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  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Checkpoint

5.1

V = i = 1 2 j = 1 2 f ( x i j * , y i j * ) Δ A = 0 V = i = 1 2 j = 1 2 f ( x i j * , y i j * ) Δ A = 0

5.2

a. 26 b. Answers may vary.

5.3

1340 3 1340 3

5.4

4 ln 5 ln 5 4 ln 5 ln 5

5.5

π 2 π 2

5.6

Answers to both parts a. and b. may vary.

5.7

Type I and Type II are expressed as {(x,y)|0x2,x2y2x}{(x,y)|0x2,x2y2x} and {(x,y)|0y4,12yxy},{(x,y)|0y4,12yxy}, respectively.

5.8

π / 4 π / 4

5.9

{ ( x , y ) | 0 y 1 , 1 x e y } { ( x , y ) | 1 y e , 1 x 2 } { ( x , y ) | e y e 2 , ln y x 2 } { ( x , y ) | 0 y 1 , 1 x e y } { ( x , y ) | 1 y e , 1 x 2 } { ( x , y ) | e y e 2 , ln y x 2 }

5.10

Same as in the example shown.

5.11

216 35 216 35

5.12

e24+10e494e24+10e494 cubic units

5.13

814814 square units

5.14

3 4 3 4

5.15

π 4 π 4

5.16

55 72 0.7638 55 72 0.7638

5.17

14 3 14 3

5.18

8 π 8 π

5.19

π/8π/8

5.20

V=02π022(162r2)rdrdθ=64πV=02π022(162r2)rdrdθ=64π cubic units

5.21

A = 2 π / 2 π / 6 1 + sin θ 3 3 sin θ r d r d θ = 8 π + 9 3 A = 2 π / 2 π / 6 1 + sin θ 3 3 sin θ r d r d θ = 8 π + 9 3

5.22

π4π4

5.23

B z sin x cos y d V = 8 B z sin x cos y d V = 8

5.24

E 1 d V = 8 x = −3 x = 3 y = 9 x 2 y = 9 x 2 z = 9 x 2 y 2 z = 9 x 2 y 2 1 d z d y d x = 36 π . E 1 d V = 8 x = −3 x = 3 y = 9 x 2 y = 9 x 2 z = 9 x 2 y 2 z = 9 x 2 y 2 1 d z d y d x = 36 π .

5.25

(i) z=0z=4x=0x=4zy=x2y=4zf(x,y,z)dydxdz,z=0z=4x=0x=4zy=x2y=4zf(x,y,z)dydxdz, (ii) y=0y=4z=0z=4yx=0x=yf(x,y,z)dxdzdy,y=0y=4z=0z=4yx=0x=yf(x,y,z)dxdzdy, (iii) y=0y=4x=0x=yz=0z=4yf(x,y,z)dzdxdy,y=0y=4x=0x=yz=0z=4yf(x,y,z)dzdxdy, (iv) x=0x=2y=x2y=4z=0z=4yf(x,y,z)dzdydx,x=0x=2y=x2y=4z=0z=4yf(x,y,z)dzdydx, (v) x=0x=2z=0z=4x2y=x2y=4zf(x,y,z)dydzdxx=0x=2z=0z=4x2y=x2y=4zf(x,y,z)dydzdx

5.26

f ave = 8 f ave = 8

5.27

8 8

5.28

E f ( r , θ , z ) r d z d r d θ = θ = 0 θ = π r = 0 r = 2 sin θ z = 0 z = 4 r sin θ f ( r , θ , z ) r d z d r d θ . E f ( r , θ , z ) r d z d r d θ = θ = 0 θ = π r = 0 r = 2 sin θ z = 0 z = 4 r sin θ f ( r , θ , z ) r d z d r d θ .

5.29

E={(r,θ,z)|0θ2π,0z1,zr2z2}E={(r,θ,z)|0θ2π,0z1,zr2z2} and V=r=0r=1z=rz=2r2θ=0θ=2πrdθdzdr.V=r=0r=1z=rz=2r2θ=0θ=2πrdθdzdr.

5.30

E2={(r,θ,z)|0θ2π,0r1,rz4r2}E2={(r,θ,z)|0θ2π,0r1,rz4r2} and V=r=0r=1z=rz=4r2θ=0θ=2πrdθdzdr.V=r=0r=1z=rz=4r2θ=0θ=2πrdθdzdr.

