Calculus Volume 3

# Chapter 5

### Checkpoint

5.1

$V=∑i=12∑j=12f(xij*,yij*)ΔA=0V=∑i=12∑j=12f(xij*,yij*)ΔA=0$

5.2

a. 26 b. Answers may vary.

5.3

$−13403−13403$

5.4

$4−ln5ln54−ln5ln5$

5.5

$π2π2$

5.6

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

5.7

Type I and Type II are expressed as ${(x,y)|0≤x≤2,x2≤y≤2x}{(x,y)|0≤x≤2,x2≤y≤2x}$ and ${(x,y)|0≤y≤4,12y≤x≤y},{(x,y)|0≤y≤4,12y≤x≤y},$ respectively.

5.8

$π/4π/4$

5.9

${(x,y)|0≤y≤1,1≤x≤ey}∪{(x,y)|1≤y≤e,1≤x≤2}∪{(x,y)|e≤y≤e2,lny≤x≤2}{(x,y)|0≤y≤1,1≤x≤ey}∪{(x,y)|1≤y≤e,1≤x≤2}∪{(x,y)|e≤y≤e2,lny≤x≤2}$

5.10

Same as in the example shown.

5.11

$2163521635$

5.12

$e24+10e−494e24+10e−494$ cubic units

5.13

$814814$ square units

5.14

$3434$

5.15

$π4π4$

5.16

$5572≈0.76385572≈0.7638$

5.17

$143143$

5.18

$8π8π$

5.19

$π/8π/8$

5.20

$V=∫02π∫022(16−2r2)rdrdθ=64πV=∫02π∫022(16−2r2)rdrdθ=64π$ cubic units

5.21

$A=2∫−π/2π/6∫1+sinθ3−3sinθrdrdθ=8π+93A=2∫−π/2π/6∫1+sinθ3−3sinθrdrdθ=8π+93$

5.22

$π4π4$

5.23

$∭BzsinxcosydV=8∭BzsinxcosydV=8$

5.24

$∭E1dV=8∫x=−3x=3∫y=−9−x2y=9−x2∫z=−9−x2−y2z=9−x2−y21dzdydx=36π.∭E1dV=8∫x=−3x=3∫y=−9−x2y=9−x2∫z=−9−x2−y2z=9−x2−y21dzdydx=36π.$

5.25

(i) $∫z=0z=4∫x=0x=4−z∫y=x2y=4−zf(x,y,z)dydxdz,∫z=0z=4∫x=0x=4−z∫y=x2y=4−zf(x,y,z)dydxdz,$ (ii) $∫y=0y=4∫z=0z=4−y∫x=0x=yf(x,y,z)dxdzdy,∫y=0y=4∫z=0z=4−y∫x=0x=yf(x,y,z)dxdzdy,$ (iii) $∫y=0y=4∫x=0x=y∫z=0z=4−yf(x,y,z)dzdxdy,∫y=0y=4∫x=0x=y∫z=0z=4−yf(x,y,z)dzdxdy,$ (iv) $∫x=0x=2∫y=x2y=4∫z=0z=4−yf(x,y,z)dzdydx,∫x=0x=2∫y=x2y=4∫z=0z=4−yf(x,y,z)dzdydx,$ (v) $∫x=0x=2∫z=0z=4−x2∫y=x2y=4−zf(x,y,z)dydzdx∫x=0x=2∫z=0z=4−x2∫y=x2y=4−zf(x,y,z)dydzdx$

5.26

$fave=8fave=8$

5.27

$88$

5.28

$∭Ef(r,θ,z)rdzdrdθ=∫θ=0θ=π∫r=0r=2sinθ∫z=0z=4−rsinθf(r,θ,z)rdzdrdθ.∭Ef(r,θ,z)rdzdrdθ=∫θ=0θ=π∫r=0r=2sinθ∫z=0z=4−rsinθf(r,θ,z)rdzdrdθ.$

