2
x
y sin x 2
0
1 0
0
dx
0
1 1 1 cos 1 1 1 cos 1 0.2298 2 2 2
4
65.
0
y
2 dx dy ln 52 2.590 x 1y 1
0
y
x2 32
4
x = y3
(8, 2)
2
x
2
1 1 26 27 1 9 9 9
x
1664 15.848 105
x 2y xy 2 dy dx
x 2 −2
(c) Both integrals equal 67520693 97.43
69.
dy
0
13
x232
0
y
2
1
sinx 2 dy dx
x 42y ⇔ x2 32y ⇔ y (b)
3 32
67. (a) x y3 ⇔ y x13
8
x2 2
12 13 23 1 y
1 x sin x 2 dx cos x 2 2
x3 3y 2 dy dx
0
0
0
2x
1 y3
0
x
0
1
0
1 y3 y 2 dy
2
x1 y3 dx dy
2
1 2
1
sin x 2 dx dy
63.
y
0
61.
2
x1 y3 dy dx
Double Integrals and Volume
4
6
8
x = 4 2y
2
4x2
exy dy dx 20.5648
71.
0
1cos
0
6r 2 cos dr d
0
15 2
73. An iterated integral is a double integral of a function of two variables. First integrate with respect to one variable while holding the other variable constant. Then integrate with respect to the second variable. 75. The region is a rectangle.
Section 13.2
77. True
Double Integrals and Volume
For Exercise 1–3, xi yi 1 and the midpoints of the squares are
12, 21 , 32, 12 , 52, 12 , 72, 12 , 12, 32 , 32, 32 , 52, 32 , 72, 32 .
y 4 3 2
1 x 1
2
1. f x, y x y 8
f x , y x y 1 2 3 4 2 3 4 5 24 i
i
i
i
i1
4
0
2
0
4
x y dy dx
0
xy
y2 2
2 0
4
dx
0
2x 2 dx x 2 2x
4 0
24
3
4
137
138
Chapter 13
Multiple Integration
3. f x, y x 2 y 2 8
2
f x , y x y 4 i
i
i
i
i1
4
0
4
5.
0
10 26 50 10 18 34 58 52 4 4 4 4 4 4 4
2
4
x 2 y 2 dy dx
0
y3 3
x2y
0
2 0
4
dx
2x 2
0
8 2x3 8x dx 3 3 3
4 0
160 3
4
f x, ydy dx 32 31 28 23 31 30 27 22 28 27 24 19 23 22 19 14
0
400 Using the corner of the ith square furthest from the origin, you obtain 272.
2
7.
0
1
y
2
1 2x 2y dy dx
0
y 2xy y
1
2
dx
0
0
3
2
2
2 2x dx
0
1
2x x 2
2 x
1
0
2
3
8
6
9.
0
3
6
x y dx dy
y2
0
6
0
1 2 x xy 2
3
dy
y
y2
(3, 6) 6
9 5 3y y 2 dy 2 8
5 9 3 y y 2 y3 2 2 24
4
2
6 0
x 2
36
a2 x2
a
11.
x y dy dx
a a2 x2
a
a
1 xy y 2 2
a2 x2
a2 x2
4
6
y
dx
a
a
a
2xa2 x 2 dx
2 a2 x232 3
5
13.
0
3
3
xy dx dy
0
0
25 2
x y dy dx
y 5
1 2 xy 2
5
dx
0
4 3 2
3
x dx
1
0
x 1
254 x
3
2
0
225 4
a
−a
0
0
0
a
5
3
a
−a
2
3
4
5
x
Section 13.2
2
15.
0
y
4
y 2 2 dx dy y2 x y
2
y 2 2 dx dy y2 x y
2
2
0
3
4y
4
17.
0
4y
dx
2
x
1
2
ln
5x 2
1
0
x 1
0
dx
12 ln 2x
2
5
0
ln
5 2 y
2y ln x dy dx
4
ln x y2
(1, 3)
3
4x2
dx
2
4x
1
ln x4 x22 4 x2 dx
x
4
0
3x4
5
x dy dx
0
4
1
25x 2
0
0
4
0
2
0
y dy dx 2
4
0
y2 4
25y2
3
dy
4y3
25 18
x= 4y 3
9y y 25 18 3 1
3
3
0
x 1
2
25
y
dx
4
0
3
dx 4
2
0
1 x 1
2
23.
0
y
2
4 x ydx dy
0
4x
0
2
4y
0
2y2
x2 xy 2
y3 y3 6 3
8 8 8 4 6 3
2
2
3
4
y
dy
y
0
y2 y2 dy 2
(4, 3)
1
9 y 2 dy
0
2
25 − y 2
x=
2
3
4
5 4
1 2 x 2
0
x dx dy
4y3
3
4
y
25y 2
3
x dy dx
21.
3
26 25
3
0
0
19.
2
2
1 5 ln 2 2
1
y=x
dx
ln
2x 2
4x
x=2
3
2x
lnx 2 y 2
0
y = 2x
4
4x2
1
2y ln x dx dy
x
y
y dy dx x2 y 2
2
1 2
1 2
2x
Double Integrals and Volume
2
1
y=x
0
x 1
2
3
4
5
4
139
140
Chapter 13
23x4
6
25.
0
0
Multiple Integration
6
12 2x 3y dy dx 4
0
6
0
3 1 12y 2xy y 2 4 2
dx
5
0
3
y = − 23 x + 4
4
1 2 x 2x 6 dx 6
181 x
y
23x4
3 2 1
6
x 2 6x
x
0
1
−1
2
3
4
12
1
27.
y
0
1
1 xy dx dy
0
0
1
0
y
x 2y x 2
y
dy
0 1
y3 y dy 2
y=x x
y2 y4 2 8
29.
0
0
1
0
3 8
1 dy dx x 12 y 12
0
x 11 y 1 2
0
dx
0
1 1 dx x 12 x 1
0
4x2
2
31. 4
1
0
4 x 2 y 2 dy dx 8
0
1
33. V
0
1
0
x
2
35. V
xy dy dx
0
0
x
1 2 xy 2
18 x
1
0
0
1 2
1
x 2 dy dx
0
2
x3 dx
0
dx
32 3
4 0
4x3
3 2 0
y
2
x 2y
0
1 8
4
dx
4
0
y
4
1
y=x 3 2 1
x 1
x
−1
37. Divide the solid into two equal parts.
1
0
y=x
1 x 2 dy dx
0
1
2
dx
0
x 1
1
x1 x 2 dx
0
1
x
y1 x 2
0
2
y
x
V2
1
2 1 x 232 3
1 0
2 3
2
3
4x 2 dx
1
5
6
Section 13.2
0
xy
0
2
1 2 y 2
4x 2
0
0
0
1 1 4 x 232 2x x3 3 6
2 0
2
4
1 x4 x 2 2 x 2 dx 2
x 2 y 2 dy dx
1 x 24 x 2 4 x 232 dx, 3
4
dx
16 cos2
0
x 2 sin
32 cos4 d 3
32 3
4 3 16
4 16
16 3
8
y
y 2
x 2 + y2 = 4
4 − x2
y=
1
1 x
−1
1 −1
x 1
2
4x2
2
43. V 4
0
0.5x1
2
4 x 2 y 2 dy dx 8
45. V
0
0
0
2 dy dx 1.2315 1 x2 y 2
47. f is a continuous function such that 0 ≤ f x, y ≤ 1 over a region R of area 1. Let f m, n the minimum value of f over R and f M, N the maximum value of f over R. Then
f m, n
dA ≤
R
Since
f x, y dA ≤ f M, N
R
dA.
