Calculus Volume 3

# Chapter 6

### Checkpoint

6.1

$12i−j12i−j$

6.2 6.3

Rotational 6.4

$6565$ m/sec

6.5

No.

6.6 6.7

$1.49063×10−18,4.96876×10−19,9.93752×10−19N1.49063×10−18,4.96876×10−19,9.93752×10−19N$

6.8 6.9

No

6.10

$∇f=v∇f=v$

6.11

$Py=x≠Qx=−2xyPy=x≠Qx=−2xy$

6.12

No

6.13

$22$

6.14

$13+26+3π413+26+3π4$

6.15

Both line integrals equal $−1000303.−1000303.$

6.16

$417417$

6.17

$∫CF·Tds∫CF·Tds$

6.18

$−26−26$

6.19

0

6.20

$182π2182π2$ kg

6.21

3/2

6.22

$2π2π$

6.23

0

6.24

Yes

6.25

The region in the figure is connected. The region in the figure is not simply connected.

6.26

2

6.27

If $C1C1$ and $C2C2$ represent the two curves, then $∫C1F•dr≠∫C2F•dr.∫C1F•dr≠∫C2F•dr.$

6.28

$f(x,y)=exy3+xyf(x,y)=exy3+xy$

6.29

$f(x,y,z)=4x3+sinycosz+zf(x,y,z)=4x3+sinycosz+z$

6.30

$f(x,y,z)=Gx2+y2+z2f(x,y,z)=Gx2+y2+z2$

6.31

It is conservative.

6.32

$−10π−10π$

6.33

Negative

6.34

$452452$

6.35

$4343$

6.36

$3π23π2$

6.37

$g(x,y)=−xcosyg(x,y)=−xcosy$

6.38

No

6.39

$105π105π$

6.40

$y−z2y−z2$

6.41

Yes

6.42

All points on line $y=1.y=1.$

6.43

$−i−i$

6.44

$curlv=0curlv=0$

6.45

No

6.46

Yes

6.47

Cylinder $x2+y2=4x2+y2=4$

6.48

Cone $x2+y2=z2x2+y2=z2$

6.49

$r(u,v)=〈ucosv,usinv,u〉,r(u,v)=〈ucosv,usinv,u〉,$ $0

6.50

Yes

6.51

$≈43.02≈43.02$

6.52

With the standard parameterization of a cylinder, Equation 6.18 shows that the surface area is $2πrh.2πrh.$

6.53

$2π(2+sinh−1(1))2π(2+sinh−1(1))$

6.54

24

6.55

0

6.56

$38.401π≈120.64038.401π≈120.640$

6.57

$N(x,y)=〈−y1+x2+y2,−x1+x2+y2,11+x2+y2〉N(x,y)=〈−y1+x2+y2,−x1+x2+y2,11+x2+y2〉$

6.58

0

6.59

400 kg/sec/m

6.60

$−440π3−440π3$

6.61

Both integrals give $−13645.−13645.$

6.62

$−π−π$

6.63

$3232$

6.64

$curlE=〈x,y,−2z〉curlE=〈x,y,−2z〉$

6.65

Both integrals equal $6π.6π.$

6.66

30

6.67

$9ln(16)9ln(16)$

6.68

$≈6.777×109≈6.777×109$

### Section 6.1 Exercises

1.

Vectors

3.

False

5. 7. 9. 11. 13. 15.

$F(x,y)=sin(y)i+(xcosy−siny)jF(x,y)=sin(y)i+(xcosy−siny)j$

17.

$F(x,y,z)=(2xy+y)i+(x2+x+2yz)j+y2kF(x,y,z)=(2xy+y)i+(x2+x+2yz)j+y2k$

19.

$F(x,y)=(2x1+x2+2y2)i+(4y1+x2+2y2)jF(x,y)=(2x1+x2+2y2)i+(4y1+x2+2y2)j$

21.

$F(x,y)=(1−x)i−yj(1−x)2+y2F(x,y)=(1−x)i−yj(1−x)2+y2$

23.

$F(x,y)=(yi−xj)x2+y2F(x,y)=(yi−xj)x2+y2$

25.

$F(x,y)=yi−xjF(x,y)=yi−xj$

27.

$F(x,y)=−10(x2+y2)3/2(xi+yj)F(x,y)=−10(x2+y2)3/2(xi+yj)$

29.

$E=c|r|2r=c|r|r|r|E=c|r|2r=c|r|r|r|$

31.

$c′(t)=(cost,−sint,e−t)=F(c(t))c′(t)=(cost,−sint,e−t)=F(c(t))$

33.

H

35.

d. $−F+G−F+G$

37.

a. $F+GF+G$

### Section 6.2 Exercises

39.

