Post on 23-Oct-2014
C H A P T E R 13 Multiple IntegrationSection 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 IntegrationSection 13.1 Iterated Integrals and Area in the PlaneSolutions to Odd-Numbered Exercisesx
1.0
2x
y dy
2xy
1 2 y 2
x 0
3 2 x 2
2y
3.1
y dx x
2y
y ln x1
y ln 2y
0
y ln 2y
4
x2
5.0
x 2y dy
1 2 2 x y 2
4 0
x2
4x 2 2
x4
y
7.ey
y ln x dx x
1 y ln2 x 2
y ey
1 y ln2y 2x3 x3
ln2ey
y ln y 2
2
y2
x3
x3
9.0
ye u1
y x
dy dy, dv
xye e
y x 0
x0
e xe
y x
dy
x4 e
x2
x 2e
y x 0
x2 1
e
x2
x 2e
x2
y, du2
y x
dy, v 1 2 y 2
y x
1
2
1
1
11.0 0
x
y dy dx0
xy
dx0 0
2x
2 dx
x2
2x0
3
1
x
1
x
1
13.0 0
1
x2 dy dx0
y 1
x20
dx0
x 1
x2 dx
1 2 1 2 3
1
x2
3 2 0
1 3
2
4
2
15.1 0
x2
2y 2
1 dx dy1 2 1
1 3 x 3 64 3
4
2xy 2 8y 2
x0
dy 4 32
4 dy
191
6y 2 dy
4 19y 3
2y
3
2 1
20 3
1
1 0
y2
1
17.0
x
y dx dy0 1 0
1 2 x 2 1 1 2
xy0
1
y2
dy 1 y 2 1 3 y 6 1 2 1 2 31
y2
y 1
y 2 dy
y2
3 2 0
2 3
2
4 0
y2
19.0
2 4 y
2
dx dy 2
0
2x 4 y2sin
4 0
y2
2
2
dy0
2 dy
2y0
4
2
sin
2
21.0 0
r dr d0
r2 22
2
d0 0
1 sin2 2 1 42
d 1 cos 2 42 2
1 4
cos 20
d
2
2
sin 20
32
1 8
133
134
Chapter 131 x
Multiple Integration1 x
23.1 0
y dy dx1
y2 2
dx0
1 2
1
1 dx x2
1 2x
01
1 2
1 2
25.1 1
1 dx dy xy
1
1 ln x y
dy1 1
1 y
1 0 y
dy
Divergesy
8
3
8
3
8
8
27. A0 3 0 8
dy dx0 3
y0 8
dx0 3
3 dx
3x0 3
24
8 6
A0 0
dx dy0
x0
dy0
8 dy
8y0
24
4
2 x2 4 6 8
2
4 0
x2
2
4
x2
y
29. A0 2
dy dx0
y0
dx
4 3
y = 4 x2
40
x2 x3 34 y 2 0
dx 16 3 dx dy
2 1 x 1 2 3
4x4
1
A0 4 0 4 0
x0
y
4
4
dy0
4
y dy0
4
y
1 2
1 dy
2 4 34 2 0 x
y
3 2
4 0
2 8 34
16 32 x2
2
1
4
x2
31. A2 x 1 2 4
dy dxx2
y = 4 x2
y
33.03
dy dx0 4
y0
dx
(1, 3)
y2 1 x 2
dx
42
4 x 8 x x 3
x dx x2 24 0
y=x+21
0
42 1
x
2
x
2 dx
4x2 1 x 1 2
8 3
4
2 0
y
2
22
x 1 2 x 2
x 2 dx 1 3 x 3 dx dy1 2 4
dx dy0
8 3
2x3 y
9 24 0 4 y
Integration steps are similar to those above.y 4
2
A0 3 4 y y 2
23 4
dx dyy
32
y = (2
x )2
x0 3 4 y
dy
23
x0 4
dy
1 x1 2 3 4
y0
2 2y
4 2 4 3
y dy3
23
4 4 4 3
y dy4
1 2 y 2
y
3 2 0
y
3 2 3
9 2
Section 13.13 2x 3 5 5 0 5 x x
Iterated Integrals and Area in the Planea 0 0 a b a a2 x2 a b a a2 x2
135
35. A0 3 0
dy dx3 2x 3 5
dy dx y3 5 0
A 37. 4
dy dx0 2
y0
dx d
y0 3 0 0
dx 53
dx x
b a
a20
x 2 dx
ab0
cos2
2x dx 33
a sin , dx ab 22
a cos d cos 2 d ab 2 1 sin 2 22 0
x dx 1 2 x 25
10
1 2 x 32 5
5x0 y
53
ab 4 Therefore, A A 4b 0 0 a b
A0 2 3y 2 5
dx dyy
ab.b2 y2
x0 2 3y 2
dy 3y dy 2 5y 5 2 y 42
dx dy
ab 4
50 2
y
Therefore, A above.y
ab. Integration steps are similar to those
52y
5y dy 2
50b
y= b a
a2 x2
a
x
4 32
y= 2x 3
y=5x
1 x1 2 3 4 5
1
4
y
2
4
x2
39.0 0
f x, y dx dy, 0 x y, 0 y 44 0y
41.2 0 2
f x, y dy dx, 0 y 4 4y 3
4
x2,
2 x 2
4
y2
f x, y dy dxx 0 y2
dx dy
3 2 1 x 1 2 3 4 2 1 1 1 x 1 2
10
ln y
1
1
43.1 0
f x, y dx dy, 0 x ln y, 1 y 10ln 10 0y
45.1 x2
f x, y dy dx, x 2 y 1, 1 x 11 y
10
f x, y dy dxex
f x, y dx dy0y4
y
83
6 4 2 x 1 2 32 1 x 1 2 2
1361
Chapter 132 2 1
Multiple Integration1 1 1 y2 1 1 0 x2
47.0y
dy dx0 0 0
dx dy
2
49.0 y2y
dx dy1
dy dx
2
3
1
21
x
1x
1
1
2
3
2
x
4
4 0
x
2
4 y
y
2
1
1
2y
51.0y
dy dx0 2
dy dx0
dx dy
4
53.0y
dy dxx 2 0 0
dx dy
1
3
22
1x1 2 3 4
1
1
x
1
2
1
3
y
1
x
55.0 y2
dx dy0 x3
dy dx
5 122
x= 3 yy
x = y2
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 x 50 x2 5
x 2y 2 dy dx0
1 2 x 50 3
x2
3 2
1 5 x dx 3
15625 245 0 y 5 2 0 50 y2 5
x 2y 2 dx dy0 5
x 2y 2 dx dy0
1 5 y dy 3
5 5
2
1 50 3
y2
3 2
y2 dy
15625 18
15625 18
15625 18
15625 24y
y=
50 x 2
(0, 5 2 )5
(5, 5) y=xx 5
Section 13.22 2 2 y 2
Double Integrals and Volume
137
59.0 x
x 1
y3 dy dx0 0 2
x 1 1 2
y3 dx dy0
1 1 2 1 3 2 1 3x
y3
x2 23 2
y
dy0 2
10
y3 y 2 dy
y3
0
1 27 9
1 1 9
26 9
1
1
1
x
1
61.0 y
sin x 2 dx dy0 1 0
sin x 2 dy dx0
y sin x 20 1 0
dx 1 cos 1 24
x sin x 2 dx0
1 cos x 2 2
1 1 2y 0
1 1 2
cos 1
0.2298
2
2x
63.0 x2
x3
3y 2 dy dx
1664 105
15.848
65.0
x
2 1 y
1
dx dy
ln 5
2
2.590
67. (a) x x8
y3 y
x1
3
y
4 2y x2x1 3
32y y
x2 32
4
x = y32
(8, 2)
(b)0 x2 32
x 2y
xy 2 dy dx2