R
dA 1 and 0 ≤ f m, n ≤ f M, N ≤ 1, we have 0 ≤ f m, n1 ≤
R
Therefore, 0 ≤
f x, y dA ≤ f M, N1 ≤ 1.
R
f x, y dA ≤ 1.
R
1
49.
12
0
12
ex 2 dx dy
y2
0
2x
1
ex 2 dy dx
0
51.
arccos y
0
sin x1 sin2 x dx dy
0
12
2xex 2 dx
ex 2
12
0
2
e14
cos x
sin x1 sin2 x dy dx
0
2
1 sin2 x12 sin x cos x dx
0
e14 1 1
0
0
0.221
12 23 1 sin
2
y
y
y = 2x 1
2
1 2
1
y = cos x
x 1 2
1
141
0
0
0
2
4x 2
2
41. V 4
x y dy dx
2
4x 2
2
39. V
Double Integrals and Volume
π 2
π
x
2
x32
0
1 22 1 3
142
Chapter 13 1 8
53. Average
Multiple Integration
4
0
2
x dy dx
0
1 8
4
2x dx
0
8 x2
4 0
1 4
55. Average
2
57. See the definition on page 946.
1 4
2
0
2
x 2 y 2 dx dy
0
2
0
x3 xy 2 3
2 0
14 83 y 32 y
2
3
0
59. The value of
dy
1 4
2
0
8 3
f x, y dA would be kB.
R
1 1250
61. Average
1 1250
325
300
250
100x 0.6y 0.4 dx dy
200
325
100y 0.4
300
x1.6 1.6
63. f x, y ≥ 0 for all x, y and
250 200
5
f x, y dA
0
0
2
P0 ≤ x ≤ 2, 1 ≤ y ≤ 2
2
0
2
1
dy
325
128,844.1 1250
1 dy dx 10 1 dy dx 10
5
0 2
0
y 0.4 dy 103.0753
300
y1.4
1.4 325 300
25,645.24
1 dx 1 5 1 1 dx . 10 5
65. f x, y ≥ 0 for all x, y and
3
f x, y dA
0
3
0
1
P0 ≤ x ≤ 1, 4 ≤ y ≤ 6
0
6
3
1 9 x y dy dx 27
1 y2 9y xy 27 2 6
4
6 3
3
dx
1 9 x y dy dx 27
0
1
0
1 1 x x2 x dx 2 9 2 18
3 0
1
2 7 4 x dx . 27 27
67. Divide the base into six squares, and assume the height at the center of each square is the height of the entire square. Thus,
z
V 4 3 6 7 3 2100 2500m3.
7
(15, 15, 7) (5, 5, 3)
(15, 5, 6)
(5, 15, 2)
(25, 5, 4)
20 y
30
(25, 15, 3)
x
1
69.
0
2
6
sin x y dy dx
0
m 4, n 8
71.
4
2
y cos x dx dy
0
(a) 1.78435
(a) 11.0571
(b) 1.7879
(b) 11.0414
m 4, n 8
8 2y 2 dy 3
Section 13.3
73. V 125
75. False
z
(4, 0, 16) 16
Matches d.
Change of Variables: Polar Coordinates
1
(4, 4, 16)
V8
0
5
(4, 4, 0)
1
0
1
f x dx
x
0
1
0
1
e t dt dx 2
1
0
t
t
1 2
et dt dx
x 1
1
e t dx dt 2
0
2
te t dt
0
1 2 et 2
Section 13.3
1 x 2 y 2 dx dy
0
y
5
x
1y 2
(0, 4, 0)
(4, 0, 0)
77. Average
143
1 0
x
1 1 e 1 1 e 2 2
1
Change of Variables: Polar Coordinates
1. Rectangular coordinates
3. Polar coordinates
5. R r, : 0 ≤ r ≤ 8, 0 ≤ ≤
7. R r, : 0 ≤ r ≤ 3 3 sin , 0 ≤ ≤ 2 Cardioid
2
9.
0
6
3r 2 sin dr d
0
2
0
2
r 3 sin
6 0
π 2
d
216 sin d
0
0
4
216 cos
2
11.
0
3
9 r 2 r dr d
2
2
0
0
31 9 r
3
2 3 2
2
0
2
π 2
d
2
5 3 5
0
55 6
0 1
2
13.
0
1sin
r dr d
2
2 1sin 0
0
0
2r
2
0
8
3 2 9 32 8
2
3
d
π 2
1 1 sin 2 d 2
1
2
1 1 sin cos cos 2 2
2
sin 2 8 sin2 0 1
1
0 1
2
144
Chapter 13
y dx dy
x 2 y 23 2 dy dx
a2 y 2
a
15.
0
0
a
r 2 sin dr d
0
0
2xx 2
xy dy dx
2
0
2
2
x 2 y 2 dy dx
0
2
r 4 dr d
243 5
2
r3 cos sin dr d 4
d
2
8x 2
cos5 sin d
4
x 2 y 2 dy dx
4x 2
0
x y dy dx
2
2
0
0
4y 2
1 2
25.
1y 2
0
8 3
r cos r sin r dr d
0
2
0
y arctan dx dy x
2
1 2
4
0
2
29. V
1
r3 sin 2 dr d
0
2
5
r 2 dr d
0
0
2
0
2
2
cos sin r2 dr d
0
2
83 sin cos
0
π 2
y arctan dx dy x
16 3
(
1 , 2
1 2
2
( ( 2, 2)
r dr d
3 3 2 d 2 4
4
0
3 2 64
0 1
2
r cos r sin r dr d
0
31. V 2
0
2
1 2
0 3
1
0
y
2
1
4
0
27. V
4y 2
1
2 3
162 d 3
0
cos sin d
0
π 2
r 2 dr d
42 3
2
0
4
0
2
a3 3
4 cos6 6
22
0
0
243 10
0
0
2
0
0
23.
3
0
3
2
a3 cos
sin d
2 cos
22
x
a3 3
0
0
0
0
0
21.
2
9x 2
2
19.
0
0
3
17.
Multiple Integration
128 3
4 cos
0
2
0
1 8
2
sin 2 d
0
1 cos 2 16
2
0
1 8
250 3
16 r2 r dr d 2
2
0
1 sin 1 cos2 d
4 cos
3 16 r 1
2 3
0
128 cos3 cos 3 3
d
2
0
2 3
2
64 sin3 64 d
0
64 3 4 9
Section 13.3
33. V
2
4
16 r 2 r dr d
a
0
2
3 16 r 1
2 3
4 a
0
d
Change of Variables: Polar Coordinates
1 16 a2 32 3
One-half the volume of the hemisphere is 64 3. 2 64 16 a23 2 3 3
16 a23 2 32 16 a2 322 3 3 a2 16 322 3 16 8 2 3 3 2 24 2 2 2.4332 a 4 4 2
35. Total Volume V
2
4
50e r2 4
4 0
0
2
0
0
2
25er 4 r dr d
2
d
50e4 1 d
0
1 e4 100 308.40524 Let c be the radius of the hole that is removed. 1 V 10
2
c
25er
r dr d
4
2
0
0
2
2
0
c
50e
r 2 4
0
d
50ec 4 1 d ⇒ 30.84052 1001 ec 2
4
2
0
⇒ ec
4
2
0.90183
c2 0.10333 4
c2 0.41331 c 0.6429 ⇒ diameter 2c 1.2858
37. A
6 cos
0
0
2
39.