True

41.

False

43.

False

45.

$∫C(x−y)ds=10∫C(x−y)ds=10$

47.

$∫Cxy4ds=81925∫Cxy4ds=81925$

49.

$W=8W=8$

51.

$W=3π4W=3π4$

53.

$W=πW=π$

55.

$∫CF·dr=4∫CF·dr=4$

57.

$∫Cyzdx+xzdy+xydz=−1∫Cyzdx+xzdy+xydz=−1$

59.

$∫C(y2)dx+(x)dy=2456∫C(y2)dx+(x)dy=2456$

61.

$∫Cxydx+ydy=1903∫Cxydx+ydy=1903$

63.

$∫Cy2x2−y2ds=2ln5∫Cy2x2−y2ds=2ln5$

65.

$W=−66W=−66$

67.

$W=−10π2W=−10π2$

69.

$W=2W=2$

71.

a. $W=11;W=11;$ b. $W=11;W=11;$ c. Yes

73.

$W=2πW=2π$

75.

$∫CF·dr=255+1120∫CF·dr=255+1120$

77.

$∫Cy2dx+(xy−x2)dy=6.15∫Cy2dx+(xy−x2)dy=6.15$

79.

$∫γxeyds≈7.157∫γxeyds≈7.157$

81.

$∫γ(y2−xy)dx≈−1.379∫γ(y2−xy)dx≈−1.379$

83.

$∫CF·dr≈−1.133∫CF·dr≈−1.133$

85.

$∫CF·dr≈22.857∫CF·dr≈22.857$

87.

$flux=−13flux=−13$

89.

$flux=−20flux=−20$

91.

$flux=0flux=0$

93.

$m=4πρ5m=4πρ5$

95.

$W=0W=0$

97.

$W=k2W=k2$

### Section 6.3 Exercises

99.

True

101.

True

103.

$∫CF·dr=24∫CF·dr=24$

105.

$∫CF·dr=e−3π2∫CF·dr=e−3π2$

107.

Not conservative

109.

Conservative, $f(x,y)=3x2+5xy+2y2f(x,y)=3x2+5xy+2y2$

111.

Conservative, $f(x,y)=yex+xsin(y)f(x,y)=yex+xsin(y)$

113.

$∮C(2ydx+2xdy)=32∮C(2ydx+2xdy)=32$

115.

$F(x,y)=(10x+3y)i+(3x+10y)jF(x,y)=(10x+3y)i+(3x+10y)j$

117.

F is not conservative.

119.

F is conservative and a potential function is $f(x,y,z)=xyez.f(x,y,z)=xyez.$

121.

F is conservative and a potential function is $f(x,y,z)=z2–z–xy.f(x,y,z)=z2–z–xy.$

123.

F is conservative and a potential function is $f(x,y,z)=x2y+y2z.f(x,y,z)=x2y+y2z.$

125.

F is conservative and a potential function is $f(x,y)=ex2yf(x,y)=ex2y$

127.

$∫CF·dr=e2+1∫CF·dr=e2+1$

129.

$∫CF·dr=41∫CF·dr=41$

131.

$∮C1G·dr=−8π∮C1G·dr=−8π$

133.

$∮C2F·dr=7∮C2F·dr=7$

135.

$∫CF·dr=150∫CF·dr=150$

137.

$∫CF·dr=−1∫CF·dr=−1$

139.

$4×1031erg4×1031erg$

141.

$∫CF·ds=0.4687∫CF·ds=0.4687$

143.

$circulation=πa2and flux=0circulation=πa2and flux=0$

### Section 6.4 Exercises

147.

$∫C2xydx+(x+y)dy=323∫C2xydx+(x+y)dy=323$

149.

$∫Csinxcosydx+(xy+cosxsiny)dy=112∫Csinxcosydx+(xy+cosxsiny)dy=112$

151.

$∮C(−ydx+xdy)=π∮C(−ydx+xdy)=π$

153.

$∫Cxe−2xdx+(x4+2x2y2)dy=0∫Cxe−2xdx+(x4+2x2y2)dy=0$

155.

$∮Cy3dx−x3dy=−24π∮Cy3dx−x3dy=−24π$

157.

$∮C−x2ydx+xy2dy=8π∮C−x2ydx+xy2dy=8π$

159.

$∮C(x2+y2)dx+2xydy=0∮C(x2+y2)dx+2xydy=0$

161.

$A=19πA=19π$

163.

$A=38πA=38π$

165.

$∫C+(y2+x3)dx+x4dy=0∫C+(y2+x3)dx+x4dy=0$

167.

$A=9π8A=9π8$

169.

$A=835A=835$

171.

$∫C(x2y−2xy+y2)ds=3∫C(x2y−2xy+y2)ds=3$

173.