5.29

$E={(r,θ,z)|0≤θ≤2π,0≤z≤1,z≤r≤2−z2}E={(r,θ,z)|0≤θ≤2π,0≤z≤1,z≤r≤2−z2}$ and $V=∫r=0r=1∫z=rz=2−r2∫θ=0θ=2πrdθdzdr.V=∫r=0r=1∫z=rz=2−r2∫θ=0θ=2πrdθdzdr.$

5.30

$E2={(r,θ,z)|0≤θ≤2π,0≤r≤1,r≤z≤4−r2}E2={(r,θ,z)|0≤θ≤2π,0≤r≤1,r≤z≤4−r2}$ and $V=∫r=0r=1∫z=rz=4−r2∫θ=0θ=2πrdθdzdr.V=∫r=0r=1∫z=rz=4−r2∫θ=0θ=2πrdθdzdr.$

5.31

$V(E)=∫θ=0θ=2π∫ϕ=0φ=π/3∫ρ=0ρ=2ρ2sinφdρdφdθV(E)=∫θ=0θ=2π∫ϕ=0φ=π/3∫ρ=0ρ=2ρ2sinφdρdφdθ$

5.32

Rectangular: $∫x=−2x=2∫y=−4−x2y=4−x2∫z=−4−x2−y2z=4−x2−y2dzdydx−∫x=−1x=1∫y=−1−x2y=1−x2∫z=−4−x2−y2z=4−x2−y2dzdydx.∫x=−2x=2∫y=−4−x2y=4−x2∫z=−4−x2−y2z=4−x2−y2dzdydx−∫x=−1x=1∫y=−1−x2y=1−x2∫z=−4−x2−y2z=4−x2−y2dzdydx.$
Cylindrical: $∫θ=0θ=2π∫r=1r=2∫z=−4−r2z=4−r2rdzdrdθ.∫θ=0θ=2π∫r=1r=2∫z=−4−r2z=4−r2rdzdrdθ.$
Spherical: $∫φ=π/6φ=5π/6∫θ=0θ=2π∫ρ=cscφρ=2ρ2sinφdρdθdφ.∫φ=π/6φ=5π/6∫θ=0θ=2π∫ρ=cscφρ=2ρ2sinφdρdθdφ.$

5.33

$9π8kg9π8kg$

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

$xc=Mym=1/151/6=25andyc=Mxm=1/121/6=12xc=Mym=1/151/6=25andyc=Mxm=1/121/6=12$

5.38

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

5.39

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

5.40

$5435=1.5435435=1.543$

5.41

$(32,98,12)(32,98,12)$

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 $u=3x−y3u=3x−y3$ and $v=y3v=y3$

5.44

$J(u,v)=∂(x,y)∂(u,v)=|∂x∂u∂x∂v∂y∂u∂y∂v|=|1102|=2J(u,v)=∂(x,y)∂(u,v)=|∂x∂u∂x∂v∂y∂u∂y∂v|=|1102|=2$

5.45

$∫0π/2∫01r3drdθ∫0π/2∫01r3drdθ$

5.46

$x=12(v+u)x=12(v+u)$ and $y=12(v−u)y=12(v−u)$ and $∫−44∫−224u2(12)dudv.∫−44∫−224u2(12)dudv.$

5.47

$12(sin2−2)12(sin2−2)$

5.48

$∫03∫02∫12(v3+vw3u)dudvdw=2+ln8∫03∫02∫12(v3+vw3u)dudvdw=2+ln8$

### 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. $fave≃0.175;fave≃0.175;$ 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.

$2π.2π.$

13.

40.

15.

$812+3923.812+3923.$

17.

$e−1.e−1.$

19.

$15−1029.15−1029.$

21.

0.

23.

$(e−1)(1+sin1−cos1).(e−1)(1+sin1−cos1).$

25.

$34ln(53)+2ln22−ln2.34ln(53)+2ln22−ln2.$

27.