0
r dr d
1cos
r dr d
0
3
0
2 sin 3
0
1 2 1 2
r dr d
18 cos d 9 2
0
41. 3
1 cos 2 d 9
0
2
1 sin 2 2
0
9
1 2 cos cos2 d
0 2
2
2 1 2 cos 1 cos
d 21 2 sin 21 21 sin 2 2
0
0
3 2
3
0
4 sin2 3 d 3
3
1 cos 6 d 3
0
43. Let R be a region bounded by the graphs of r g1 and r g2, and the lines a and b. When using polar coordinates to evaluate a double integral over R, R can be partitioned into small polar sectors.
1 sin 6 6
3
0
3 2
45. r-simple regions have fixed bounds for .
-simple regions have fixed bounds for r.
145
146
Chapter 13
Multiple Integration
47. You would need to insert a factor of r because of the r dr d nature of polar coordinate integrals. The plane regions would be sectors of circles.
2
49.
5
4
r1 r 3 sin dr d 56.051
0
Note: This integral equals
2
5
sin d
4
r1 r3 dr
0
51. Volume base height
53. False
z 16
8 12 300
Let f r, r 1 where R is the circular sector 0 ≤ r ≤ 6 and 0 ≤ ≤ . Then,
Answer (c)
r 1 dA > 0
but
R
6
4
4 6
x
55. (a) I 2
e x
2 dA
2 y2
2
4
0
y
er
22
r dr d 4
2
0
0
e r 22
0
2
d 4
d 2
0
(b) Therefore, I 2.
49x 2
7
57.
4000e0.01 x
2 y2
7 49x 2
2
dy dx
7
4000e0.01r r dr d 2
0
0
2
0
200,000e
7
0.01r 2
0
d
2 200,000 e0.49 1 400,000 1 e0.49 486,788
4
59. (a)
3x
2
23
4
61. A
43
4 csc
2 csc
3x
x
f dy dx
(2, 2) 2 ,2 3
(
1
1
2
0
0
2
0
4
xy dy dx
0
Center of Mass and Moments of Inertia
3
xy2 2
3 0
4
dx
0
2
r cos r sin r dr d
0
9 9x2 x dx 2 4
2
0
4
36
0
2
cos sin
0
2
r 3 dr d
4 cos sin d
0
4
sin2 2
2
0
(4, 4) 4 ,4 3
(
3
43 x
f r dr d
y=x
3x
4
4
f dy dx
y=
r r2
r22 r12 1
r2 r1 r r 2 2 2
4
3. m
f dy dx
2
Section 13.4 1. m
5
2
3
(c)
f dx dy
y3
2
(b)
y
y
2
3
(
( x
4
5
r 1 0 for all r.
Section 13.4
a
5. (a)
m
b
0
ky dy dx
0
a
My
k dy dx kab
m
(b)
0
0
a
b
0
a
b
0
Mx
Center of Mass and Moments of Inertia
kx dy dx
0
Mx
0
My
0
b
ky 2 dy dx
kab3 3
kxy dy dx
ka2b2 4
b
0
x
My b2 a m kab 2
My ka2b24 a x m kab22 2
y
Mx kab22 b m kab 2
y
ka2
a2, b2
x, y
a
(c) m
x, y
b
0
a
kx dy dx k
0
a
0
b
Mx
0
ka2b2 4
kx 2 dy dx
ka3b 3
b
My
0
0
Mx kab33 2 b m kab22 3
a2, 23b
1 xb dx ka2b 2
kxy dy dx
0
a
My ka3b3 2 2 a x m ka b2 3 y
Mx ka2b24 b 2 m ka b2 2
x, y
7. (a)
kab2 2
0
a
ka2b 2
ky dy dx
0
a
kab2 2
b
23 a, b2
k m bh 2
y
y=
b x by symmetry 2
b2
Mx
0
2hxb
h
b
ky dy dx
0
2hx b y=−
2h xb b
ky dy dx
b2 0
x b
kbh2 kbh2 kbh2 12 12 6 y
x, y
Mx kbh26 h m kbh2 3
b2, h3
—CONTINUED—
2 h (x − b ) b
147
148
Chapter 13
Multiple Integration
7. —CONTINUED—
b2
(b) m
0
2hxb
0
b2
Mx
0
0
0 2hxb
0
0
2hxb
kxy dy dx
kb2h2 12
2h xb b
b
kx dy dx
0
kx dy dx
b2 0
1 2 1 1 kb h kb2h kb2h 12 6 4
b2
0
2hxb
2h xb b
b
kxy dy dx
0
kxy dy dx
b2 0
1 2 2 5 1 kh b kh2b2 kh2b2 32 96 12
b2
0
kbh3 12
kb2 2
b2
My
ky 2 dy dx
b2 0
Mx kbh312 h m kbh26 2
2h xb b
b
kxy dy dx
y
Mx
2hxb
b
2
kx dy dx
0
2h xb b
kx2 dy dx
b2 0
11 7 1 3 kb h kb3h kb3h 32 96 48
x
My 7kb3h48 7 b m kb2h4 12
y
Mx kh2b212 h m kb2h4 3
9. (a) The x-coordinate changes by 5: x, y
a2 5, 2b
a 2b (b) The x-coordinate changes by 5: x, y 5, 2 3
a5
(c) m
5
a5
Mx
5
a5
My
5
x
kbh2 6
b2 0
My h 12 b m kbh26 2
ky dy dx
2h xb b
b
ky 2 dy dx
x
(c) m
2h xb b
b2 0
2hxb
b2
My
b
ky dy dx
b
0
1 25 2 kx dy dx k a 5 b kb 2 2
b
0
1 25 2 2 2 kxy dy dx k a 5 b kb 4 4
b
kx 2 dy dx
0
1 125 3 k a 5 b kb 3 3
My 2 a 15a 75 m 3 a 10
M b y x m 2
11. (a)
x 0 by symmetry m
a2k 2
yk dy dx
Mx 2a3k m 3
a2k 3
a
Mx y
a 0
a
(b) m Mx
a2 x 2
a 0 a
My
a2 x 2
a 0 a
2
a2 x 2
a2 x 2
a 0
2
4a
k a y y dy dx
a4k
16 3 24
k a y y 2 dy dx
a5k
15 32 120
kx a y y dy dx 0
x
My 0 m
y
Mx a 15 32 m 5 16 3
2a3k 3
Section 13.4
x
4
13. m
0
32k 3
kxy dy dx
0
Mx
0
0
1 1x 2
kx
Mx
dy dx 32k
2y
0
x
My 32k m 1
y
Mx 256k m 21
k dy dx
1 0 1 1x 2
1
x
My
m
256k kxy 2 dy dx 21
0
4
x 0 by symmetry 1
x
4
15.