$∫Cxdx+ydyx2+y2=2π∫Cxdx+ydyx2+y2=2π$

175.

$W=2252W=2252$

177.

$W=12πW=12π$

179.

$W=2πW=2π$

181.

$∮Cy2dx+x2dy=13∮Cy2dx+x2dy=13$

183.

$∫C1+x3dx+2xydy=3∫C1+x3dx+2xydy=3$

185.

$∫C(3y−esinx)dx+(7x+y4+1)dy=36π∫C(3y−esinx)dx+(7x+y4+1)dy=36π$

187.

$∮CF·dr=2∮CF·dr=2$

189.

$∮C(y+x)dx+(x+siny)dy=0∮C(y+x)dx+(x+siny)dy=0$

191.

$∮Cxydx+x3y3dy=2221∮Cxydx+x3y3dy=2221$

193.

$∮CF·dr=15π4∮CF·dr=15π4$

195.

$∫Csin(x+y)dx+cos(x+y)dy=4∫Csin(x+y)dx+cos(x+y)dy=4$

197.

$∫CF·dr=π∫CF·dr=π$

199.

$∮CF·n^ds=4∮CF·n^ds=4$

201.

$∮CF·nds=32π∮CF·nds=32π$

203.

$∫C[−y3+sin(xy)+xycos(xy)]dx+[x3+x2cos(xy)]dy=4.7124∫C[−y3+sin(xy)+xycos(xy)]dx+[x3+x2cos(xy)]dy=4.7124$

205.

$∮C(y+ex)dx+(2x+cos(y2))dy=13∮C(y+ex)dx+(2x+cos(y2))dy=13$

### Section 6.5 Exercises

207.

False

209.

True

211.

True

213.

$curlF=i+x2j+y2kcurlF=i+x2j+y2k$

215.

$curlF=(xz2−xy2)i+(x2y−yz2)j+(y2z−x2z)kcurlF=(xz2−xy2)i+(x2y−yz2)j+(y2z−x2z)k$

217.

$curlF=i+j+kcurlF=i+j+k$

219.

$curlF=−yi−zj−xkcurlF=−yi−zj−xk$

221.

$curlF=0curlF=0$

223.

$divF=3yz2+2ysinz+2xe2zdivF=3yz2+2ysinz+2xe2z$

225.

$divF=2(x+y+z)divF=2(x+y+z)$

227.

$divF=1x2+y2divF=1x2+y2$

229.

$divF=a+bdivF=a+b$

231.

$divF=x+y+zdivF=x+y+z$

233.

Harmonic

235.

$div(F×G)=2z+3xdiv(F×G)=2z+3x$

237.

$divF=2r2divF=2r2$

239.

$curlr=0curlr=0$

241.

$curlrr3=0curlrr3=0$

243.

$curlF=2xx2+y2kcurlF=2xx2+y2k$

245.

$divF=0divF=0$

247.

$divF=2−2e−6divF=2−2e−6$

249.

$divF=0divF=0$

251.

$curlF=j−3kcurlF=j−3k$

253.

$curlF=2j−kcurlF=2j−k$

255.

$a=3a=3$

257.

F is conservative.

259.

$divF=coshx+sinhy−xydivF=coshx+sinhy−xy$

261.

$(bz−cy)i(cx−az)j+(ay−bx)k(bz−cy)i(cx−az)j+(ay−bx)k$

263.

$curlF=2ωcurlF=2ω$

265.

$F×GF×G$ does not have zero divergence.

267.

$∇·F=−200k[1+2(x2+y2+z2)]e−x2+y2+z2∇·F=−200k[1+2(x2+y2+z2)]e−x2+y2+z2$

### Section 6.6 Exercises

269.

True

271.

True

273.

$r(u,v)=〈u,v,2−3u+2v〉r(u,v)=〈u,v,2−3u+2v〉$ for $−∞≤u<∞−∞≤u<∞$ and $−∞≤v<∞.−∞≤v<∞.$

275.

$r(u,v)=〈u,v,13(16−2u+4v)〉r(u,v)=〈u,v,13(16−2u+4v)〉$ for $|u|<∞|u|<∞$ and $|v|<∞.|v|<∞.$

277.

$r(u,v)=〈3cosu,3sinu,v〉r(u,v)=〈3cosu,3sinu,v〉$ for $0≤u≤π2,0≤v≤30≤u≤π2,0≤v≤3$

279.

$A=87.9646A=87.9646$

281.

$∬SzdS=8π∬SzdS=8π$

283.

$∬S(x2+y2)zdS=16π∬S(x2+y2)zdS=16π$

285.

$∬SF·NdS=4π3∬SF·NdS=4π3$

287.