$18[(23−3)π+6ln2].18[(23−3)π+6ln2].$

29.

$14e4(e4−1).14e4(e4−1).$

31.

$4(e−1)(2−e).4(e−1)(2−e).$

33.

$−π4+ln(54)−12ln2+arctan2.−π4+ln(54)−12ln2+arctan2.$

35.

$12.12.$

37.

$12(2cosh1+cosh2−3).12(2cosh1+cosh2−3).$

49.

a. $f(x,y)=12xy(x2+y2)f(x,y)=12xy(x2+y2)$ b. $V=∫01∫01f(x,y)dxdy=18V=∫01∫01f(x,y)dxdy=18$ c. $fave=18;fave=18;$
d. 53.

a. For $m=n=2,m=n=2,$ $I=4e−0.5≈2.43I=4e−0.5≈2.43$ b. $fave=e−0.5≃0.61;fave=e−0.5≃0.61;$
c. 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.

$27202720$

63.

Type I but not Type II

65.

$π2π2$

67.

$16(8+3π)16(8+3π)$

69.

$1000310003$

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.

$π2π2$

77.

$00$

79.

$2323$

81.

$41204120$

83.

$−63−63$

85.

$ππ$

87.

a. Answers may vary; b. $2323$

89.

a. Answers may vary; b. $7373$

91.

$8π38π3$

93.

$e−32e−32$

95.

$2323$

97.

$∫01∫x−11−xxdydx=∫−10∫0y+1xdxdy+∫01∫01−yxdxdy=13∫01∫x−11−xxdydx=∫−10∫0y+1xdxdy+∫01∫01−yxdxdy=13$

99.

$∫−11∫−1–y21–y2ydxdy=∫−11∫−1–x21–x2ydydx=0∫−11∫−1–y21–y2ydxdy=∫−11∫−1–x21–x2ydydx=0$

101.

$∬D(x2−y2)dA=∫−11∫y4−11−y4(x2−y2)dxdy=4644095∬D(x2−y2)dA=∫−11∫y4−11−y4(x2−y2)dxdy=4644095$

103.

$4545$

105.

$5π325π32$

109.

$11$

111.

$22$

113.

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

115.

a. $43;43;$ b. $2π;2π;$ c. $6π−436π−43$

117.

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

119.

$P[X+Y≤6]=1+32e2−5e6/5≈0.45;P[X+Y≤6]=1+32e2−5e6/5≈0.45;$ there is a $45%45%$ chance that a customer will spend $66$ minutes in the drive-thru line.

### Section 5.3 Exercises

123.

$D={(r,θ)|4≤r≤5,π2≤θ≤π}D={(r,θ)|4≤r≤5,π2≤θ≤π}$

125.

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

127.

$D={(r,θ)|0≤r≤4sinθ,0≤θ≤π}D={(r,θ)|0≤r≤4sinθ,0≤θ≤π}$

129.

$D={(r,θ)|3≤r≤5,π4≤θ≤π2}D={(r,θ)|3≤r≤5,π4≤θ≤π2}$

131.

$D={(r,θ)|3≤r≤5,3π4≤θ≤5π4}D={(r,θ)|3≤r≤5,3π4≤θ≤5π4}$

133.

$D={(r,θ)|0≤r≤tanθsecθ,0≤θ≤π4}D={(r,θ)|0≤r≤tanθsecθ,0≤θ≤π4}$

135.

$00$

137.

$63π1663π16$

139.

$3367π183367π18$

141.

$35π257635π2576$

143.

$7576π2(21−e+e4)7576π2(21−e+e4)$

145.

$54ln(3+22)54ln(3+22)$

147.

$16(2−2)16(2−2)$

149.

$∫0π∫02r5drdθ=32π3∫0π∫02r5drdθ=32π3$

151.