x 2 4 6 8
x = 4 2y
(c) Both integrals equal 67520 693
97.43
2
4 0
x2
2
1 0
cos
69.0
exy dy dx
20.5648
71.0
6r 2 cos dr d
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. 77. True
Section 13.2For Exercise 13, 1 1 , , 2 2 3 1 , , 2 2 xi
Double Integrals and Volumeyi 1 and the midpoints of the squares are 7 1 , , 2 2 1 3 , , 2 2 3 3 , , 2 2 5 3 , , 2 2 7 3 , . 2 24 32
y
5 1 , , 2 2
1 x1 2 3 4
1. f x, y8
x
y xi yi x 1 24
f xi, yii 1 4 0 2
3
4 y2 2
22
3
44
5
244
y dy dx0
xy
dx0 0
2x
2 dx
x2
2x0
24
0
138
Chapter 13 x2 y2 xi yi x20 0
Multiple Integration
3. f x, y8
f xi, yii 1 4 2
2 4
10 44
26 4 x2y0
50 4 y3 3
10 42
18 44
34 4 2x 2
58 4 8 dx 3
52 2x3 3 8x 34 0
y 2 dy dx
dx0 0
160 3
4
4
5.0 0
f x, y dy dx
32 400
31
28
23
31
30
27
22
28
27
24
19
23
22
19
14
Using the corner of the ith square furthest from the origin, you obtain 272.2 1 2y
1
7.0 0
1
2x
2y dy dx0 2
y
2xy
y
2 0
dx3 2
20
2x dx1
2
2x 86 3 6
x20
x
1
2
3
9.0 y 2
x
y dx dy0 6 0
1 2 x 2 9 2 3y 3 2 y 2
3
xyy 2
dy6
y
(3, 6)
5 2 y dy 8 5 3 y 246 0
4
9 y 2 36a a2 a2 x2
2
x2 4 6
a
11.a x2
x
y dy dxa a
xy
1 2 y 2
a2 a2
x2 x2
y
dx
a
2x a2a
x 2 dxa
a
a
x
2 2 a 35 3 3 5
x2
3 2 a
0
a
13.0 0
xy dx dy0 3 0 0
x y dy dx5
y
1 2 xy 23
5
dx0
4 3 2
25 2
x dx0 3 0
1 x 1 2 3 4 5
25 2 x 4
225 4
Section 13.22 y y 2
Double Integrals and Volumey
139
15.0
y x2 y
4 2 dx dy 2
2 y 2
y x2 y
2 2 dx dy 0
2x x 2
y x2 ln x 2 y
2 dy dx
4 3
y = 2x x=2
1 2 1 2
2x
y2x
dx
2
0 2
1
y=xx1 2 3 4
ln0
5x 22
ln
2x 2
dx
1 5 ln 2 2
dx0 2
5 1 ln x 2 2
ln0
5 2y
4
4 4 y
y
1
4 4 1
x2
17.3
2y ln x dx dy0 x
2y ln x dy dx4 x2
4 3
(1, 3)
ln x0 1
y24 x 2
dx x2 4 x2
2 1
ln x 40
dx1 3 4
x
26 25
4
3x 4
5
25 0
x2
3
25 4y 3
y2
y
19.0 0
x dy dx4
x dy dx0 3 0
x dx dy 1 2 x 23 25 4y 3 y2
5 4
x= x= 4y 3 (4, 3)
25 y 2
dy y 2 dy
3 2 1
25 18
90
x 1 2 3 4 5
25 9y 184 2 0
1 3 y 3
3
250
21.0
y dy dx 2
4 0 4
y2 4
2
y
dx0
4 3
dx0
4
2
1 x1 2 3 4
2
y
2
23.0 0
4
x
y dx dy0 2
4x 4y0
x2 2 y2 2 y3 6
y
xy0
dy2
y
y2 dy y3 3 42 01 2 1
2y2 8 8 6
y=xx
8 3
1406
Chapter 132 3x 0 4
Multiple Integration6 2 3x 0 4y
25.0
12
2x 4
3y
dy dx0 6 0
1 12y 4 1 2 x 6 2x x2
2xy
3 2 y 2
dx
5 4
y = 2x + 4 3
6 dx6
3 2 1 x 1 2 3 4 5 6
1 3 x 18 121 y 1
6x01
y
27.0 0
1
xy dx dy0 1
x
x 2y 2
y
dy01
y0
y3 dy 2 y4 81 0
y=xx1
y2 2 3 8 10 0
29.
x4 0 x2
1
2
y
1
2
dy dx0
1 x 12
y
1
dx0 0
1 x 12
dx
1 x 10
1
2
31. 40
4
x2
y 2 dy dx
8
1
x
2
4
33. V0 1 0 0
xy dy dx 1 2 xy 21 0 x
35. V0 0
x 2 dy dx2 4 2
dx0
1 2
1
x3 dx0 0
x 2y0 2 0
dx0
4x 2 dx
1 4 x 8y
1 8
4x3 3y
32 3
4
1
y=x3 2 1 x 1 2 3
x1
1
37. Divide the solid into two equal parts.1 x
y
y=x
V
20 1 0
1
x 2 dy dxx
1
20 1
y 1
x20
dxx1
20
x 1 2 1 3
x 2 dx1
x2
3 2 0
2 3
Section 13.22 4 0 x2 2 4 0 x2
Double Integrals and Volume
141
39. V0 2
x 1 2 y 2 x2
y dy dx4 0 x2
41. V
40 2
x2 x2 40 2
y 2 dy dx x2 1 4 3 32 cos4 3 x23 2
xy0 2
dx 1 2 x dx 2
4 40
dx,
x
2 sin
x 40
2
16 cos2 32 3 3 16
d
1 4 3y
x2
3 2
2x
1 3 x 6
2 0
16 3
4 16 8y
4
2
y=
4 x21
x 2 + y2 = 4
11x
1 1
x1 2
2
4 0
x2
2
0.5x 0
1
43. V
40
4
x2
y 2 dy dx
8
45. V0
1
2 x2
y2
dy dx
1.2315
47. f is a continuous function such that 0 f x, y 1 over a region R of area 1. Let f m, n and f M, N the maximum value of f over R. Then f m, nR
the minimum value of f over R
dA R
f x, y dA f M, NR
dA. f x, y dA f M, N 1 1.R
SinceR
dA
1 and 0 f m, n f M, N 1, we have 0 f m, n 1 f x, y dA 1.R
Therefore, 0
1
1 2
1 2
2x
1
arccos y
49.0 y 2
e
x 2 dx
dy0 1 2 0
e
x 2 dy
dx
51.0 0
sin x 12 cos x
sin2 x dx dy
2xe0
x 2 dx 0 1 2 0 2
sin x 1
sin2 x dy dx
e e 1y
x2 0 1 4 0
1 1 0.221y
sin2 x
1 2
sin x cos x dx2
e
1 4
1 2
2 1 3
sin2 x
3 2 0
1 2 2 3
1
y = 2x1 2
y = cos x1 2
1
x1 2 1
2
x
142
Chapter 13 1 84 0 2
Multiple Integration 1 84
53. Average
x dy dx0
2x dx0
x2 8
4
20
55. Average
1 4 1 4
2 0 2 0
2
x20
y 2 dx dy2
x3 3
xy 20
dy2 0
1 4 8 3
2 0
8 3
2y 2 dy
1 8 y 4 3
2 3 y 3
57. See the definition on page 946.
59. The value ofR
f x, y dA would be kB.