5.31

V ( E ) = θ = 0 θ = 2 π ϕ = 0 φ = π / 3 ρ = 0 ρ = 2 ρ 2 sin φ d ρ d φ d θ V ( E ) = θ = 0 θ = 2 π ϕ = 0 φ = π / 3 ρ = 0 ρ = 2 ρ 2 sin φ d ρ d φ d θ

5.32

Rectangular: x=−2x=2y=4x2y=4x2z=4x2y2z=4x2y2dzdydxx=−1x=1y=1x2y=1x2z=4x2y2z=4x2y2dzdydx.x=−2x=2y=4x2y=4x2z=4x2y2z=4x2y2dzdydxx=−1x=1y=1x2y=1x2z=4x2y2z=4x2y2dzdydx.
Cylindrical: θ=0θ=2πr=1r=2z=4r2z=4r2rdzdrdθ.θ=0θ=2πr=1r=2z=4r2z=4r2rdzdrdθ.
Spherical: φ=π/6φ=5π/6θ=0θ=2πρ=cscφρ=2ρ2sinφdρdθdφ.φ=π/6φ=5π/6θ=0θ=2πρ=cscφρ=2ρ2sinφdρdθdφ.

5.33

9 π 8 kg 9 π 8 kg

5.34

Mx=81π64Mx=81π64 and My=81π64My=81π64

5.35

x=Mym=81π/649π/8=98x=Mym=81π/649π/8=98 and y=Mxm=81π/649π/8=98.y=Mxm=81π/649π/8=98.

5.36

x=Mym=1/201/12=35x=Mym=1/201/12=35 and y=Mxm=1/241/12=12y=Mxm=1/241/12=12

5.37

x c = M y m = 1 / 15 1 / 6 = 2 5 and y c = M x m = 1 / 12 1 / 6 = 1 2 x c = M y m = 1 / 15 1 / 6 = 2 5 and y c = M x m = 1 / 12 1 / 6 = 1 2

5.38

Ix=x=0x=2y=0y=xy2xydydx=6415Ix=x=0x=2y=0y=xy2xydydx=6415 and Iy=x=0x=2y=0y=xx2xydydx=6435.Iy=x=0x=2y=0y=xx2xydydx=6435. Also, I0=x=0x=2y=0y=x(x2+y2)xydydx=12821.I0=x=0x=2y=0y=x(x2+y2)xydydx=12821.

5.39

Rx=63535,Rx=63535, Ry=61515,Ry=61515, and R0=4427.R0=4427.

5.40

54 35 = 1.543 54 35 = 1.543

5.41

( 3 2 , 9 8 , 1 2 ) ( 3 2 , 9 8 , 1 2 )

5.42

The moments of inertia of the tetrahedron QQ about the yz-plane,yz-plane, the xz-plane,xz-plane, and the xy-planexy-plane are 99/35,36/7,and243/35,99/35,36/7,and243/35, respectively.

5.43

T−1(x,y)=(u,v)T−1(x,y)=(u,v) where u=3xy3u=3xy3 and v=y3v=y3

5.44

J ( u , v ) = ( x , y ) ( u , v ) = | x u x v y u y v | = | 1 1 0 2 | = 2 J ( u , v ) = ( x , y ) ( u , v ) = | x u x v y u y v | = | 1 1 0 2 | = 2

5.45

0 π / 2 0 1 r 3 d r d θ 0 π / 2 0 1 r 3 d r d θ

5.46

x=12(v+u)x=12(v+u) and y=12(vu)y=12(vu) and 24−uu4u2(12)dvdu.24−uu4u2(12)dvdu.

5.47

1 2 ( sin 2 2 ) 1 2 ( sin 2 2 )

5.48

0 3 0 2 1 2 ( v 3 + v w 3 u ) d u d v d w = 2 + ln 8 0 3 0 2 1 2 ( v 3 + v w 3 u ) d u d v d w = 2 + ln 8

Section 5.1 Exercises

1 .

27.

3 .

0.

5 .

21.3.

7 .

a. 28 ft3ft3 b. 1.75 ft.