$∫−π/2π/2∫04rsin(r2)drdθ=πsin28∫−π/2π/2∫04rsin(r2)drdθ=πsin28$

153.

$3π43π4$

155.

$π2π2$

157.

$13(4π−33)13(4π−33)$

159.

$163π163π$

161.

$π18π18$

163.

a. $2π3;2π3;$ b. $π2;π2;$ c. $π6π6$

165.

$256π3cm3256π3cm3$

167.

$3π323π32$

169.

$4π4π$

171.

$π4π4$

173.

$12πe(e−1)12πe(e−1)$

175.

$3−π43−π4$

177.

$133π3864133π3864$

### Section 5.4 Exercises

181.

$192192$

183.

$00$

185.

$∫12∫23∫01(x2+lny+z)dzdxdy=356+2ln2∫12∫23∫01(x2+lny+z)dzdxdy=356+2ln2$

187.

$∫13∫04∫−12(x2z+1y)dzdxdy=64+12ln3∫13∫04∫−12(x2z+1y)dzdxdy=64+12ln3$

191.

$77127712$

193.

$22$

195.

$439120439120$

197.

$00$

199.

$−64105−64105$

201.

$11261126$

203.

$113450113450$

205.

$1160(63−41)1160(63−41)$

207.

$3π23π2$

209.

$12501250$

211.

$∫05∫−33∫09−y2zdzdydx=90∫05∫−33∫09−y2zdzdydx=90$

213.

$V=5.33V=5.33$ 215.

$∫01∫13∫24(y2z2+1)dzdxdy;∫01∫13∫24(y2z2+1)dzdxdy;$ $∫01∫13∫24(x2y2+1)dydzdx∫01∫13∫24(x2y2+1)dydzdx$

219.

$V=∫−aa∫−a2−z2a2−z2∫x2+z2a2dydxdzV=∫−aa∫−a2−z2a2−z2∫x2+z2a2dydxdz$

221.

$9292$

223.

$15651565$

225.

a. Answers may vary; b. $12831283$

227.

a. $∫0r∫0r2−x2∫0r2−x2−y2dzdydx;∫0r∫0r2−x2∫0r2−x2−y2dzdydx;$ b. $∫0r∫0r2−y2∫0r2−x2−y2dzdxdy,∫0r∫0r2−y2∫0r2−x2−y2dzdxdy,$ $∫0r∫0r2−z2∫0r2−x2−z2dydxdz,∫0r∫0r2−z2∫0r2−x2−z2dydxdz,$ $∫0r∫0r2−x2∫0r2−x2−z2dydzdx,∫0r∫0r2−x2∫0r2−x2−z2dydzdx,$ $∫0r∫0r2−z2∫0r2−y2−z2dxdydz,∫0r∫0r2−z2∫0r2−y2−z2dxdydz,$ $∫0r∫0r2−y2∫0r2−y2−z2dxdzdy∫0r∫0r2−y2∫0r2−y2−z2dxdzdy$

229.

$33$

231.

$25032503$

233.

$516≈0.313516≈0.313$

235.

$352352$

### Section 5.5 Exercises

241.

$9π89π8$

243.

$1818$

245.

$πe26πe26$

249.

a. $E={(r,θ,z)|0≤θ≤π,0≤r≤4sinθ,0≤z≤16−r2};E={(r,θ,z)|0≤θ≤π,0≤r≤4sinθ,0≤z≤16−r2};$ b. $∫0π∫04sinθ∫016−r2f(r,θ,z)rdzdrdθ∫0π∫04sinθ∫016−r2f(r,θ,z)rdzdrdθ$

251.

a. $E={(r,θ,z)|0≤θ≤π2,0≤r≤3,9−r2≤z≤10−r(cosθ+sinθ)};E={(r,θ,z)|0≤θ≤π2,0≤r≤3,9−r2≤z≤10−r(cosθ+sinθ)};$ b. $∫0π/2∫03∫9−r210−r(cosθ+sinθ)f(r,θ,z)rdzdrdθ∫0π/2∫03∫9−r210−r(cosθ+sinθ)f(r,θ,z)rdzdrdθ$