61. Average
1 1250 1 1250
325 300 325
250
100x 0.6y 0.4 dx dy200
100y 0.4300
x1.6 1.6
250
dy200
128,844.1 1250
325
y 0.4 dy300
103.0753
y1.4 1.4
325
25,645.24300
63. f x, y 0 for all x, y and5 2 0 2 1
f x, y dA0 2
1 dy dx 10 1 dy dx 10
5 0 2 0
1 dx 5 1 dx 10
1 1 . 5
P 0 x 2, 1 y 20
65. f x, y 0 for all x, y and3 6 3
f x, y dA0 3 0 1
1 9 27
x
y dy dx y2 26 3
1 9y 276 4
xy
dx3 0 1
1 2 2 4 27
1 x dx 9 x dx
x 2 7 . 27
x2 18
3
10
P 0 x 1, 4 y 60
1 9 27
x
y dy dx0
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, V 4 3 6 7 3 2 100 2500m3.(15, 5, 6) (25, 5, 4)7 z
(15, 15, 7) (5, 5, 3) (5, 15, 2)20 y
30 x
(25, 15, 3)
1
2
6
2
69.0 0
sin (a) 1.78435 (b) 1.7879
x
y dy dx
m
4, n
8
71.4 0
y cos (a) 11.0571 (b) 11.0414
x dx dy
m
4, n
8
Section 13.3
Change of Variables: Polar Coordinates
143
73. V
125(4, 0, 16) 16
z
75. False(4, 4, 16)1 1 0 y2
Matches d.
V
80
1
x2
y 2 dx dy
(4, 0, 0)5 x
(0, 4, 0)5 y
(4, 4, 0)
1
1
x
1
1
t
77. Average0
f x dx0 1 0 t 1
e t dt dx0 1 x
2
et dt dx1
2
e t dx dt0 1 0 0
2
te t dt 1 1 1 2x
2
1 t2 e 2
1 e 2
e
1
Section 13.3
Change of Variables: Polar Coordinates3. Polar coordinates
1. Rectangular coordinates 5. R r, : 0 r 8, 0
7. R
r,
: 0 r 3
3 sin , 0
2
Cardioid
2
6
2
6
9.0 0
3r 2 sin dr d0 2
r 3 sin0
d
2
216 sin d0 204
216 cos0
0
2
3
2
11.0 2
9
r 2 r dr d0
1 9 32 0
3
r2
3 2 2
d
2
5 5 3 5 5 6
0 1 2 3
2
1 0
sin
2
13.0
r dr d0 2 0
r2 2 1 1 2
1 0
sin
d
2
sin
2
d 1 2 1 cos 2 sin 1 2 1 2 sin 82 00 1 2
1 8 3 32
2
sin 9 8
cos
2
144a
Chapter 13a2 0 y2
Multiple Integration2 a
15.0
y dx dy0 0
r 2 sin dr d
a3 3
2
sin d0
a3 3
2
cos0
a3 3
3
9 0
x2
2
3
17.0
x2
y2
3 2
dy dx0 0
r 4 dr d
243 5
2
d0
243 10
2
2x 0
x2
2
2 cos
2
19.0
xy dy dx0 0
r3 cos sin dr d
40
cos5
sin d
4 cos6 6 2
2 0
2 3
2
x
2
2 0
8
x2
4
2 0
2
21.0 0
x2
y 2 dy dx2
x2
y 2 dy dx0 4 0
r 2 dr d 16 2 d 30 1 2 3
4 2 3
2
4 0
x2
2
2
2
2
23.0
x
y dy dx0 0 2
r cos 8 3
r sin
r dr d0 0
cos2
sin 16 3 2
r2 dr d
cos0
sin
d
8 sin 3
cos0
1
2
4 1
y2 y2
25.0
y arctan dx dy x
2 1 2 4 0 4 0 y 2
4
y2
y arctan dx dy x
(
1 , 2
1 2
(( 2, 2)
r dr d1
3 d 2
3 2 4
4 0
3 2 64
0 1 2
2
1
27. V0 0 2 0
r cos 1 21
r sin
r dr d 1 82
r3 sin 2 dr d0
sin 2 d0
1 cos 2 16
2 0
1 8
2
5
29. V0 0
r 2 dr d
250 3
2
4 cos
2
31. V
20 0 2
16
r2 r dr d
20
1 3
16
r2
3
4 cos
d0 2 0
2 3 64 3 9
2
64 sin30
64 d
128 3
10
sin
1
cos2
d
128 3
cos
cos3 3
4
Section 13.32 4 2 4
Change of Variables: Polar Coordinates
145
33. V0 a
16
r 2 r dr d0
1 3
16 3.
r2
3
da
1 3
16
a2
3
2
One-half the volume of the hemisphere is 64 2 16 3 16 16 a2 a23 2
64 3 32 322 163
3 2
a2 a2 a
322 44
3
16 2
83 2 4 23 2 2.4332
23 2
2
4
35. Total Volume
V0 2 0
25e
r2 4
r dr d
4
50e0 2
r2 4 0
d
50 e0
4
1 d 308.40524
1
e
4
100
Let c be the radius of the hole that is removed. 1 V 102 0 2 c 2 c
25e0
r2 4
r dr d0
50e
r2 4 0
d ec2 4
50 e0
c2 4
1 d 30.84052 ec2 4
100 1 0.90183 0.10333 0.41331 0.6429 2c
c2 4 c2 c diameter6 cos
1.2858
37. A0 0
r dr d0
18 cos
2
d
90
1
cos 2
d
9
1 sin 2 2
90
2
1 0
cos
39.0
r dr d
1 2 1 2
2
10 2
2 cos
cos2 1
d cos 2 23
10
2 cos
d
1 2
2 sin
1 2
1 sin 2 2
2 0
3 2
3
2 sin 3
41. 30 0
r dr d
3 2
3
4 sin2 3 d0
30
1
cos 6
d
3
1 sin 6 6
3 0
43. Let R be a region bounded by the graphs of r r g2 , and the lines a and b.
g1
and
45. r-simple regions have fixed bounds for . -simple regions have fixed bounds for r.
When using polar coordinates to evaluate a double integral over R, R can be partitioned into small polar sectors.
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 5
49.4 0
r 1
r 3 sin
dr d2
56.0515
Note: This integral equals4
sin
d0
r 1
r3 dr
51. Volume
base 8
height 12 30016
z
53. False r 1 where R is the circular sector Let f r, 0 r 6 and 0 . Then, rR
Answer (c)
1 dA > 0
but
r
1
0 for all r.