9 .

a. 0.1120.112 b. fave0.175;fave0.175; here f(0.4,0.2)0.1,f(0.4,0.2)0.1, f(0.2,0.6)−0.2,f(0.2,0.6)−0.2, f(0.8,0.2)0.6,f(0.8,0.2)0.6, and f(0.8,0.6)0.2.f(0.8,0.6)0.2.

11 .

2 π . 2 π .

13 .

40.

15 .

81 2 + 39 2 3 . 81 2 + 39 2 3 .

17 .

e 1 . e 1 .

19 .

15 10 2 9 . 15 10 2 9 .

21 .

0.

23 .

( e 1 ) ( 1 + sin 1 cos 1 ) . ( e 1 ) ( 1 + sin 1 cos 1 ) .

25 .

3 4 ln ( 5 3 ) + 2 ln 2 2 ln 2 . 3 4 ln ( 5 3 ) + 2 ln 2 2 ln 2 .

27 .

1 8 [ ( 2 3 3 ) π + 6 ln 2 ] . 1 8 [ ( 2 3 3 ) π + 6 ln 2 ] .

29 .

1 4 e 4 ( e 4 1 ) . 1 4 e 4 ( e 4 1 ) .

31 .

4 ( e 1 ) ( 2 e ) . 4 ( e 1 ) ( 2 e ) .

33 .

π 4 + ln ( 5 4 ) 1 2 ln 2 + arctan 2 . π 4 + ln ( 5 4 ) 1 2 ln 2 + arctan 2 .

35 .

1 2 . 1 2 .

37 .

1 2 ( 2 cosh 1 + cosh 2 3 ) . 1 2 ( 2 cosh 1 + cosh 2 3 ) .

49 .

a. f(x,y)=12xy(x2+y2)f(x,y)=12xy(x2+y2) b. V=0101f(x,y)dxdy=18V=0101f(x,y)dxdy=18 c. fave=18;fave=18;
d.

In xyz space, a plane is formed at z = 1/8, and there is another shape that starts at the origin, increases through the plane in a line roughly running from (1, 0.25, 0.125) to (0.25, 1, 0.125), and then rapidly increases to (1, 1, 1).
53 .

a. For m=n=2,m=n=2, I=4e−0.52.43I=4e−0.52.43 b. fave=e−0.50.61;fave=e−0.50.61;
c.

In xyz space, a plane is formed at z = 0.61, and there is another shape with maximum roughly at (0, 0, 0.92), which decreases along all the sides to the points (plus or minus 1, plus or minus 1, 0.12).
55 .

a. 2n+1+142n+1+14 b. 1414

59 .

56.5°56.5° F; here f(x1*,y1*)=71,f(x1*,y1*)=71, f(x2*,y1*)=72,f(x2*,y1*)=72, f(x2*,y1*)=40,f(x2*,y1*)=40, f(x2*,y2*)=43,f(x2*,y2*)=43, where xi*xi* and yj*yj* are the midpoints of the subintervals of the partitions of [a,b][a,b] and [c,d],[c,d], respectively.

Section 5.2 Exercises

61 .

27 20 27 20

63 .

Type I but not Type II

65 .

π 2 π 2

67 .

1 6 ( 8 + 3 π ) 1 6 ( 8 + 3 π )

69 .

1000 3 1000 3

71 .

Type I and Type II

73 .

The region DD is not of Type I: it does not lie between two vertical lines and the graphs of two continuous functions g1(x)g1(x) and g2(x).g2(x). The region DD is not of Type II: it does not lie between two horizontal lines and the graphs of two continuous functions h1(y)h1(y) and h2(y).h2(y).

75 .

π 2 π 2

77 .

0 0

79 .

2 3 2 3

81 .

41 20 41 20

83 .

−63 −63

85 .

π π

87 .

a. Answers may vary; b. 2323

89 .

a. Answers may vary; b. 7373

91 .

8 π 3 8 π 3

93 .

e 3 2 e 3 2

95 .

2 3 2 3

97 .

0 1 x 1 1 x x d y d x = −1 0 0 y + 1 x d x d y + 0 1 0 1 y x d x d y = 1 3 0 1 x 1 1 x x d y d x = −1 0 0 y + 1 x d x d y + 0 1 0 1 y x d x d y = 1 3

99 .

−1 1 1 y 2 1 y 2 y d x d y = −1 1 1 x 2 1 x 2 y d y d x = 0 −1 1 1 y 2 1 y 2 y d x d y = −1 1 1 x 2 1 x 2 y d y d x = 0

101 .