Center of Mass and Moments of Inertia
1 0
k 2
k ky dy dx 2 8
M 2 k 2 y x 2 m 8 k 4
3
32k 3
y
3 8 32k 7
2
1 1 + x2
y=
y 3
y=
x
2
x
−1
1
1 x 1
2
3
4
−1
17. y 0 by symmetry
16y 2
4
m
4 0
x
kx dx dy
8192k 15
4 0
L by symmetry 2
sin xL
L
m
0
16y 2
4
My
19. x
kx 2 dx dy
My 524,288k m 105
524,288k 105
15
8192k
sin xL
L
Mx
0
64 7
y
y
ky dy dx
0
ky 2 dy dx
0
Mx 4kL m 9
4
16
kL 9
y
x = 16 − y 2
8
2
4 x 4
y = sin π x L
1
8
−4
x
−8
21. m Mx
a2k 8
ky dA
π 2
4
0
R
My
L
L 2
kx dA
R
a
kr 2 sin dr d
0
4
0
ka3 2 2 6
a
kr 2 cos dr d
0
x
My ka32 m 6
y
Mx ka3 2 2 m 6
8
a2k
ka32 6
4a2 3 8
a2k
4a 2 2 3
kL 4
y=x r=a
a
0
4kL 9
149
150
Chapter 13
ex
2
23. m
0
0
2
k
1 e4 4
y 0 by symmetry
k ky 2 dy dx 1 e6 9
m
k 1 5e4 8
My
x
e
0
25.
x
e
0
My
ky dy dx
0
2
Mx
Multiple Integration
kxy dy dx
0
2 cos 3
6 0
kr dr d
k 3
kx dA
R
My k e4 5 m 8e4
k e4 1 2 e4 1 0.46
y
Mx k e6 1 m 9e6
k e4 1 9 e6 e2 0.45
e4 5
4 e6 1
4e4
6
k dA
R
x
4e4
6
2 cos 3
6 0
kr 2 cos dr d 1.17k
My 3 1.17k 1.12 m k
x π 2
y
π θ= 6
2
r = 2 cos 3θ y = e −x
1
0 1
π θ =−6
x 1
2
29. m a2
27. m bh
b
Ix
0
h
0
b
Iy
0
h
3 I bh y m 3 Iy m
b3h 3
x
Ix
a2 4
y2
dA
y 2 dA
R
Iy
3 1 h bh 3 1
2
3
2
x 2 dA
b b 3 3
I0 Ix Iy
3 h h 3 3
xy
4
a4 4
r3 cos2 dr d
a4 4
0
2
a
0
0
a4
r3 sin2 dr d
a
0
R
b2
bh
2
0
x2
dA
2
0
R
I0 Ix Iy xy
a 4 a 4 4 2
mI a4 1a 4
x
2
a 2
33. ky
R
Iy
Ix
b3h 3
x 2dy dx
0
x
31. m
bh3 3
y 2 dy dx
r3
sin2
0
a4 dr d 16
a
r3
cos2
0
a4 dr d 16
mI 16a 4a 4
2
b
mk
0
a
0
a 2
kab2 2
b
kab4 4
y3 dy dx
0 b
Iy k
0
y dy dx
0
Ix k
a
a4 a4 a4 16 16 8 x
a
a
x 2y ydy dx
0
I0 Ix Iy
ka3b2 6
3kab4 2kb2a3 12
m kakabb 26 a3 a3 33 a I b kab 4 b 2 y b m kab 2 2 2 2 x
Iy
3 2
x
4 2
2
2
2
Section 13.4 35. kx
37. kxy
2
4x
2
mk
0
0
4x 2
Ix k
0
xy 2 dy dx
0 4x 2
2
Iy k
0
x3 dy dx
0
32k 3
Ix
0
Iy
0
2 23 3 3
8 mI 32k3 4k 3
6
Iy
x
kxy3 dy dx 16k
x
kx3 y dy dx
0
I0 Ix Iy
4 m 16k3 4k 3
4
x
26 3
32k 3
kxy dy dx
0
4
16k 3
x
0
4
I0 Ix Iy 16k
y
4
m
x dy dx 4k
0
2
x
Center of Mass and Moments of Inertia
512k 5
592k 5
x
3 48 4 15 32k 5 5 m 512k 5
y
mI 16k1 32k3 32
Iy
x
39. kx
x
1
m
0
x
0
kxy 2 dy dx
2
x
x
1
Iy
3k 20
x
1
Ix
kx dy dx
2
0
x
kx3 dy dx 2
I0 Ix Iy
3k 56
k 18
55k 504
x
m 18k 203k
30
y
mI 563k 203k
70
Iy
x
b
b
x a 2 dy dx 2k
0
b
2k
2k
b
0
b
x a 2b2 x2 dx
b
x 2b2 x 2 dx 2a
b
b
xb2 x2 dx a2
b
b2 x 2 dx
b4 a2b2 kb2 2 0
b 4a2 8 2 4
4
43. I
14
b2 x 2
b
41. I 2k
9
x
0
4
kx x 6 2 dy dx
0
9 x
kxx x 2 12x 36 dx k
2
92
24 72 72 52 x x 7 5
4 0
42,752k 315
6
2
151
152
Chapter 13
a 2x 2
a
45. I
0
Multiple Integration
0
0
a
k 4
a
a
0
3
0
a2 x2
k 4 a y 4
a2 x 2
dx
0
dx
0
a4 4a3a2 x2 6a2 a2 x2 4a a2 x2 a2 x2 a4 2a2x2 x4 a4 dx
0
k 4
a
k a y dy dx
0
a4 4a3y 6a2 y2 4ay3 y4
0
k 4
a2 x 2
a
k a y y a 2 dy dx
7a4 8a 2x 2 x 4 8a3a2 x 2 4ax 2a 2 x 2 dx
k 8a2 3 x5 x a x x 4a3 xa2 x 2 a2 arcsin 7a4x x 2x 2 a2 a2 x 2 a4 arcsin 4 3 5 a 2 a
k 8 1 1 7 17 7a5 a5 a5 2a5 a5 a5k 4 3 5 4 16 15
a 0
49. x, y kxy.
47. x, y ky. y will increase
Both x and y will increase 51. Let x, y be a continuous density function on the planar lamina R. The movements of mass with respect to the x- and y-axes are
y x, y dA and My
Mx
R
x x, y dA.