253.

a. $E={(r,θ,z)|0≤r≤3,0≤θ≤π2,0≤z≤rcosθ+3},E={(r,θ,z)|0≤r≤3,0≤θ≤π2,0≤z≤rcosθ+3},$ $f(r,θ,z)=1rcosθ+3;f(r,θ,z)=1rcosθ+3;$ b. $∫03∫0π/2∫0rcosθ+3rrcosθ+3dzdθdr=9π4∫03∫0π/2∫0rcosθ+3rrcosθ+3dzdθdr=9π4$

255.

a. $y=rcosθ,z=rsinθ,x=z,y=rcosθ,z=rsinθ,x=z,$ $E={(r,θ,z)|1≤r≤3,0≤θ≤2π,0≤z≤1−r2},f(r,θ,z)=z;E={(r,θ,z)|1≤r≤3,0≤θ≤2π,0≤z≤1−r2},f(r,θ,z)=z;$ b. $∫13∫02π∫01−r2zrdzdθdr=356π3∫13∫02π∫01−r2zrdzdθdr=356π3$

257.

$ππ$

259.

$π3π3$

261.

$ππ$

263.

$4π34π3$

265.

$V=π12≈0.2618V=π12≈0.2618$ 267.

$∫01∫0π∫r2rzr2cosθdzdθdr∫01∫0π∫r2rzr2cosθdzdθdr$

269.

$180π10180π10$

271.

$81π(π−2)1681π(π−2)16$

277.

a. $f(ρ,θ,φ)=ρsinφ(cosθ+sinθ),f(ρ,θ,φ)=ρsinφ(cosθ+sinθ),$ $E={(ρ,θ,φ)|1≤ρ≤2,0≤θ≤π,0≤φ≤π2};E={(ρ,θ,φ)|1≤ρ≤2,0≤θ≤π,0≤φ≤π2};$ b. $∫0π∫0π/2∫12ρ3cosφsinφdρdφdθ=15π8∫0π∫0π/2∫12ρ3cosφsinφdρdφdθ=15π8$

279.

a. $f(ρ,θ,φ)=ρcosφ;f(ρ,θ,φ)=ρcosφ;$ $E={(ρ,θ,φ)|0≤ρ≤2cosφ,0≤θ≤π2,0≤φ≤π4};E={(ρ,θ,φ)|0≤ρ≤2cosφ,0≤θ≤π2,0≤φ≤π4};$ b. $∫0π/2∫0π/4∫02cosφρ3sinφcosφdρdφdθ=7π24∫0π/2∫0π/4∫02cosφρ3sinφcosφdρdφdθ=7π24$

281.

$π4π4$

283.

$9π(2−1)9π(2−1)$

285.

$∫0π/2∫0π/2∫04ρ6sinφdρdφdθ∫0π/2∫0π/2∫04ρ6sinφdρdφdθ$

287.

$V=4π33≈7.255V=4π33≈7.255$ 289.

$343π32343π32$

291.

$∫02π∫24∫−16−r216−r2rdzdrdθ;∫02π∫24∫−16−r216−r2rdzdrdθ;$ $∫π/65π/6∫02π∫2cscφ4ρ2sinρdρdθdφ∫π/65π/6∫02π∫2cscφ4ρ2sinρdρdθdφ$

293.

$P=32P0π3P=32P0π3$ watts

295.

$Q=kr4πμCQ=kr4πμC$

### Section 5.6 Exercises

297.

$272272$

299.

$242242$

301.

$7676$

303.

$8π8π$

305.

$π2π2$

307.

