Solucionario Larson Cap 13

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


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


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


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


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)


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


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


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


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


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