R
If m is the mass of the lamina, then the center of mass is
x, y
m , m . My Mx
53. See the definition on page 968
L L 55. y , A bL, h 2 2
b
Iy
0
b
0
ya y
L
0
y
L 2
57. y
2L bL L .A ,h 3 2 3
b2
2
y L2 3 3
L
Iy 2
dy dx
0
L 0
dx
L3b 12
Iy L L3b12 L hA 2
L2 bL 3
ya
2 3
2Lxb
y 2L3
b2
y
0
b2
2 3
0
2L 3
2
dy dx
3 L
dx
2L xb
L 2Lx 2L 27 b 3
b 2Lx 2L 2 L3x 3 27 8L b 3 L3b36
2L L 2 3 L b6 2
dx 3
4 b2 0
L3b 36
Section 13.5
Section 13.5
Surface Area
1. f x, y 2x 2y
3. f x, y 8 2x 2y
R triangle with vertices 0, 0, 2, 0, 0, 2
R x, y: x 2 y 2 ≤ 4
fx 2, fy 2
fx 2, fy 2
1 fx fy 3 2
2
S
0
1 fx 2 fy 2 3
2
2x
2
3 dy dx 3
0
x2 2
2 0
4x 2
2
2 x dx
S
2 4x 2
0
3 2x
6
2
2
3r dr d 12
0
0
y = 4 − x2
R
1
x
−1
y = −x + 2
1 −1
R
1
3 dy dx
y
y
2
Surface Area
y = − 4 − x2 x 1
2
5. f x, y 9 x 2
y
R square with vertices, 0, 0, 3, 0, 0, 3, 3, 3
3
fx 2x, fy 0
2
R
1 fx 2 fy 2 1 4x 2
3
S
0
3
1
3
1 4x 2 dy dx
0
3 1 4x 2 dx
x
1
34 2x 1 4x
3
3
y
R rectangle with vertices 0, 0, 0, 4, 3, 4, 3, 0
4
3 fx x1 2, fy 0 2
3
3
0
4
4 9x
2
0
4 9x
3
dx
dy dx
27 4 9x
3
3 2
0
1 94 x
0
4
R
2
1 fx 2 fy 2
4
4 9x
2
1
2
x 1
2
3
4
4 31 31 8 27
9. f x, y ln sec x
R x, y: 0 ≤ x ≤
y
, 0 ≤ y ≤ tan x 4
2
y = tan x
fx tan x, fy 0
1
R
1 fx 2 fy 2 1 tan2 x sec x
S
4
0
3
0 4 6 37 ln 6 37
ln 2x 1 4x 2
2
7. f x, y 2 x3 2
S
2
0
tan x
0
sec x dy dx
4
0
4
sec x tan x dx sec x
0
2 1
π 4
π 2
x
153
154
Chapter 13
Multiple Integration
11. f x, y x 2 y 2
y
R x, y: 0 ≤ f x, y ≤ 1
x 2 + y2 = 1
1
0 ≤ x 2 y 2 ≤ 1, x 2 y 2 ≤ 1 x
1
x y fx , fy x 2 y 2 x 2 y 2
1 x
1 fx 2 fy 2
1x 2
1
S
1 1x 2
2 dy dx
2
x2 y2 2 2 2 y x y2
2
0
1
2 r dr d 2
0
13. f x, y a2 x 2 y 2
y
R x, y: x 2 y 2 ≤ b2, b < a
a
x y fx , fy a2 x 2 y 2 a2 x 2 y 2
b x
b
S
2
2
2
b b2 x 2
a2
a dy dx x2 y 2
2
0
b
0
a a2 r 2
15. z 24 3x 2y 16
3 2x12
12
8
0
b
x
a
−b
r dr d 2a a a2 b2
y
1 fx 2 fy 2 14
S
−b
x2 y2 a 1 2 a x 2 y 2 a2 x 2 y 2 a2 x 2 y 2
1 fx fy 2
x 2 + y 2 ≤ b2
b
14 dy dx 48 14
8
0 4 x 4
8
12
16
17. z 25 x 2 y 2 1 fx 2 fy 2
9x
3
S2
1 25 x
x2 2
y2
y2 25 x 2 y 2
5
2
2
3
0
0
5 25 r 2
y
R triangle with vertices 0, 0, 1, 0, 1, 1
1
S
0
x
0
5 4x 2 dy dx
1 27 5 5 12
x
−1 −2
19. f x, y 2y x 2 1 fx 2 fy 2 5 4x 2
1
−2 −1
r dr d 20
y=x
1
R
x 1
x 2 + y2 = 9
2
25 x 2 y 2
5 dy dx 2 y 2 2 25 x 9x
3
2
y
1
2
Section 13.5 21. f x, y 4 x 2 y 2
Surface Area
23. f x, y 4 x 2 y 2
R x, y: 0 ≤ f x, y
R x, y: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
0 ≤ 4 x 2 y 2, x 2 y 2 ≤ 4
fx 2x, fy 2y
fx 2x, fy 2y
1 fx 2 fy 2 1 4x 2 4y 2
1 fx 2 fy 2 1 4x 2 4y 2
4x 2
2
S
2 4x 2 2
1 4x 2 4y 2 dy dx
2
1 4r 2 r dr d
1
0
1 4x2 4y2 dy dx 1.8616
0
17 17 1
0
0
1
S
6
y
x 2 + y2 = 4
1
x
−1
1 −1
25. Surface area > 4 6 24.
27. f x, y ex R x, y: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
Matches (e)
fx ex, fy 0
z
1 fx 2 fy 2 1 e2x
10
1
S
1
0
1 e2x dy dx
0
1
5
5
y
1 e2x 2.0035
0
x
29. f x, y x3 3xy y3
31. f x, y ex sin y
R square with vertices 1, 1, 1, 1, 1, 1, 1, 1
fx ex sin y, fy ex cos y
fx 3x 2 3y 3x 2 y, fy 3x 3y 2 3 y 2 x
1 fx2 fy2 1 e2x sin2 y e2x cos2 y
S
1
1
1
1
1 9x 2 y2 9 y 2 x2 dy dx
S
33. f x, y exy R x, y: 0 ≤ x ≤ 4, 0 ≤ y ≤ 10 fx yexy, fy xexy 1 fx 2 fy 2 1 y 2e2xy x 2e2xy 1 e2xy x 2 y 2
4
S
0
10
0
1 e2xyx 2 y 2 dy dx
1 e2x
2
4x2
2
4x2
1 e2x dy dx
155
156
Chapter 13
Multiple Integration 37. f x, y 1 x 2; fx
35. See the definition on page 972.
S
x 12 x 2
, fy 0
1 fx2 fy2 dA
R
1
x
16
0
0
1 1 x 2
1
16
0
502 x 2
50
39. (a) V
0
20 50
2
0
dx 161 x 21 2
1 0
xy xy dy dx 20 100 5
0
50
x
1 x 2
dy dx
x2
x x 502 x 2 502 x 2 502 x 2 dy 200 5 10
10 x 50 x 2 502 arcsin
1 x 25 x4 x3 502 x 23 2 250x x2 50 4 800 15 30
50 0
30,415.74 ft3 xy 100
(b) z 20
1 fx 2 fy 2
S
41. (a) V
1 100 1 100
502 x 2
50
0
1 100y 2
2
1002 x 2 y 2 x2 2 100 100
1002 x 2 y 2 dy dx
0
2
0
50
1002 r 2 r dr d 2081.53 ft2
0
y
f x, y
24
R
8
20
625
x2
y2
dA
where R is the region in the first quadrant
R
8
2
4
625 r 2 r dr d
x
4
2
0
R 8
25
0
4
4
25
2 625 r 23 2 3
4
d
8 0 609 609 2 3
812 609 cm3 (b) A
16 12
1 fx 2 fy 2 dA 8
R
8
R
R
25 dA 8 625 x 2 y 2
lim 200 625 r 2 b→25
1
b 4
2
0
25
4
x2 y2 dA 625 x 2 y 2 625 x 2 y 2 25
625 r 2
100 609 cm2 2
r dr d
8
12
16
20
24
16
Section 13.6
Section 13.6
3
1.
2
0
Triple Integrals and Applications
1
0
3
x y z dx dy dx
0
2
0
2
0
1
x
0
xy
0
1
x dz dy dx
0
4
1
1
0
0
1
1
2
0
zex
1
2
0
1
2
4
0
0
2zex y
4
7.
1
4
0
1x
x cos y dz dy dx
0
2
4
0
4x
0
0
4x 2
0
0
4
1
0
13.