$22$

309.

a. $Mx=815,My=1625;Mx=815,My=1625;$ b. $x−=125,y−=65;x−=125,y−=65;$
c. 311.

a. $Mx=21625,My=43225;Mx=21625,My=43225;$ b. $x−=185,y−=95;x−=185,y−=95;$
c. 313.

a. $Mx=3685,My=15525;Mx=3685,My=15525;$ b. $x−=9295,y−=38895;x−=9295,y−=38895;$
c. 315.

a. $Mx=16π,My=8π;Mx=16π,My=8π;$ b. $x−=1,y−=2;x−=1,y−=2;$
c. 317.

a. $Mx=0,My=0;Mx=0,My=0;$ b. $x−=0,y−=0;x−=0,y−=0;$
c. 319.

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

$m=13m=13$

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. 339.

a. $x−=322π,y−=3(2−2)2π,z−=0;x−=322π,y−=3(2−2)2π,z−=0;$ b. the solid $QQ$ and its center of mass are shown in the following figure. 343.

$n=−2n=−2$

349.

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

351.

$Mxy=π(f(0)−f(a)+af′(a))Mxy=π(f(0)−f(a)+af′(a))$

355.

$Ix=Iy=Iz≃0.84Ix=Iy=Iz≃0.84$

### 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. 359.

a. $T(u,v)=(g(u,v),h(u,v)),x=g(u,v)=2u−v,T(u,v)=(g(u,v),h(u,v)),x=g(u,v)=2u−v,$ 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. 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 $2u1−v1=2u2−v22u1−v1=2u2−v2$ 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.

$u=x−2y3,v=x+y3u=x−2y3,v=x+y3$

371.

$u=ex,v=e−x+yu=ex,v=e−x+y$

373.

$u=x−y+z2,v=x+y−z2,w=−x+y+z2u=x−y+z2,v=x+y−z2,w=−x+y+z2$

375.

$S={(u,v)|u2+v2≤1}S={(u,v)|u2+v2≤1}$

377.

$R={(u,v,w)|u2−v2−w2≤1,w>0}R={(u,v,w)|u2−v2−w2≤1,w>0}$

379.

$3232$

381.

$−1−1$

383.

$2uv2uv$

385.

$vu2vu2$

387.

$22$

389.

a. $T(u,v)=(2u+v,3v);T(u,v)=(2u+v,3v);$ b. The area of $RR$ is
$A(R)=∫03∫y/3(6−y)/3dxdy=∫01∫01−u|∂(x,y)∂(u,v)|dvdu=∫01∫01−u6dvdu=3.A(R)=∫03∫y/3(6−y)/3dxdy=∫01∫01−u|∂(x,y)∂(u,v)|dvdu=∫01∫01−u6dvdu=3.$

391.

$−14−14$

393.

$−1+cos2−1+cos2$

395.

$π15π15$

397.

$315315$

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 $10−46;10−46;$ the boundary curves of $RR$ are graphed in the following figure. 405.

$88$

409.

a. $R={(x,y)|y2+x2−2y−4x+1≤0};R={(x,y)|y2+x2−2y−4x+1≤0};$ b. $RR$ is graphed in the following figure; c. $3.163.16$

411.

a. $T0,2∘T3,0(u,v)=(u+3v,2u+7v);T0,2∘T3,0(u,v)=(u+3v,2u+7v);$ 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; d. $3232$

413.

$26623π≃282.45in326623π≃282.45in3$

415.

$A(R)≃83,999.2A(R)≃83,999.2$

### Chapter Review Exercises

417.

True.

419.

False.

421.

0

423.

$1414$

425.

1.475

427.

$523π523π$

429.

$π16π16$

431.

93.291

433.

$(815,815)(815,815)$

435.

$(0,0,85)(0,0,85)$

437.

$1.452π×10151.452π×1015$ ft-lb

439.

$y=−1.238×10−7x3+0.001196x2−3.666x+7208;y=−1.238×10−7x3+0.001196x2−3.666x+7208;$ average temperature approximately $2800°C2800°C$

441.

$π3π3$