6 x
4
4 6
y
2
2
2
55. (a) I 2 (b) Therefore, I7 49 49 x2
e
x2
y2 2 dA
40 0
e
r2 2
r dr d
40
e
r2 2 0
d
40
d
2
2 .2 7 2
7
57.7
4000ex2
0.01 x 2
y2
dy dx0 0
4000e 2
0.01r 2
r dr d0
200,000e 400,000 1y
0.01r 2 0
d 486,788
200,000 e
0.49
1
e
0.49
4
y
59. (a)2 2 y 3
f dx dy3x 4 3 x 3x 4 4
5
y=
3x
y=x
(b)2 3 3 2 4 csc
f dy dx2
f dy dx4 3 x
f dy dx
3
( (1 2
(4, 4) 4 ,4 3
(x
1
(c)4 2 csc
f r dr d
(2, 2) 2 ,2 33
(4 5
61. A
r22 2
r12 2
r1 2
r2
r2
r1
r r
Section 13.44 3
Center of Mass and Moments of Inertia4 0
1. m0 0
xy dy dx
xy2 2
3
4
dx0 0
9 x dx 2
9x2 4
4
360
2
2
2
2
3. m0 0
r cos
r sin
r dr d0 2 0
cos sin 4 cos sin d0
r 3 dr d
4
sin2 2
2
20
Section 13.4
Center of Mass and Moments of Inertiaa b
147
a
b
5. (a)
m0 a 0 b
k dy dx
kab kab2 2 ka2b 2 a 2 b 2
(b)
m0 a 0 b
ky dy dx
kab2 2 kab3 3 ka2b2 4 a 2 2 b 3
Mx0 a 0 b
ky dy dx
Mx0 a 0 b
ky 2 dy dx
My0 0
kx dy dx My m Mx m a b , 2 2a b
My0 0
kxy dy dx My m Mx m a 2 , b 2 3 ka2b2 4 kab2 2 kab3 3 kab2 2
x y x, y
ka2
b 2 kab
x y x, y
kab2 2 kab
a
(c) m0 0 a
kx dy dxb
k0
xb dx ka2b2 4 ka3b 3
1 2 ka b 2
Mx0 a 0 b
kxy dy dx
My0 0
kx 2 dy dx My m Mx m ka3b 3 ka2b 2 ka2b2 4 ka2b 2 2 b a, 3 2 k bh 2 b by symmetry 2b 2 2hx b
x y x, y
2 a 3 b 2
7. (a)
m x Mx
y
y=h
2hx b y= 2 h (x b ) b
b
2h x
b b
ky dy dx0 0 b 2 0
ky dy dxxb
kbh2 12 y x, y Mx m b h , 2 3
kbh2 12 kbh2 6 kbh 2
kbh2 6 h 3
CONTINUED
148
Chapter 13
Multiple Integration
7. CONTINUEDb 2 2hx b b 2h x b b
(b) m0 b 2 0 2hx b
ky dy dxb 2 0 b 2h x
ky dy dxb b
kbh2 6 kbh3 12 kb2h2 12
Mx0 b 2 0 2hx b
ky 2 dy dxb 2 0 b 2h x b b
ky 2 dy dx
My0 0
kxy dy dxb 2 0
kxy dy dx
x y (c) m
My m Mx mb 2 0
kb2 2
h 12 kbh2 6
b 2 h 2b 2h x b b
kbh3 12 kbh2 62hx b
kx dy dx0 b 2 0
kx dy dx 1 2 kb h 4b 2h x b b
1 2 kb h 12b 2 2hx b
1 2 kb h 6 kxy dy dx
Mx0 0
kxy dy dxb 2 0
1 2 2 kh b 32b 2 2hx b
5 2 2 kh b 96 kx dy dx2
1 2 2 kh b 12b b 2 0 2h x b b
My0 0
kx2 dy dx 7 3 kb h 48
1 3 kb h 32 x y My m Mx m
11 3 kb h 96
7kb3h 48 kb2h 4 kh2b2 12 kb2h 4
7 b 12 h 3 a 2 a 22
9. (a) The x-coordinate changes by 5: x, y (b) The x-coordinate changes by 5: x, ya 5 b
5,
b 2
11. (a)
x m Mx
0 by symmetry a2k 2a a2 x2
2b 5, 3
(c) m5 a 5 0 b
kx dy dx kxy dy dx5 a 5 0 b
1 ka 2 1 ka 4 1 ka 3
5 b 5 b3 2 2
25 kb 2 25 2 kb 4 125 kb 3
yk dy dxa 0
2a3k 3 4a 3
Mx My5 0
y
Mx ma
2a3k 3a2 x2
2 a2k ka
kx 2 dy dx My m Mx m
5 b
(b) ma 0 a a2 x2
y y dy dx
a4k 16 24 a5k 15 120 0
3
x y
2 a2 15a 75 3 a 10 b 2
Mxa 0 a a2 x2
ka
y y 2 dy dx
32
Mya 0
kx a My m Mx m 0 a 15 5 16 32 3
y y dy dx
x y
Section 13.4
Center of Mass and Moments of Inertia
149
4
x
13. m0 4 0
kxy dy dxx
32k 3 256k 21 32k 3 8 7
15.
x m
0 by symmetry1 1 1 x2
k dy dx1 0 1 1 1 x2
Mx0 4 0 x
kxy 2
dy dx
k 2 k 2 8 2 4
My0 0
kx My m Mx m 32k 1 256k 21
2y
dy dx
Mx1 0
ky dy dx Mx my
x yy 3
3 32k 3 32k
y
k 2 8
2 k
2
y=
1 1 + x2
y=2
x1 x 1
1 x1 2 3 4
1
17. y m
0 by symmetry4 16 y2
19. x 8192k 15 524,288k 105 64 7 m
L by symmetry 2L 0 sin x L
kx dx dy4 0 4 16 y2
ky dy dx0 L sin x L
kL 4 4kL 9
My4 0
kx 2 dx dy My m 524,288k 105 15 8192k
Mx0 0
ky 2 dy dx Mx m 4kL 9 4 kL 16 9
xy8
yy
x = 16 y 22
4 x4 4 8 8
y = sin x L
1
xL 2
L
21. m Mx
a2k 84 a
2
ky dAR 0 4 0 a
kr 2 sin dr d
ka3 2 6 ka3 2 6
2
y=x r=a
MyR
kx dA0 0
kr 2 cos dr d 8 a2k 2 6 4a 2 3 8 a2k 4a 2 3
a
0
x y
My m Mx m
ka3 2 6 ka3 2
2
150
Chapter 13
Multiple Integration
2
e 0 e 0 e 0
x
23. m0 2x
ky dy dx ky 2 dy dx0 2x
k 1 4 k 1 9 k1 4e4 ke4
e e
4
25.6
y m
0 by symmetry6 2 cos 3
Mx My0
k dAR 6 0
kr dr d
k 3
kxy dy dx My m Mx my
5e 8 1
4
MyR
kx dA6 2 cos 3
x y
k e4 5 8e4 k e6 1 9e6
e4 2 e4 4 e6 9 e6
5 1 1 e2
0.466 0
kr 2 cos dr d x 2
1.17k
4e4 k e4 1
0.45
My m
1.17k
3 k
1.12
2
= 6r = 2 cos 3
1
y = e x
01
x1 2
=6
27. m Ix
bhb 0 b h
29. m y 2 dy dx0 h
a22 a
bh3 3 b3h 3
IxR
y 2 dA0 2 0 a
r3 sin2
dr d
a4 4 a4 4
Iy0 0
x 2dy dx Iy m Ix m a2 4
IyR
x 2 dA0 0
r3 cos2 a4 4 a4 4 a4 4 a4 2 1 a2
dr d
x y
b3h 3 bh3 3
1 bh 1 bh
b2 3 h2 3
b 3 h 3
3 b 3 3 h 3
I0 x
Ix y
Iy
Ix m
a 2
31. m Ix
33.2 a
kya b
y2R
dA0 2 0 a
r3
sin2
dr d
a4 16 a4 16
m
k0 a 0 b
y dy dx
kab2 2 kab4 4 ka3b2 6 2kb2a3 12 ka3b2 6 kab2 2 kab4 4 kab2 2 a2 3 b2 2 a 3 b 2 3 a 3 2 b 2
IyR
x2 Ix y Iy
dA0 0
r3 a4 16 a4 16 a 164
cos2 a4 8 4 a2
dr d
Ix Iy
k0 a 0 b
y3 dy dx
I0 x
k0 0
x 2y ydy dx Iy Iy m Ix m 3kab4
Ix m
a 2
I0 x y
Ix
Section 13.4 35. m kx2 4 0 4 0 4 0 x2 x2 x2
Center of Mass and Moments of Inertia kxy4 x
151
37. x dy dx0 2
k
4k 32k 3 16k 3
m0 4 0 x
kxy dy dx
32k 3 16k 512k 5
Ix
k0 2
xy 2 dy dx
Ix0 4 0 x
kxy3 dy dx
Iy I0 x
k0
x3 dy dx Iy Iy m Ix m 16k 16k 3 4k 32k 3 4k
Iy0 0
kx3 y dy dx 592k 5 512k 5 16k 1
Ix
I0 4 3 8 3 2 3 4 6 2 3 3 2 6 3 x
Ix
Iy Iy m Ix m
3 32k 3 32k 3 2
48 5
4 15 5 6 2
y
y
39. m
kx1 0 1 x2
x
kx dy dxx
3k 20 3k 56 k 18
Ix0 1
kxy 2 dy dxx2
x
Iy0 x2
kx3 dy dx 55k 504 k 18 3k 56
I0 x
Ix
Iy Iy m Ix m
20 3k 20 3k
30 9 70 14
y
b
b2 0
x2
b
41. I
2kb b
x
a 2 dy dxb
2kb
x
a
2
b2
x2 dxb
2kb
x 2 b2 b4 8
x 2 dx a2b2 2
2ab
x b2 4a2
x2 dx
a2b
b2
x 2 dx
2k
0
k b2 2 b 4
4
x
4
43. I0 0
kx x
6 2 dy dx0
kx x x 2
12x
36 dx
k
2 9 x 9
2
24 7 x 7
2
72 5 x 5
4 2 0
42,752k 315
152
Chapter 13
Multiple Integration
a
a2 x2
a
a2 0 a2 0 x2
x2
a
45. I0 0 a
ka k 4 k 4 k 4
y y
a 2 dy dx0
ka
y dy dx0
3
k a 4
y
4 0
a2
x2
dx
a40 a
4a3y
6a2 y2
4ay3
y4
dx
a40 a
4a3 a2
x2
6a2 a2
x2
4a a2
x2
a2
x2
a4
2a2x2
x4
a4 dx
7a40
8a 2x 2 8a2 3 x 3 8 5 a 3
x4 x5 5
8a3 a2
x2
4ax 2 a 2 x a
x 2 dx a x 2x 2 2 x aa 0
k 7a4x 4 k 7a5 4
4a3 x a2 1 5 a 4
x2
a2 arcsin 7 16
a2
a2
x2
a4 arcsin
1 5 a 5
2a5
a5k
17 15
47.