0
0
dz
2 0
0
a
a2 x 2
a2 x 2y 2
19. 8
0
0
4x 2
2
0
0
0
z2 2
dy dx
2
2
4
0
0
4
1
15 1 1 2 e
x1 xcos y dy dx
0
4
dx
x1 xdx
x2 x3
3 4
2
8
0
0
64 40 3 3
128 15
x sin y ln z
4
2
dy dx
1
0
x 2 ln4cos y
4x2
0
2
dx
0
x 2 ln 4 1 cos 4 x 2 dx 2.44167
3
9x2
3
9x2
2
2
9x y
dz dy dx
0
4y 2
x dx dy
2
2
18
1 10
x3 dy dx 2
1
z1 e1 dz 1 e1
1x
2 0
1 2
0
0
15.
2
dz dx dy
0
3
2
1
dz dy dx
x
dz 3z z2
2zxex dx dz
0
4y 2
2
1
4xy
2
17.
1
x4 x5 dx 2 10
4
0
4x
4
2
4
0
4x
0
x cos yz
4x 2
x 2 sin y dz dy dx z
dx
dx dz
x1 xsin y
2
x dz dy dx
0
2
2
11.
x2
4x 2
2
0
1
x
0
0
4
x
x 2y 2 2
0
9.
1 1 y y 2 yz 2 2
dy dx
2
3
0
0
2zex dy dx dz
x 2y dy dx
x
dy dx
0
1 y z dy dz 2
x
0
5.
xz
0
1
0
1
xy
x
0
1 2 x xy xz 2
0
3
3.
Triple Integrals and Applications
2
4 y 22 dy
0
a
dz dy dx 8
0
a2 x 2
a y x
256 15
a2 x2
y a2 x 2 y 2 a2 x 2 arcsin
2
a2 x 2 y 2 dy dx
2
0
4
0
0
a
4
2
8 1 16 8y 2 y 4 dy 16y y3 y5 3 5
a
0
1 a2 x 2 dx 2 a2x x3 3
a 0
4 a3 3
2
0
dx
157
158
Chapter 13
4x 2
2
21.
0
0
Multiple Integration
4x 2
2
dz dy dx
2
4 x 22 dx
0
0
0
8 1 16 8x 2 x 4 dx 16x x3 x5 3 5
23. Plane: 3x 6y 4z 12
2
0
256 15
25. Top cylinder: y 2 z2 1 Side plane: x y
z 3
z
1 2
y
3
4 x
1
x
3
0
(124z) 3
0
1
y
(124z3x) 6
dy dx dz
1
0
0
x
0
1y 2
dz dy dx
0
27. Q x, y, z: 0 ≤ x ≤ 1, 0 ≤ y ≤ x, 0 ≤ z ≤ 3
3
xyz dV
0
Q
1
0
1
3
xyz dx dy dz
y
0
0
1
0
1
0
1
0
1
1
0
y
x
xyz dy dx dz
3
0
3
0
1
y
x
0
1
R
xyz dx dz dy
x
y
1
x
xyz dy dz dx
0 3
xyz dz dx dy
0 3
0
xyz dz dy dx
9 16
29. Q x, y, z: x2 y2 ≤ 9, 0 ≤ z ≤ 4
4
xyz dV
0
Q
4
0
3
3
xyz dy dx dz
9y2
xyz dx dy dz
3 9y2 4
4
9y2
x
xyz dx dz dy
9y2
9y2
4
3 0 3
5
9x2
4
xyz dz dx dy
3 9y2 0 3
z
3 9x2
3 0 3
3
y=x
1
0
9x2
xyz dy dz dx
9x2
9x2
4
3 9x2 0
xyz dz dy dx 0
3
3
4
y
Section 13.6
dz dy dx
0
x dz dy dx
6
31.
4(2x 3)
mk
0
4(2x 3)
Myz k
0
2(y 2)(x 3)
0
8k
6
Triple Integrals and Applications
0
2(y 2)(x 3)
0
12k x
Myz 12k 3 m 8k 2
4
33.
4
0
0
4
4k
0
4
0
0
2k
0
b
x4 x dy dx
0
4
xz dz dy dx k
0
0
0
4 x2 x dy dx 2
128k 16x 8x 2 x3 dx 3
39. y will be greater than 0, whereas x and z will be unchanged. 1 m kr 2h 3 x y 0 by symmetry
r
0
z
3kh2 r2 4kh2 3r 2
r 2x 2
h
z dz dy dx
h x 2y 2 r
0
r
0
0
b
b
0
b
0
b
0
b
0
x 2y dz dy dx
kb6 6
xy 2 dz dy dx
kb6 6
xyz dz dy dx
kb6 8
0 b
0 b
Mxy k
0
r 2x 2
r 2 x 2 y 2 dy dx
0
r
r 2 x 23 2 dx
0
kr 2h2 4 Mxy kr 2h2 4 3h m kr 2h 3 4
kb5 4
b
Mxz k
0
xy dz dy dx
0
Myz kb6 6 2b 5 x m kb 4 3
37. x will be greater than 2, whereas y and z will be unchanged.
Mxy 4k
b
Myz k
Mxy z 1 m
41.
b
mk
b
4
0
35.
0
4x
4
128k 4x x 2 dx 3
Mxy k
4
4
x dz dy dx k
0
4
0
4x
mk
y
Mxz kb6 6 2b 5 m kb 4 3
z
Mxy kb6 8 b 5 m kb 4 2
159
160
43.