D ( x 2 y 2 ) d A = −1 1 y 4 1 1 y 4 ( x 2 y 2 ) d x d y = 464 4095 D ( x 2 y 2 ) d A = −1 1 y 4 1 1 y 4 ( x 2 y 2 ) d x d y = 464 4095

103 .

4 5 4 5

105 .

5 π 32 5 π 32

109 .

1 1

111 .

2 2

113 .

a. 13;13; b. 16;16; c. 1616

115 .

a. 43;43; b. 2π;2π; c. 6π436π43

117 .

0and0.865474;0and0.865474; A(D)=0.621135A(D)=0.621135

119 .

P[X+Y6]=1+32e25e6/50.45;P[X+Y6]=1+32e25e6/50.45; there is a 45%45% chance that a customer will spend 66 minutes in the drive-thru line.

Section 5.3 Exercises

123 .

D = { ( r , θ ) | 4 r 5 , π 2 θ π } D = { ( r , θ ) | 4 r 5 , π 2 θ π }

125 .

D = { ( r , θ ) | 0 r 2 , 0 θ π } D = { ( r , θ ) | 0 r 2 , 0 θ π }

127 .

D = { ( r , θ ) | 0 r 4 sin θ , 0 θ π } D = { ( r , θ ) | 0 r 4 sin θ , 0 θ π }

129 .

D = { ( r , θ ) | 3 r 5 , π 4 θ π 2 } D = { ( r , θ ) | 3 r 5 , π 4 θ π 2 }

131 .

D = { ( r , θ ) | 3 r 5 , 3 π 4 θ 5 π 4 } D = { ( r , θ ) | 3 r 5 , 3 π 4 θ 5 π 4 }

133 .

D = { ( r , θ ) | 0 r tan θ sec θ , 0 θ π 4 } D = { ( r , θ ) | 0 r tan θ sec θ , 0 θ π 4 }

135 .

0 0

137 .

63 π 16 63 π 16

139 .

3367 π 18 3367 π 18

141 .

35 π 2 576 35 π 2 576

143 .

7 288 π 2 [ 21 e 2 + e 4 ] 7 288 π 2 [ 21 e 2 + e 4 ]

145 .

5 4 ln ( 3 + 2 2 ) 5 4 ln ( 3 + 2 2 )

147 .

1 6 ( 2 2 ) 1 6 ( 2 2 )

149 .

0 π 0 2 r 5 d r d θ = 32 π 3 0 π 0 2 r 5 d r d θ = 32 π 3

151 .

π / 2 π / 2 0 4 r sin ( r 2 ) d r d θ = π sin 2 8 π / 2 π / 2 0 4 r sin ( r 2 ) d r d θ = π sin 2 8

153 .

3 π 4 3 π 4

155 .

π 2 π 2

157 .

1 3 ( 4 π 3 3 ) 1 3 ( 4 π 3 3 )

159 .

16 3 π 16 3 π

161 .

π 18 π 18

163 .

a. 2π3;2π3; b. π2;π2; c. π6π6

165 .

256 π 3 cm 3 256 π 3 cm 3

167 .

3 π 32 3 π 32

169 .

4 π 4 π

171 .

π 4 π 4

173 .

1 2 π e ( e 1 ) 1 2 π e ( e 1 )

175 .

3 π 4 3 π 4

177 .

133 π 3 864 133 π 3 864

Section 5.4 Exercises

181 .

192 192

183 .

0 0

185 .

1 2 2 3 0 1 ( x 2 + ln y + z ) d z d x d y = 35 6 + 2 ln 2 1 2 2 3 0 1 ( x 2 + ln y + z ) d z d x d y = 35 6 + 2 ln 2

187 .

1 3 0 4 −1 2 ( x 2 z + 1 y ) d z d x d y = 64 + 12 ln 3 1 3 0 4 −1 2 ( x 2 z + 1 y ) d z d x d y = 64 + 12 ln 3

191 .

77 12 77 12

193 .

2 2

195 .

439 120 439 120

197 .

0 0

199 .

64 105 64 105

201 .

11 26 11 26

203 .

113 450 113 450

205 .

1 160 ( 6 3 41 ) 1 160 ( 6 3 41 )

207 .