x, y
ky. y will increase
49.
x, y
kxy.
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 MxR
y
x, y dA and MyR
x
x, y dA.
If m is the mass of the lamina, then the center of mass is x, y My Mx , . m m
53. See the definition on page 968
55. y
L ,A 2b L
bL, h L 22
L 2 dy dx3 L
57. y
2L .A 3b 2
bL ,h 2L
L 3 2L 33 L 2
Iy0 b 0 0
y y Iy hA
Iy L3b 12 L 3
20 2Lx b b 2
y 2L 3
dy dx
L 2 3 L 2
dx0
2 3 2 3
y0 b 2 0
dx2L x b
ya
y
L3b 12 L 2 bL
L 27
2Lx b
2L 3 2L 3
3
dx L3b 36
2 L3x 3 27 ya 2L 3
b 2Lx 8L b L3b 36 L2b 6 L 2
4 b 2 0
Section 13.5
Surface Area
153
Section 13.51. f x, y R fx 12
Surface Area2y 3. f x, y R fx fy2
2x
8
2x
2y y2 4
triangle with vertices 0, 0 , 2, 0 , 0, 2 2, fy fx2 0 2 x
x, y : x 2 2, fy 1 fx2 2 4 4y
2 32
2 fyx2 2
32 2
S0
3 dy dx 3 2xy
30
2
x dx
S2 x2
3 dy dx0 0
3r dr d
12
x2 2
2
60
y = 4 x2
R1
1
2
x
y = x + 2 R
1 1
1
y = 4 x2x1 2
5. f x, y R fx 13
9
x23
y
square with vertices, 0, 0 , 3, 0 , 0, 3 , 3, 3 2x, fy fx3 2
R
0 fy 12
2
1
4x 23
1
S0 0
4x 2 dy dx0
3 1 1
4x 2 dx3
x
1
2
3
3 2x 1 4 7. f x, y R fx 13
4x 2
ln 2x
4x 20
3 6 37 4
ln 6
37
2
x3
24
y
rectangle with vertices 0, 0 , 0, 4 , 3, 4 , 3, 0 3 12 x , fy 2 fx4 0 2
R3
0 fy 4 2 9x3 2 0 2
2
13
9 x 4 40
4 2 9x 2 8
9x
1 x1 2 3 4
S0
9x
dy dx3
4
dx
4 4 27 9. f x, y R fx 1 S0 0
4 31 31 27
ln sec x x, y : 0 x 4 , 0 y tan x2
y
y = tan x1
tan x, fy fx4 2 tan x
0 fy2
1
tan2 x4
sec x4
R 4 2
x
sec x dy dx0
sec x tan x dx
sec x0
2
1
154
Chapter 13 x2 y2
Multiple Integrationy
11. f x, y R 0 fx 11
x, y : 0 f x, y 1 x2 x x2 fx2
1
x 2 + y2 = 1
y 2 1, x 2 y2 , fy fyx2 2
y2 1x
y x2 12
1
y2 x2 x2 1
y2 y2
x
2
y2 2
2
1 1
S1 x2
2 dy dx0 0
2 r dr d
13. f x, y R fx 1b
a2 x, y : x 2 a2 fx2
x2
y2a
y
y 2 b2, b < a y2 fy2
x x2
, fy 1 a x2
a2 a2 y2
y x2 x2 x2
b
x 2 + y 2 b2
y2 y22
a2b 0
y2 x2
y2
a2
a x2 2 aa
b
b b
a
x
y2 a2 b2
b
2
x
2
Sb b2 x2
a2
dy dx0
a r dr d a2 r 2
15. z 1
24 fx8
3x2 3 2x 0
2y fy12 2
y 16 12
14 14 dy dx 48 14
S0
8
4 x4 8 12 16
17. z 1
25 fx3 2
x2 fy9 9 3 0 x2
y22
y
x 2 + y2 = 9
1 5 x2 r dr d
x2 25 x2 dy dx y2 25
y2 x2 y2 25
5 x2
2
y22 1
1
x
S
23 2 x2
1 2
1
2
25
y2 20
20
5 25 x2
r2
19. f x, y R 11
2y
y
triangle with vertices 0, 0 , 1, 0 , 1, 11
y=x
fxx
2
fy 5
2
5
4x 2 1 27 12 5 5R1
S0 0
4x 2 dy dx
x
Section 13.5 x2 y2 x2 y2
Surface Area
155
21. f x, y R 0 4 fx 12
4
23. f x, y R y2 4 fx 1 1 4x 2 4x 2 4y 2 S1 0
4
x, y : 0 f x, y x2 2x, fy fx2 4 4 2 x2
x, y : 0 x 1, 0 y 1 2x, fy fx1 2
y 2, x 2 2y fy2
2y fy 12
1
4x 2
4y 2 1.8616
4x2
4y2 dy dx
S2 2 0 x2
1
4y 2 dy dx 17 17 6 1
0
10y
4r 2 r dr d
x 2 + y2 = 4
1
1 1
x
1
25. Surface area > 4 Matches (e)z 10
6
24.