Chapter 13
m
Multiple Integration
128k 3
x y 0 by symmetry z 42 x 2 y 2
42 x 2
4
Mxy 4k
0
0
2k
0
2
1024k 3
4
0
42 x2
0
dx
4k 3
4
4 x 23 2 dx 2
0
let x 4 sin
cos4 d
by Wallis’s Formula
5 y 12
3
y
mk
0
0
Myz k
0
0
0
0
4 x 4
(5 12)y
8
12
16
20
y dz dy dx 1200k
(5 12)y
z dz dy dx 250k
0
x
Myz 1000k 5 m 200k
y
Mxz 1200k 6 m 200k
z
Mxy 250k 5 m 200k 4
a
a
0
a
a
0
0
Ix Iy Iz
a
a
a 0
a
dz ka
0
0
2ka5 by symmetry 3
y 2 z2xyz dx dy dz
0
0
0
ka2 2
a
y 4z y2z3 4 2
Ix Iy Iz
a 0
dz
ka4 8
ka8 by symmetry 8
y 2 z2 dy dz
0
1 1 3 1 2 a az dz ka a3z az3 3 3 3
a
(b) Ix k
0
a
y 2 z2 dx dy dz ka
1 3 y z2y 3
ka
a
47. (a) Ix k
0
x dz dy dx 1000k
0
(3 5)x12
20
y = − 35 x + 12
8
0
0
Mxy k
16 12
(5 12)y
(3 5)x12
Mxz k
dz dy dx 200k
0
0
20
20
(5 12)y
(3 5)x12
20
3
128k 2
(3 5)x12
20
1 16y x 2y y3 3
0
Mxy 64k m 1
45. f x, y
42 x 2 y 2 dy dx 2k
0
64k z
z dz dy dx
0
42 x 2
4
42 x 2y 2
ka2 2
a
0
0
2ka 3
a
y3z yz3 dy dz
0
a
0
5
a
a2z 2z3 dz
ka8 a2z 4
2 2
2z4 4
a 0
ka8 8
Section 13.6
4
4
0
0
4
4
0
k
0
0
4
0
4
4
0
4
0
4
0
y3 4 x 3
4 0
4
4
0
4
0
4
0
0
4
0
4x
4
0
0
4 0
a
a2 x 2
2k 3
L 2
k
L 2
L 2
2
L 2
4 0
1024k 3
x 2y y34 x dx
0
4
8x 2 644 x dx
dx k
0
L 2 a a2 x 2
4
yx 2 y 2 dz dy dx k
2 3
0
1 x 2y4 x y4 x3 dy dx 3
4 1 32 8x 4x 2 x3 dx 8k 32x 4x 2 x3 x4 3 4 0
L 2
8 644 x 4 x3 dx 3
4
4
0
2048k 3 4
x 2y 2 y 4 4 x 2 4
0
51. Ixy k
0
0
4
k
8k
4
yx 2 z2 dz dy dx k
Iz k
0
4
64 4 x dx 256k 3
1 4 1 1 4x 2 x3 4 x3 dx 8k x3 x4 4 x4 3 3 4 12
8k
4
0
512k 3
1 y34 x y4 x3 dy dx 3
0
dx k
4x
Iy k
0
4
y y 2 z2 dz dy dx k
2 k 324 x2 4 x4 3 4
4x 2
0
0
0
0
dx k
y4 y2 4 x 4 x3 4 6
4
4
x 2 y 24 x dy dx
0
4x
(b) Ix k
0
4
x 2 y 2 dz dy dx k
x 2y
4
1 x 24 x 4 x3 dy dx 3
4x
4
k
0
0
k
0
4
0
Iz k
4
4
x 2 z2 dz dy dx k
1 4 1 1 4x2 x3 4 x3 dx 4k x3 x 4 4 x4 3 3 4 12
0
4
64 4 4 x 4 x3 dx 3 3
256k
0
0
4k
0
4
4x
Iy k
0
0
4
dx k
0
1 3 y 24 x 4 x dy dx 3
0
4
32 1 4 x2 4 x4 3 3
4
4
y 2 z2 dz dy dx k
y3 y 4 x 4 x3 3 3
k
4x
49. (a) Ix k
Triple Integrals and Applications
L 2
z2 dz dx dy k
a
L 2 a
4 0
2048k 3
2 2 a x2 a2 x 2 dx dy 3
x 1 x a2 x a2 x 2 a2 arcsin x2x 2 a2 x 2 a2 a4 arcsin 2 a 8 a a4 a4 a4Lk dy 4 16 4
Since m a2Lk, Ixy ma2 4. —CONTINUED—
a a
dy
161
162
Chapter 13
Multiple Integration
51. —CONTINUED—
L2
Ixz k
a2 x 2
a
L2 a a2 x 2 L2
2k
L2
a
a2 x 2
L2 a a2 x 2
a
L2 a
y2 x xa2 x 2 a2 arcsin 2 a
L2
L2
Iyz k
y 2 dz dx dy 2k
a a
L2
L2 a
2
Iy Ixy Iyz
ma2 ma2 ma2 4 4 2
Iz Ixz Iyz
mL2 ma2 m 3a2 L2 12 4 12
1
L2
a a
dy
ka4 4
L2
dy
L2
ka4L ma2 4 4
2
ma mL m 3a2 L2 4 12 12
2ka2 L3 1 mL2 3 8 12
y 2 dy
x 2a2 x 2 dx dy
1 x x2x 2 a2a2 x 2 a4 arcsin a L2 8
Ix Ixy Ixz
1
dy ka2
a
L2
2k
53.
L2
x 2 dz dx dy 2k
y 2a2 x 2 dx dy
1x
x 2 y 2x 2 y 2 z2 dz dy dx
1 1 0
55. See the definition, page 978. See Theorem 13.4, page 979.
57. (a) The annular solid on the right has the greater density. (b) The annular solid on the right has the greater movement of inertia. (c) The solid on the left will reach the bottom first. The solid on the right has a greater resistance to rotational motion.
Section 13.7
4
1.
0
2
0
Triple Integrals in Cylindrical and Spherical Coordinates
2
4
r cos dr d dz
0
0
4
0
2
3.
2 cos2
0
0
2
0
2 cos r2
2
d dz
0
2
4
2 cos d dz
0
0
4r2
r sin dz dr d
2
0
0
2 cos2
2 sin
2
r4 r 2sin dr d
2
4
0
0
0
z
0
2 sin d d d
0
4
7.
cos
2
0
rer d dr dz e4 3
1 3
2
0
0
2
8 cos4 4 cos8 sin d
2
0
4
0
cos3 sin d d
2dz 8
0
0
0
5.
4
dz
1 12
2r
2 cos2
r4 sin 4
2
cos4
4
0
d
d
0
8 cos5 4 cos9 5 9
0
2
2
0
8
52 45
Section 13.7
2
9.
0
3
2
r
e
0
r dz dr d
2
3
0
0
2
2
11.
2
2
2 sin d d d
0
2
2
4 sec
0
0
2
0
2
0
2
2
6
sin d d
cos
0
323 3
2
z 4
2
6
d
d
4
4
x
y
0
643 3
2
a cos
2
2
cot csc
arctan12 0
3 sin2 cos d d d 0
3 sin2 cos d d d 0
2
0
0
0
a cos
ra2 r 2 dr d
0
a2 r2
r dz dr d
2
0
4 2 2a 3 a3 3 4 3 2 3 9
ra2 r 2 dr d
a cos
0
2a3 3 4 9
2
2
krr dz dr d
0
2
kr 29 r cos 2r sin dr d
0
2
0
9r cos 2r sin
k 3r 3
r4 r4 cos sin 4 2
0
k 24 4 cos 8 sin d
0
0
1 sin3 d
2a3 cos3 cos 3 3
d
2
0
0
1 a2 r 232 3
0
0
0
2
21. m
0
a cos
0
2
4 1 1 sin3 d a3 cos sin2 2 3 3
a cos
0
r dz dr d 4
0
a2 r2
0
4 a3 3
2a3 3
r 2 cos dz dr d 0
a sec
0
3 sin2 cos d d d
x
0
2a cos
0
2
64 3
y
3
a
4
2
64 3
3
1 2
a a2 r2
a
0
19. V 2
1
0
2
17. V 4
2
r2
arctan12
0
(b)
0
r 2 cos dz dr d 0
2
15. (a)
d
4
0
0
(b)
3
3
1 e9 4
13. (a)
z
1 1 e9d 2
4
6
0
2
1 2 er 2
0
rer dr d
0
0
Triple Integrals in Cylindrical and Spherical Coordinates
k 24 4 sin 8 cos k 48 8 8 48k
2
0
2 0
d
163
164
Chapter 13
23. z h
h h x2 y2 r0 r r0 r0
2
V4
0
Multiple Integration
4h r0
0
2
r0
r0r r 2 dr d
1 r02h 2 3
2
0
0
r3 dz dr d
r0
0
r0r3 r 4 dr d
2
0
2
b
r3 dz dr d
0
2
b
r3 dr d
0
a
2
kh
h
a
4kh
1 k r04h 10
b4 a4 d
0
Since the mass of the core is m kV k3 r02h from Exercise 23, we have k 3m r02h. Thus,
kb4 a4h 2
1 Iz k r04h 10
kb2 a2b2 a2h 2
1
1 3m r04h 10 r02h
3 mr 2 10 0
31. V
2
0
0
r 2 z dz dr d
0
Mxy k r03h230 h m k r03h6 5
Iz 4k
0
29. m kb2h a2h khb2 a2
0
4kh r05 r0 20
h(r0 r)r0
r0
1 k r03 h2 30
0
2
4kh r0
h(r0 r)r0
r0
2
0
z
r 2 dz dr d
0
Mxy 4k
27. Iz 4k
0
1 k r03h 6
r0 d 6
4h r03 r0 6
h(r0 r)r0
r0
0
3
0
2
m 4k
0
2
4h r0
r dz dr d
0
kx 2 y 2 kr x y 0 by symmetry
hr0 rr0
r0
0
25.