3 π 2 3 π 2

209 .

1250 1250

211 .

0 5 −3 3 0 9 y 2 z d z d y d x = 90 0 5 −3 3 0 9 y 2 z d z d y d x = 90

213 .

V=5.33V=5.33

A complex shape that starts at the origin and reaches its maximum at (negative 2, negative 2, 8). The shape is truncated by the x = y plane, the x = 0 plane, the y = negative 2 plane, the z = 0 plane, and a complex triangular-like shape with curved edges and sides (negative 2, negative 2, 8), (0, 0, 0), and (0, negative 2, 4).
215 .

011324(y2z2+1)dzdxdy;011324(y2z2+1)dzdxdy; 011324(x2y2+1)dydzdx011324(x2y2+1)dydzdx

219 .

V = a a a 2 z 2 a 2 z 2 x 2 + z 2 a 2 d y d x d z V = a a a 2 z 2 a 2 z 2 x 2 + z 2 a 2 d y d x d z

221 .

9 2 9 2

223 .

156 5 156 5

225 .

a. Answers may vary; b. 12831283

227 .

a. 0r0r2x20r2x2y2dzdydx;0r0r2x20r2x2y2dzdydx; b. 0r0r2y20r2x2y2dzdxdy,0r0r2y20r2x2y2dzdxdy, 0r0r2z20r2x2z2dydxdz,0r0r2z20r2x2z2dydxdz, 0r0r2x20r2x2z2dydzdx,0r0r2x20r2x2z2dydzdx, 0r0r2z20r2y2z2dxdydz,0r0r2z20r2y2z2dxdydz, 0r0r2y20r2y2z2dxdzdy0r0r2y20r2y2z2dxdzdy

229 .

3 3

231 .

250 3 250 3

233 .

5 16 0.313 5 16 0.313

235 .

35 2 35 2

Section 5.5 Exercises

241 .

9 π 8 9 π 8

243 .

1 8 1 8

245 .

π e 2 6 π e 2 6

249 .

a. E={(r,θ,z)|0θπ,0r4sinθ,0z16r2};E={(r,θ,z)|0θπ,0r4sinθ,0z16r2}; b. 0π04sinθ016r2f(r,θ,z)rdzdrdθ0π04sinθ016r2f(r,θ,z)rdzdrdθ

251 .

a. E={(r,θ,z)|0θπ2,0r3,9r2z20r(cosθ+sinθ)};E={(r,θ,z)|0θπ2,0r3,9r2z20r(cosθ+sinθ)}; b. 0π/2039r220r(cosθ+sinθ)f(r,θ,z)rdzdrdθ0π/2039r220r(cosθ+sinθ)f(r,θ,z)rdzdrdθ

253 .

a. E={(r,θ,z)|0r3,0θπ2,0zrcosθ+3},E={(r,θ,z)|0r3,0θπ2,0zrcosθ+3}, f(r,θ,z)=1rcosθ+3;f(r,θ,z)=1rcosθ+3; b. 030π/20rcosθ+3rrcosθ+3dzdθdr=9π4030π/20rcosθ+3rrcosθ+3dzdθdr=9π4

255 .

a. y=rcosθ,z=rsinθ,x=z,y=rcosθ,z=rsinθ,x=z, E={(r,θ,z)|1r3,0θ2π,0z1r2},f(r,θ,z)=z;E={(r,θ,z)|1r3,0θ2π,0z1r2},f(r,θ,z)=z; b. 1302π01r2zrdzdθdr=256π31302π01r2zrdzdθdr=256π3

257 .

π π

259 .

π 3 π 3

261 .

π 2 π 2

263 .

4 π 3 4 π 3

265 .

V=π120.2618V=π120.2618

A quarter section of an ellipsoid with width 2, height 1, and depth 1.
267 .

0 1 0 π r 2 r z r 2 cos θ d z d θ d r 0 1 0 π r 2 r z r 2 cos θ d z d θ d r

269 .

180 π 10 180 π 10

271 .