27. f x, y R fx 11
ex x, y : 0 x 1, 0 y 1
ex, fy fx1 2
0 fy 12
1
e2x
S0 15 x 5 y
e2x dy dx
0
10
e2x
2.0035
29. f x, y R fx S1
x3
3xy
y3 1, 1 , 3x 9 y2 3y 1,2
31. f x, y 1 , 1, 3 y2
e ex
x
sin y e 1 1x
square with vertices 1, 1 , 3x2 1
1 x
fx 1
sin y, fy fy24 4 x2
cos y e e2x 2x
3y 11
3x
2
y , fy y2
fx22
sin2 y
e
2x
cos2 y
1
9 x2
x 2 dy dx S2 x2
1
e
2x
dy dx
33. f x, y R fx 14
exy x, y : 0 x 4, 0 y 10
yexy, fy fx2 10
xexy fy 12
1
y 2e2xy y 2 dy dx
x 2e2xy
1
e2xy x 2
y2
S0 0
e2xy x 2
156
Chapter 13
Multiple Integration x 12 fy2 dA dy dx1
35. See the definition on page 972.
37. f x, y SR 1
1 1x 0
x 2; fx fx2 1 1 x x2 x2
x2
, fy
0
160 1
160
1
dx
16 1
x2
1 2 0
16
50
502 0
x2
39. (a) V0
20
xy 100
x 5
y
dy dx
50
20 5020
x2 x2
x 502 200 502 arcsin x 50
x2
x 5 25 2 x 4
502 x4 800
x2
502 x 2 dy 10 x23 2
10 x 50 30,415.74 ft3 (b) z 1 S 20 fx 1 100 1 100 xy 1002
1 502 15
250x
x3 30
50 0
fy
2
1x2
y2 1002 x2
x2 1002 y 2 dy dx
1002
x2 100
y2
50 0 2 0 0
502
100250
10020
r 2 r dr d
2081.53 ft2
y
41. (a) VR
f x, y
24 20
8R 2
62525
x2
y2
dA
where R is the region in the first quadrant
16 12
R8
80 4 2
625 2 625 3 609 609 609 1 cm3 fx2
r 2 r dr d25
4 x 4 8 12 16 20 24
40
r2
3 2 4
d
8 0 3 812 (b) AR
2
fy
2
dA
8R 2 25 4
1
625 25 625
x2 x2 r dr d
y2
625
y2 x2
y2
dA
8R
25 625 x 2 200 625
y
dA 2b
80
r2
b25
lim
r24
2
100
609 cm2
Section 13.6
Triple Integrals and Applications
157
Section 13.63 2 0 1
Triple Integrals and Applications3 2 0 2 0
1.0 0
x
y
z dx dy dx0 3 0
1 2 x 2 1 2 y
1
xy
xz0
dy dx3
z dy dz0
1 y 2
1 2 y 2
2
3
yz0
dz
3z
z20
18
1
x 0
xy
1
x
xy
3.0 0
x dz dy dx0 1 0 0 x
xz0
dy dx1
x 2y dy dx0 0
x 2y 2 2
x
1
dx0 0
x4 dx 2
x5 10
1 0
1 10
4
1 0
x
4
1
x
4
1
5.1 0
2ze
x2
dy dx dz1 4 0
2ze ze1 x2
x2
y0
dx dz1 4 0
2zxe e1
x2
dx dz 1 e1
1
dz0 1
z1
dz
z2 2
4 1
15 1 2
1 e
4
2 0
1 0
x
4 0 4 0
2
1
x
4
2
7.0
x cos y dz dy dx x10
x cos y z0 2
dy dx0 4 0
x1 x10
x cos y dy dx x2 2 x3 34
x sin y0
dx
x dx
80
64 3
40 3
2
4 4
x2 x2
x2
2
4 4
x2
9.0 0
x dz dy dx0 x2
x3 dy dx
128 15
2
4 0
x2
4 1
11.0
x 2 sin y dz dy dx z
2 0 2 0
4
x2
4
x 2 sin y ln z1 4 x2
dy dx2
x 2 ln40
cos y0
dx0
x 2 ln 4 1
cos 4
x 2 dx
2.44167
4
4 0
x
4 0
x
y
3
9 9
x2 x2
9 0
x
2
y
2
13.0
dz dy dx
15.3
dz dy dx
2
4
y2
x
2
4
y2
17.2 0 0
dz dx dy2 0
x dx dy 1 22 2
42
y 2 2 dy0
16
8y 2
y 4 dy
16y
8 3 y 3
1 5 y 5
2 0
256 15
a
a2 0
x2 0
a2
x2 y2
a
a2 0
x2
19. 80
dz dy dx
80 a
a2
x2
y 2 dy dx y a2a 0 a2 x2
40
y a2a
x2
y2
a2
x 2 arcsin 1 3 x 3
x2
dx0
4
2
a20
x 2 dx
2
a2x
4 3 a 3
1582
Chapter 134 0 x2 4 0 x2
Multiple Integration2 2
21.0
dz dy dx0
4
x 2 2 dx0
16
8x 2
x 4 dx
16x
8 3 x 3
1 5 x 5
2 0
256 15
23. Plane: 3xz 3
6y
4z
12
25. Top cylinder: y 2 Side plane: xz
z2 y
1
1 2 4 x 1 3 y
x
1
y
3 0
(12 0
4z) 3
(12 0
4z
3x) 6
dy dx dz1 0 x 0 0 1 y2
dz dy dx
27. Q
x, y, z : 0 x 1, 0 y x, 0 z 33 1 0 1 3 1 0 1 0 1 0 1 0 1 0 3 0 3 0 1 y x 0 x
y
xyz dV0 Q y
xyz dx dy dz0 0 1
xyz dy dx dz
1
y=x
xyz dx dz dyy x
Rx1
xyz dy dz dx0 3
xyz dz dx dy0 3
xyz dz dy dx0
9 16
29. Q
x, y, z : x24
y2 9, 0 z 45
z
3 3 3 3 4
9 9 9 9 9 9 9 9 y2
x2
xyz dVQ 0 4 0 3 x2 y2
xyz dy dx dz
xyz dx dy dzy24 3 3 4 y
y2
x
xyz dx dz dy3 0 3 3 3 4 y2 4
xyz dz dx dyy2 0 9 9 9 9 x2 x2
xyz dy dz dx3 0 3 3 x2 4
xyz dz dy dxx2 0
0
Section 13.66 4 0 (2x 3) 2 0 (y 2) (x 3)
Triple Integrals and Applications
159
31.
m
k0
dz dy dx
8k6 4 0 (2x 3) 2 0 (y 2) (x 3)
Myz
k0
x dz dy dx
12k x Myz m 12k 8k 3 2
4
4 0 4
4 0
x
4
4
b
b 0 b 0 b 0 b 0
b
33.
m
k0
x dz dy dx 4x0 4 4 0 4 4 0 x
k0 0
x4
x dy dx
35.
m
k0 b 0 b
xy dz dy dx
kb5 4 kb6 6 kb6 6 kb6 8
4k
x 2 dx
128k 34 4
Myz k0 0
k0 0 b b
x 2y dz dy dx
Mxy
k0
xz dz dy dx
x 128k 3
4 2
x
2
dy dx
Mxz Mxy x y z
k0 b 0 b
xy 2 dz dy dx
2k0
16x 1
8x 2
x3 dx
k0 0
xyz dz dy dx kb6 6 kb5 4 kb6 6 kb5 4 kb6 8 kb5 4 2b 3 2b 3 b 2
z
Mxy m
Myz m Mxz m Mxy m
37. x will be greater than 2, whereas y and z will be unchanged. 39. y will be greater than 0, whereas x and z will be unchanged. 1 k r 2h 3 y 4k0 0 r 0 r 0 h r2 x2 x2 y2 r
41.
m x Mxy
0 by symmetryr r2 x2 h
z dz dy dx
3kh2 r2 4kh2 3r 2
r2
x2
y 2 dy dx
r20
x2
3 2
dx
k r 2h2 4 z Mxy m k r 2h2 4 k r 2h 3 3h 4
160
Chapter 13 128k 3 y 4 4k0 4 0 42 0 2 x2 2 4
Multiple Integration
43.