4 sin
2 sin d d d 16 2
1 ma2 b2 2
33. m 8k
0
2
0
2ka4
2
0
0
2
2
0
ka4
a
3 sin d d d sin d d
0
2
sin d
0
ka4
2
ka4cos
0
Section 13.7
35.
Triple Integrals in Cylindrical and Spherical Coordinates
2 m kr 3 3
37. Iz 4k
x y 0 by symmetry
2
Mxy 4k
2
0
1 kr 4 2
kr 4 4
0
2
2
0
3 cos sin d d d sin 2 d d
1 k r 4 cos 2 8 z
2
cos5 sin3 d d
0
2
4
cos5 1 cos2 sin d
k 192
2
1
6
1 cos8 8
41.
1
g2
g1
h2r cos , r sin
h1r cos , r sin
zz (b) 0: sphere of radius 0
43. (a) r r0: right circular cylinder about z-axis
0: plane parallel to z-axis
0: plane parallel to z-axis
z z0: plane parallel to xy-plane
0: cone
0
4
f r cos , r sin , zr dz dr d
y x
tan
zz
2
x2 y 2 r 2
y r sin
a
2
Mxy kr 44 3r m 2kr33 8
39. x r cos
45. 16
2
5 k 6 cos
1 k r 4 4
0
4 sin3 d d d
0
0
4
cos
sin 2 d
0
2
2
2 k 5
0
2
4
4 k 5
r
0
2
a2 x 2
0
a2 x 2y 2
0
a2 x2 y 2z2
0
dw dz dy dx
a2 x 2
a
16
0
0
a
0
0
2
16
0
2
0
4
a
0
2
0
a4
2
0
a2 r 2
a2 r 2 z2 dzr dr d
0
a
0
8
a2 x 2 y 2 z2 dz dy dx
0
2
16
a2 x 2y 2
1 z za2 r 2 z2 a2 r 2 arcsin 2 a2 r 2
2 a r 2r dr d 2 a2r 2 r 4 2 4
d
a4 2 2
a 0
d
a2 r2
0
r dr d
165
166
Chapter 13
Multiple Integration
Section 13.8
Change of Variables: Jacobians
1 1. x u v 2
3. x u v 2 yuv
1 y u v 2
x y y x 11 12v 1 2v u v u v
y x 1 x y u v u v 2
12 1212
1 2
5. x u cos v sin y u sin v cos x y y x cos2 sin2 1 u v u v 7. x eu sin v y eu cos v y x x y eu sin veu sin v eu cos veu cos v e2u u v u v 9. x 3u 2v y 3v v
y 3
u
x 2v x 2 y3 3 3
x, y
u, v
0, 0
0, 0
3, 0
1, 0
2, 3
0, 1
v
(0, 1)
1
(1, 0) u
1
x 2y 3 9
1 11. x u v 2 1 y u v 2 y x 1 x y u v u v 2
21 1212 21
1
4x 2 y 2 dA
1
1 1
R
1
12 dv du
14 u v
4
1 u v2 4
2
1
1 1
1
u2 v 2 dv du
1
2 u2
u3 u 1 du 2 3 3 3
13. x u v
R
1
8 3
4
y x x y 10 11 1 u v u v
3
yx y dA
1
v
yu
0
4
0
3
2
3
uv1 dv du
0
8u du 36
1 u 1
2
3
4
Section 13.8
15.
xy2
e
Change of Variables: Jacobians
dA
y = x1
y
R
4
1 4 x R: y , y 2x, y , y 4 x x
y = 2x 3
y = 4x
x u x, y y u, v u
1 v12 x v 2 u32 y 1 v12 v 2 u12
R 1 x
1 1 2 u12v12 1 1 1 1 4 u u 2u 1 u12 2 v12
Transformed Region: y
y = 41 x
2
y x vu, y uv ⇒ u , v xy x
1
2
3
4
3
4
v
1 ⇒ yx 1 ⇒ v 1 x
3
S 2
4 y ⇒ yx 4 ⇒ v 4 x y 2x ⇒ y
u
y 2 ⇒ u2 x
1
y 1 1 x ⇒ ⇒ u 4 x 4 4
2
exy2 dA
4
14 1
R
2u1 dv du e u 2
ev2
vxy0
u x y 8,
vxy4
1 x u v 2
1 y u v 2
1 e2 e12 du u 14
17. u x y 4,
2
du
1
14
e2 e12ln u
2
e2 e12 ln 2 ln
14
1 e12 e2ln 8 0.9798 4
y
6
x−y=0
x+y=8
4
2
1 x, y u, v 2
v2 4
x−y=4
x+y=4
x
8
x yexy dA
4
R
1 2
2
4
uev
0
4
6
12 dv du
8
ue4 1 du
4
19. u x 4y 0,
vxy0
u x 4y 5,
vxy5
1 x u 4v, 5
1 y u v 5
14 u e 2
4
1
8 4
12e4 1
y
x−y=0 2
x + 4y = 5
y x 1 x y u v u v 5
1
51 1545 15
5
x yx 4y dA
uv
0
R
5
5
0
0
x
−1
3 −1
x + 4y = 0 −2
1 du dv 5
1 2 32 u v 5 3
5 0
dv
2 3 5 23v
5
32
0
100 9
4
x−y=5
167
168
Chapter 13
Multiple Integration
1 1 21. u x y, v x y, x u v, y u v 2 2 x y y x 1 u v u v 2
a
x y dA
u
u
0
R
12 dv du
a
u
u u du
0
y
5 u 2
a
52
0
2 a52 5
y
v=u a a
x+y=a x 2a
a
23.
x −a
v = −u
x2 y 2 2 1, x au, y bv a2 b
(a)
x2 y 2 21 a2 b
u2 v 2 1 v
au2 bv2 2 1 a2 b
y
1
u2 v 2 1
b
S R
u x
a
(b)
x, y x y y x u, v u v u v ab 00 ab
(c) A
ab dS
S
ab12 ab 25. Jacobian
x y y x x, y u, v u v u v
27. x u1 v, y uv1 w, z uvw
1v x, y, z v1 w u, v, w vw
u 0 u1 w uv uw uv
1 v u2v1 w u2vw u uv21 w uv2w 1 vu2v uuv 2 u2v
29. x sin cos , y sin sin , z cos
sin cos x, y, z sin sin , , cos
sin sin sin cos 0
cos cos cos sin sin
cos
sin cos sin sin cos cos sin
sin2 cos2 sin2 sin2 2
2
2
2
cos
2 sin cos sin2 cos2 sin
sin2 cos2 sin2 2 sin cos2 2 sin3
2 sin cos2 sin2 2 sin
1
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