81 π ( π 2 ) 16 81 π ( π 2 ) 16

277 .

a. f(ρ,θ,φ)=ρsinφ(cosθ+sinθ),f(ρ,θ,φ)=ρsinφ(cosθ+sinθ), E={(ρ,θ,φ)|1ρ2,0θπ,0φπ2};E={(ρ,θ,φ)|1ρ2,0θπ,0φπ2}; b. 0π0π/212ρ3cosφsinφdρdφdθ=15π80π0π/212ρ3cosφsinφdρdφdθ=15π8

279 .

a. f(ρ,θ,φ)=ρcosφ;f(ρ,θ,φ)=ρcosφ; E={(ρ,θ,φ)|0ρ2cosφ,0θπ2,0φπ4};E={(ρ,θ,φ)|0ρ2cosφ,0θπ2,0φπ4}; b. 0π/20π/402cosφρ3sinφcosφdρdφdθ=7π240π/20π/402cosφρ3sinφcosφdρdφdθ=7π24

281 .

π π

283 .

9 π ( 2 1 ) 9 π ( 2 1 )

285 .

0 π / 2 0 π 0 4 ρ 6 sin φ d ρ d φ d θ 0 π / 2 0 π 0 4 ρ 6 sin φ d ρ d φ d θ

287 .

V=4π337.255V=4π337.255

A sphere of radius 1 with a hole drilled into it of radius 0.5.
289 .

343 π 32 343 π 32

291 .

02π2416r216r2rdzdrdθ;02π2416r216r2rdzdrdθ; π/65π/602π2cscφ4ρ2sinρdρdθdφπ/65π/602π2cscφ4ρ2sinρdρdθdφ

293 .

P=32P0π3P=32P0π3 watts

295 .

Q = k r 4 π μ C Q = k r 4 π μ C

Section 5.6 Exercises

297 .

27 2 27 2

299 .

24 2 24 2

301 .

76 76

303 .

8 π 8 π

305 .

π 2 π 2

307 .

2 2

309 .

a. Mx=815,My=1625;Mx=815,My=1625; b. x=125,y=65;x=125,y=65;
c.

A triangular region R bounded by the x and y axes and the line y = negative x/2 + 3, with a point marked at (12/5, 6/5).
311 .

a. Mx=21625,My=43225;Mx=21625,My=43225; b. x=185,y=95;x=185,y=95;
c.

A rectangle R bounded by the x and y axes and the lines x = 6 and y = 3 with point marked (18/5, 9/5).
313 .

a. Mx=3685,My=15525;Mx=3685,My=15525; b. x=38895,y=9295;x=38895,y=9295;
c.

A trapezoid R bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5 with the point marked (92/95, 388/95).
315 .

a. Mx=16π,My=8π;Mx=16π,My=8π; b. x=1,y=2;x=1,y=2;
c.

A circle with radius 2 centered at (1, 2), which is tangent to the x axis at (1, 0) and has pointed marked at the center (1, 2).
317 .

a. Mx=0,My=0;Mx=0,My=0; b. x=0,y=0;x=0,y=0;
c.

An ellipse R with center the origin, major axis 2, and minor axis 0.5, with point marked at the origin.
319 .

a. Mx=2,My=0;Mx=2,My=0; b. x=0,y=1;x=0,y=1;
c.

A square R with side length square root of 2 rotated 45 degrees, with corners at the origin, (2, 0), (1, 1), and (negative 1, 1). A point is marked at (0, 1).
321 .

a. Ix=24310,Iy=4865,andI0=2432;Ix=24310,Iy=4865,andI0=2432; b. Rx=355,Ry=655,andR0=3Rx=355,Ry=655,andR0=3

323 .

a. Ix=259227,Iy=64827,andI0=324027;Ix=259227,Iy=64827,andI0=324027; b. Rx=6217,Ry=3217,andR0=31057Rx=6217,Ry=3217,andR0=31057

325 .

a. Ix=88,Iy=1560,andI0=1648;Ix=88,Iy=1560,andI0=1648; b. Rx=41819,Ry=741019,Rx=41819,Ry=741019, and R0=2195719R0=2195719

327 .

a. Ix=128π3,Iy=56π3,andI0=184π3;Ix=128π3,Iy=56π3,andI0=184π3; b. Rx=433,Ry=213,Rx=433,Ry=213, and R0=693R0=693

329 .

a. Ix=π32,Iy=π8,andI0=5π32;Ix=π32,Iy=π8,andI0=5π32; b. Rx=14,Ry=12,andR0=54Rx=14,Ry=12,andR0=54

331 .

a. Ix=73,Iy=13,andI0=83;Ix=73,Iy=13,andI0=83; b. Rx=426,Ry=66,andR0=233Rx=426,Ry=66,andR0=233

333 .

m = 1 3 m = 1 3

337 .

a. m=9π4;m=9π4; b. Mxy=3π2,Mxz=818,Myz=818;Mxy=3π2,Mxz=818,Myz=818; c. x=92π,y=92π,z=23;x=92π,y=92π,z=23; d. the solid QQ and its center of mass are shown in the following figure.