m x z Mxy
0 by symmetry x242 x2 0 4
y242 x2 y2
z dz dy dx 1 3 y 342 0 x2
2k0
42
x2
y 2 dy dx
2k0
16y
x 2y
dx
4k 3
4
40
2
x2
3 2
dx
1024k 3 64 k z Mxy m
cos40
d
let x
4 sin
by Walliss Formula 64k 1 3 128k 3 2
45. f x, y m k
5 y 1220 0 20 0 (3 5)x 0 (3 5) 0 (3 5)x 0 12 x 12 (3 5)x 12 (5 12)y20
y
dz dy dx0 (5 12)y
200k
16 12 8
y = 3 x + 12 5
Myz Mxz Mxy x y z
k0 20 0 12
x dz dy dx(5 12)y
1000k
4 x 4 8 12 16 20
k0 20 0 (5 12)y
y dz dy dx
1200k
k0 0
z dz dy dx 1000k 200k 1200k 200k 250k 200k 5 6 5 4
250k
Myz m Mxz m Mxy m
a
a 0 a
a
a
a
47. (a) Ix
k0 0
y2 1 3 y 3
z2 dx dy dza a
ka0 0
y2 az2
z2 dy dz dz ka 1 3 az 3 1 3 az 3a 0
ka0
z2y0
dz
ka0
1 3 a 3
2ka 3
5
Ix (b) Ix
Iya
Iza 0 a 0 a
2ka5 by symmetry 3 y2 z2 xyz dx dy dz y2z3 2a
k0 0
ka2 2a
a 0
a
y3z0
yz3 dy dz ka4 a2z2 8 2 2z4 4a 0
ka2 2 Ix Iy
y 4z 4
dz0
ka4 8
a2z0
2z3 dz
ka8 8
Iz
ka8 by symmetry 8
Section 13.64 4 0 4 0 x 4 4
Triple Integrals and Applications
161
49. (a) Ix
k0 4
y2 y3 4 3 32 4 34 4 0 4 4 0 x
z2 dz dy dx y 4 34
k0 0 4
y2 4 k0
x x
1 4 3 4 4 3
x
3
dy dx3
k0
x x2
x x
3 0 4 4 0
dx
64 4 3
x
dx
k Iy k0
1 4 3
256k4 4
x2 4x20 4 4 0 4 0 x
z2 dz dy dx 1 4 3 x3
k0 0
x2 4 4k4
x 1 4 x 4 y2 4
1 4 3 1 4 12
x
3
dy dx4 4 0
4k Iz k0 4
x3 x2
dx k
4 3 x 34
x
512k 3
y 2 dz dy dx4
x20 0 4
x dy dx x dx 256k x x3
k0 4 4 0
x 2y4 0 x
y3 4 3 y y2
x0
dx
k0
4x 24 4
64 4 3 y3 4 x x
(b) Ix
k0 4
z2 dz dy dx y2 4 6 2 4 3 x4
k0 0 4
1 y4 3 8 4 3
dy dx dx
k0
y4 4 4 32 4 xx
x2
x4
3 0 4 0
dx
k0
64 4
3
k4
2048k 34 4
4 0 4
4 0
Iy
k0
y x2 4x 20 4 4 0 4 0 x
z2 dz dy dx 1 4 3 x3
k0 0
x 2y 4 8k4
x
1 y4 3 1 4 12 x4
x4 0
3
dy dx 1024k 3
8k Iz k0 4
x3 y x2
dx
4 3 x 34
1 4 x 4
y 2 dz dy dx4
k0 0 4
x 2y 8x 20
y3 4 64 4
x dx x dx
k0 4
x 2y 2 2 32 8x
y4 4 4 4x 2
x0
dx
k
8k0
x3 dx
8k 32x
4x 2
4 3 x 3
1 4 x 4
4 0
2048k 3
L 2
a a
a2 a2
x2
L 2
a a
51. Ixy
kL 2 x2
z2 dz dx dyL 2
kL 2
2 2 a 3 1 x 2x 2 8
x2
a2
x 2 dx dy x aa
2 3 2k 3 Since m
kL 2 L 2
a2 x a2 2 a4 4 a4 16
x2
a2 arcsin a4 Lk 4
x a
a2
x2
a2
a4 arcsin
dya
2L 2
dy
a2Lk, Ixy
ma2 4.
CONTINUED
162
Chapter 13
Multiple Integration
51. CONTINUEDL 2 a a a2 a2 x2 L 2 a
Ixz
kL 2 L 2 x2
y 2 dz dx dy y2 x a2 2a a a2 a2 x2
2kL 2 a
y 2 a2 x aL 2 a
x 2 dx dyL 2
2kL 2 L 2
x2
a2 arcsin
dya a
k a2L 2
y 2 dy
2k a2 L3 3 8
1 mL2 12
Iyz
kL 2 L 2 x2
x 2 dz dx dy 1 x 2x 2 2 8 ma 42
2kL 2 a
x 2 a2 x aa
x 2 dx dy ka4 4L 2
2kL
a2 mL 122
a2
x2
a4 arcsin L2
dya
dyL 2
ka4 L 4
ma2 4
Ix Iy Iz
Ixy Ixy Ixz
Ixz Iyz Iyz
m 3a2 12 ma2 2 m 3a2 12
ma2 4 mL2 12
ma2 4 ma2 4
L2
1
1
1
x
53.1 1 0
x2
y2
x2
y2
z2 dz dy dx
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 13.74 2 0 2
Triple Integrals in Cylindrical and Spherical Coordinates4 0 4 0 0 0 2 2
1.0 0
r cos dr d dz
r2 cos 2
2
d dz0 4 2 4
2 cos d dz0
2 sin0
dz0
2dz
8
2
2 cos2 0
4 0
r2
2 0 2
2 cos2
2
3.0
r sin dz dr d0
r4 8 cos40
r 2 sin dr d0
2r 2 8 cos5 5
r4 sin 4 4 cos9 9
2 cos2
d0 2 0
4 cos8
sin d
52 45
2
4 0
cos 2
5.0 0
sin
d d d
1 3
2 0 0
4
cos3
sin
d d
1 12
2
4
cos40 0
d
8
4
z 0 0
2
7.0
rer d dr dz
e4
3
Section 13.72 3 0 e 0r2
Triple Integrals in Cylindrical and Spherical Coordinates
163
2
3
9.0
r dz dr d0 2 0 2 0 0
re 1 e 2 1 1 2 e9
r2
dr d3 r2 01 3
z
d9
2
e
d2 3 x
1
3 y
42 2 6 4 2 0 0
1
11.