A quarter cylinder in the first quadrant with height 1 and radius 3. A point is marked at (9/(2 pi), 9/(2 pi), 2/3).
339 .

a. x=322π,y=3(22)2π,z=0;x=322π,y=3(22)2π,z=0; b. the solid QQ and its center of mass are shown in the following figure.

A wedge from a cylinder in the first quadrant with height 2, radius 1, and angle roughly 45 degrees. A point is marked at (3 times the square root of 2/(2 pi), 3 times (2 minus the square root of 2)/(2 pi), 0).
343 .

n = −2 n = −2

349 .

a. ρ(x,y,z)=x2+y2;ρ(x,y,z)=x2+y2; b. 16π716π7

351 .

M x y = π ( f ( 0 ) f ( a ) + a f ( a ) ) M x y = π ( f ( 0 ) f ( a ) + a f ( a ) )

355 .

I x = I y = I z 0.84 I x = I y = I z 0.84

Section 5.7 Exercises

357 .

a. T(u,v)=(g(u,v),h(u,v)),x=g(u,v)=u2T(u,v)=(g(u,v),h(u,v)),x=g(u,v)=u2 and y=h(u,v)=v3.y=h(u,v)=v3. The functions gg and hh are continuous and differentiable, and the partial derivatives gu(u,v)=12,gu(u,v)=12, gv(u,v)=0,hu(u,v)=0andhv(u,v)=13gv(u,v)=0,hu(u,v)=0andhv(u,v)=13 are continuous on S;S; b. T(0,0)=(0,0),T(0,0)=(0,0), T(1,0)=(12,0),T(0,1)=(0,13),T(1,0)=(12,0),T(0,1)=(0,13), and T(1,1)=(12,13);T(1,1)=(12,13); c. RR is the rectangle of vertices (0,0),(12,0),(12,13),and(0,13)(0,0),(12,0),(12,13),and(0,13) in the xy-plane;xy-plane; the following figure.

A rectangle with one corner at the origin, horizontal length 0.5, and vertical height 0.34.
359 .

a. T(u,v)=(g(u,v),h(u,v)),x=g(u,v)=2uv,T(u,v)=(g(u,v),h(u,v)),x=g(u,v)=2uv, and y=h(u,v)=u+2v.y=h(u,v)=u+2v. The functions gg and hh are continuous and differentiable, and the partial derivatives gu(u,v)=2,gu(u,v)=2, gv(u,v)=−1,gv(u,v)=−1, hu(u,v)=1,hu(u,v)=1, and hv(u,v)=2hv(u,v)=2 are continuous on S;S; b. T(0,0)=(0,0),T(0,0)=(0,0), T(1,0)=(2,1),T(1,0)=(2,1), T(0,1)=(−1,2),T(0,1)=(−1,2), and T(1,1)=(1,3);T(1,1)=(1,3); c. RR is the parallelogram of vertices (0,0),(2,1),(1,3),and(−1,2)(0,0),(2,1),(1,3),and(−1,2) in the xy-plane;xy-plane; see the following figure.

A square of side length square root of 5 with one corner at the origin and another at (2, 1).
361 .

a. T(u,v)=(g(u,v),h(u,v)),x=g(u,v)=u3,T(u,v)=(g(u,v),h(u,v)),x=g(u,v)=u3, and y=h(u,v)=v3.y=h(u,v)=v3. The functions gg and hh are continuous and differentiable, and the partial derivatives gu(u,v)=3u2,gu(u,v)=3u2, gv(u,v)=0,gv(u,v)=0, hu(u,v)=0,hu(u,v)=0, and hv(u,v)=3v2hv(u,v)=3v2 are continuous on S;S; b. T(0,0)=(0,0),T(0,0)=(0,0), T(1,0)=(1,0),T(1,0)=(1,0), T(0,1)=(0,1),T(0,1)=(0,1), and T(1,1)=(1,1);T(1,1)=(1,1); c. RR is the unit square in the xy-plane;xy-plane; see the figure in the answer to the previous exercise.

363 .

TT is not one-to-one: two points of SS have the same image. Indeed, T(−2,0)=T(2,0)=(16,4).T(−2,0)=T(2,0)=(16,4).

365 .

TT is one-to-one: We argue by contradiction. T(u1,v1)=T(u2,v2)T(u1,v1)=T(u2,v2) implies 2u1v1=2u2v22u1v1=2u2v2 and u1=u2.u1=u2. Thus, u1=u2u1=u2 and v1=v2.v1=v2.

367 .

TT is not one-to-one: T(1,v,w)=(−1,v,w)T(1,v,w)=(−1,v,w)

369 .

u = x 2 y 3 , v = x + y 3 u = x 2 y 3 , v = x + y 3

371 .

u = e x , v = e x + y u = e x , v = e x + y

373 .

u = x y + z 2 , v = x + y z 2 , w = x + y + z 2 u = x y + z 2 , v = x + y z 2 , w = x + y + z 2

375 .

S = { ( u , v ) | u 2 + v 2 1 } S = { ( u , v ) | u 2 + v 2 1 }

377 .

R = { ( u , v , w ) | u 2 v 2 w 2 1 , w > 0 } R = { ( u , v , w ) | u 2 v 2 w 2 1 , w > 0 }

379 .

3 2 3 2

381 .

−1 −1

383 .

2 u v 2 u v

385 .

v u 2 v u 2

387 .

2 2

389 .

a. T(u,v)=(2u+v,3v);T(u,v)=(2u+v,3v); b. The area of RR is
A(R)=03y/3(6y)/3dxdy=0101u|(x,y)(u,v)|dvdu=0101u6dvdu=3.A(R)=03y/3(6y)/3dxdy=0101u|(x,y)(u,v)|dvdu=0101u6dvdu=3.

391 .

1 4 1 4

393 .

−1 + cos 2 −1 + cos 2

395 .

π 15 π 15

397 .

31 5 31 5

399 .

T(r,θ,z)=(rcosθ,rsinθ,z);S=[0,3]×[0,π2]×[0,1]T(r,θ,z)=(rcosθ,rsinθ,z);S=[0,3]×[0,π2]×[0,1] in the rθz-spacerθz-space

403 .

The area of RR is 1046;1046; the boundary curves of RR are graphed in the following figure.

Four lines are drawn, namely, y = 3, y = 2, y = 3/(x squared), and y = 2/(x squared). The lines y = 3 and y = 2 are parallel to each other. The lines y = 3/(x squared) and y = 2/(x squared) are curves that run somewhat parallel to each other.
405 .

8 8

409 .

a. R={(x,y)|y2+x22y4x+10};R={(x,y)|y2+x22y4x+10}; b. RR is graphed in the following figure;

A circle with radius 2 and center (2, 1).


c. 3.163.16

411 .

a. T0,2T3,0(u,v)=(u+3v,2u+7v);T0,2T3,0(u,v)=(u+3v,2u+7v); b. The image SS is the quadrilateral of vertices (0,0),(3,7),(2,4),and(4,9);(0,0),(3,7),(2,4),and(4,9); c. SS is graphed in the following figure;

A four-sided figure with points the origin, (2, 4), (4, 9), and (3, 7).


d. 3232

413 .

2662 3 π 282.45 in 3 2662 3 π 282.45 in 3

415 .

A ( R ) 83,999.2 A ( R ) 83,999.2

Review Exercises

417 .

True.

419 .

False.

421 .

0

423 .

1 4 1 4

425 .

1.475

427 .

52 3 π 52 3 π

429 .

π 16 π 16

431 .

93.291

433 .

( 8 15 , 8 15 ) ( 8 15 , 8 15 )

435 .

( 0 , 0 , 8 5 ) ( 0 , 0 , 8 5 )

437 .

1.452π×10151.452π×1015 ft-lb

439 .

y=−1.238×10−7x3+0.001196x23.666x+7208;y=−1.238×10−7x3+0.001196x23.666x+7208; average temperature approximately 2800°C2800°C

441 .

π 3 π 3

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