sin
d d d
64 3 64 3
2 0 2
2
sin6
d d4
z
2
cos0 2 6
dx 4 4 y
32 3 3 64 3 32 2 0 4
d0
13. (a)0 2 r2
r 2 cos dz dr darctan 1 2 0 4 sec 3 0 0
02 2 arctan 1 2 cot 0 csc 3 0
(b)
sin2
cos d d d
sin2
cos d d d
0
2
a 0 4 2 0
a a
a2
r2
15. (a)0 2a cos
r 2 cos dz dr d3 0 a sec
0
(b)
sin2
cos d d d
0
2
a cos 0 2 0
a2
r2
2
a cos
17. V
40
r dz dr d 4 3 a 3
40 0
r a2 1 cos 3
r 2 dr d2
4 3 a 3
10
sin3
d
sin2
20
4 3 a 3 2
2 3
2a 3 3 9
4
a cos
a2 0
r2
2
2 0 2
9 0
r cos
2r sin
19. V
20 0 a cos
r dz dr d
21. m0 2
kr r dz dr d kr 2 90 0 2
20 0
r a2 1 2 a 3 10
r 2 dr da cos 3 2 0
r cos r4 cos 4 4 cos
2r sin r4 sin 2 8 sin2
dr d2
20
r2
d0 2
k 3r 3 k 240
d0
2a3 3 2a3 3
sin3 cos 4
d cos3 3
d
0
k 24 k 48
4 sin 8 8
8 cos0
2a3 3 9
48k
164
Chapter 13 h r02
Multiple Integration h r r0 0 r dz dr d
23. z V
h 40
x2r0 0 2 h r0 0 r0
y2r r0
r
25. x m
k x2 y 4k0 0
y2
kr
0 by symmetry2 r0 h(r0 0 r) r0
r 2 dz dr d
4h r0 4h r0
r0r0 2 0 0
r 2 dr d
r0 d 6 2 1 r 2h 3 0
3
1 k r03h 62 r0 0 h(r0 0 r) r0
Mxy
4k0
r 2 z dz dr d
4h r03 r0 6
1 k r03 h2 30 z Mxy m k r03h2 30 k r03h 6 h 5
2
r0 0 2
h(r0 0 r0
r) r0
27. Iz
4k0
r3 dz dr d
29. m Iz
k b2h2 b a 2
a2hh
k h b2
a2
4kh r0
r0r30 0
r 4 dr d
4k0 0 b
r3 dz dr d
4kh r05 r0 20 1 k r04h 10
4kh 2 kh0 0 2 a
r3 dr d
b4 a4 h 2
a4 d
1 Since the mass of the core is m kV k 3 r02h from 2h. Exercise 23, we have k 3m r0 Thus,
k b4 k b2 1 m a2 2
Iz
1 k r04h 10 1 10 3m r02h r04h
a2 b2 2 b2
a2 h
3 mr 2 10 0
2
4 sin 2
2
2 0 2
a 3
31. V0 0 0
sin d d d
16
2
33. m
8k0 0 2
sin
d d d
2ka40 2 0
sin
d d
k a40
sin cos
d2
k a4 k a4
0
Section 13.7
Triple Integrals in Cylindrical and Spherical Coordinates2 2 0 2 4 0 2 2 cos 4 4 0
165
35.
m x Mxy
2 k r3 3 y 4k0 0 2 0 2 0 0 2
37. Iz
4k 4 k 5 cos5 cos54
sin3
d d d d d sin2 4
0 by symmetry2 2 r 3
sin3 cos2
cos
sin
d d d 2 k 5
1
d
1 4 kr 2 kr 4 4
sin 2 d d sin 2 d0
2 k 5 k 192
1 cos6 6
1 cos8 8
1 k r 4 cos 2 8 z Mxy m k r4 4 2k r3 3
2 0
1 k r4 4
3r 8
2
g2 g1
h2 r cos , r sin
39. x y z
r cos r sin z
x2
y2 tan z
r2 y x z
41.1
f r cos , r sin , z r dz dr dh1 r cos , r sin
43. (a) r z
r0: right circular cylinder about z-axis0:
(b)
0: 0: 0:
sphere of radius cone
0
plane parallel to z-axis
plane parallel to z-axis
z0: plane parallel to xy-plane
a
a2 0
x2 0
a2
x2 y2 0
a2
x2
y 2 z2
45. 160 a a2 0 2 a 0 2 a 0 2 a 0 2 0
dw dz dy dxx2 0 a2 r2 a2 x2 y2
160
a2 a20
x2
y2
z2 dz dy dx
16 160
r2 z2
z2 dz r dr d a2 r 2 arcsin z a2 r2 0a2 r2
1 z 2 a2
a2
r2
r dr d
80
2
r 2 r dr d r4 4 a4 22 a
40 2
a2r 2 2 d
d0
a40
166
Chapter 13
Multiple Integration
Section 13.81. x y 1 u 2 1 u 2 x y u v v y x u v v
Change of Variables: Jacobians3. x y u u x y u v v2 v y x u v 1 1 1 2v 1 2v
1 2 1 2
1 2
1 2
1 2
5. x y
u cos u sin x y u v
v sin v cos y x u v cos2 sin2 1
7. x y
eu sin v eu cos v x y u v y x u v 2v eu sin v eu sin v eu cos v eu cos v e2u
9. x y v u
3u 3v y 3 x 3 x 3 1 u 2 1 u 2 x y u v 4 x2R
x, y 0, 0 3, 0 2, 3
u, v 0, 0 1, 0 0, 11
v
(0, 1)
2v 2y 9
x
2 y 3 3
(1, 0)u
1
11. x y
v v y x u v y 2 dA1 1 1 1 1
1 21 1
1 2 4 1 u 4
1 2 v
1 22
1 2 1 u 41
v
2
1 dv du 2 1 du 3 2 u3 3 u 31 1
u21
v 2 dv du1
2 u2
8 3
13. x y
u u x y u v yxR
v4
v
y x u v y dA
3
1 03 0 4
1 1 uv 1 dv du0
12
3
1
8u du0
36u1 2 3 4
Section 13.8
Change of Variables: Jacobians
167
15.R
e R: y x
xy 2
dA 2x, y 1 ,y x 4 x y , v x v1 u3 v1 u12 2 2 24
y
1 y= x y = 2x
x ,y 4 v u, y x u y u
3
y= 4 x2
1 y=4x
uv u x v y v 1 2 1 2
xy11
Rx2 3 4
x, y u, v
1 1 2 u1 2v1 1 u1 2 2 v1 2
2
1 1 4 u
1 u
1 2uv
Transformed Region: y y y y 1 yx x 4 yx x 2x y x 1 v 4 v 2 u 1 u 42 4
1 4
3
S2
u
2 1 4v 2
1
3
4
y x 4 x eR xy 2
dA14 1
e e2
1 dv du 2u2 1 2
2 1 4
e
v 2 4
2
u2
du1 14
e ln 2 ln
2
e
1 2
1 du u e2
e
ln u1 4
e
e
1 2
1 4
e
1 2
ln 8
0.9798
17. u u x
x x 1 u 2 x, y u, v x
y y v 1 2
4, 8,
v v y
x x 1 u 2
y y v
0 46
y
xy=04
x+y=8
2
x+y=42 4
xy=4x
6
8
4
y ex
y
dA4 0 8
uev 1 2 u e44
R
1 dv du 2 1 du 1 2 4 u e 48
14
12 e4
1
19. u u x
x x 1 u 5
4y 4y
0, 5, 4v ,
v v y 1 5 1 5
x x 1 u 5
y y v
0 52
y
xy=0 x + 4y = 51
1
x
3 1
4
x y u v xR
y x u v y x
1 55 5
4 5 uv
1 5 1 du dv 55 2
x + 4y = 02
xy=5
4y dA0 5 0 0
1 2 3 u 5 3
v0
dv
2 5 2 3 v 3 3
5 2 0
100 9
168 21. u
Chapter 13 x y, v y x u v xRy
Multiple Integration x y, x 1 2a u
1 u 2
v ,y
1 u 2
v
x y u v
y dA0 u
u
1 dv du 2y
a
u u du0
2 5 u 5
a 2 0
2 5 a 5
2
v=ua a
x+y=ax 2a x a
a
v = u
23. au 2 a2
x2 a2
y2 b2 bv 2 b2
1, x 1 1
au, y
bv
(a)
x2 a2
y2 b2y
1
u2
v2v
1
1
u2
v2
b Rx
Su
1
a
(b)
x, y u, v
x u a b
y v
y x u v 0 0 ab
(c) AS
ab dS ab 12
ab
25. Jacobian
x, y u, v v, y 1 v1 vw
x y u v uv 1 v w
y x u v uvw 0 uv uv 1 1 u2v v u2v 1 v u2v w u uv 2 u2vw u uv2 1 w uv2w
27. x
u1 x, y, z u, v, w
w, z u u1 w uw
29. x
sin x, y, z , ,
cos , y sin cos sin sin cos cos cos2 2 2 2 2
sin
sin , z sin sin sin cos 0
cos cos cos2
cos sin sin cos cos2 sin sin2 sin cos2 sin2 sin2 cos2 sin2 sin2
sin sin cos2 cos2
cos cos2
sin2 sin2 sin3
sin
cos2
sin sin sin
sin2