$m≈13.0639m≈13.0639$

289.

$m≈228.5313m≈228.5313$

291.

$∬SgdS=34∬SgdS=34$

293.

$∬S(x2+y−z)dS≈0.9617∬S(x2+y−z)dS≈0.9617$

295.

$∬S(x2+y2)dS=4π3∬S(x2+y2)dS=4π3$

297.

$∬Sx2zdS=10232π5∬Sx2zdS=10232π5$

299.

$∬S(z+y)dS≈10.1∬S(z+y)dS≈10.1$

301.

$m=πa3m=πa3$

303.

$∬SF·NdS=1324∬SF·NdS=1324$

305.

$∬SF·NdS=34∬SF·NdS=34$

307.

$∫08∫06(4−3y+116y2+z)(1417)dzdy∫08∫06(4−3y+116y2+z)(1417)dzdy$

309.

$∫02∫06[x2−2(8−4x)+z]17dzdx∫02∫06[x2−2(8−4x)+z]17dzdx$

311.

$∬S(x2z+y2z)dS=πa52∬S(x2z+y2z)dS=πa52$

313.

$∬Sx2yzdS=17114∬Sx2yzdS=17114$

315.

$∬SyzdS=2π4∬SyzdS=2π4$

317.

$∬S(xi+yj)·dS=16π∬S(xi+yj)·dS=16π$

319.

$m=πa7192m=πa7192$

321.

$F≈4.57lb.F≈4.57lb.$

323.

$8πa8πa$

325.

The net flux is zero.

### Section 6.7 Exercises

327.

$∬S(curlF·N)dS=πa2∬S(curlF·N)dS=πa2$

329.

$∬S(curlF·N)dS=18π∬S(curlF·N)dS=18π$

331.

$∬S(curlF·N)dS=−8π∬S(curlF·N)dS=−8π$

333.

$∬S(curlF·N)dS=0∬S(curlF·N)dS=0$

335.

$∫CF·dS=0∫CF·dS=0$

337.

$∫CF·dS=−9.4248∫CF·dS=−9.4248$

339.

$∬ScurlF·dS=0∬ScurlF·dS=0$

341.

$∬ScurlF·dS=2.6667∬ScurlF·dS=2.6667$

343.

$∬S(curlF·N)dS=−16∬S(curlF·N)dS=−16$

345.

$∫C(12y2dx+zdy+xdz)=−π4∫C(12y2dx+zdy+xdz)=−π4$

347.

$∬S(curlF·N)dS=−3π∬S(curlF·N)dS=−3π$

349.

$∫C(ck×R)·dS=2πc∫C(ck×R)·dS=2πc$

351.

$∬ScurlF·dS=0∬ScurlF·dS=0$

353.

$∮F·dS=−4∮F·dS=−4$

355.

$∬ScurlF·dS=0∬ScurlF·dS=0$

357.

$∬ScurlF·dS=−36π∬ScurlF·dS=−36π$

359.

$∬ScurlF·N=0∬ScurlF·N=0$

361.

$∮CF·dr=0∮CF·dr=0$

363.

$∬Scurl(F)·dS=84.8230∬Scurl(F)·dS=84.8230$

365.

$A=∬S(∇×F)·ndS=0A=∬S(∇×F)·ndS=0$

367.

$∬S(∇×F)·ndS=2π∬S(∇×F)·ndS=2π$

369.

$C=π(cosφ−sinφ)C=π(cosφ−sinφ)$

371.

$∮CF·dr=48π∮CF·dr=48π$

373.

$∬S(∇×F)·n=0∬S(∇×F)·n=0$

375.

0

### Section 6.8 Exercises

377.

$∫SF·nds=75.3982∫SF·nds=75.3982$

379.

$∫SF·nds=127.2345∫SF·nds=127.2345$

381.

$∫SF·nds=37.6991∫SF·nds=37.6991$

383.

$∫SF·nds=9πa42∫SF·nds=9πa42$

385.

$∬SF·dS=π3∬SF·dS=π3$

387.

$∬SF·dS=0∬SF·dS=0$

389.

$∬SF·dS=241.2743∬SF·dS=241.2743$

391.

$∬DF·dS=−π∬DF·dS=−π$

393.

$∬SF·dS=2π3∬SF·dS=2π3$

395.

$166π166π$

397.

$−1283π−1283π$

399.

$−703.7168−703.7168$

401.

20

403.

$∬SF·dS=8∬SF·dS=8$

405.

$∬SF·NdS=18∬SF·NdS=18$

407.

$∬S‖R‖R·nds=4πa4∬S‖R‖R·nds=4πa4$

409.

$∭Rz2dV=4π15∭Rz2dV=4π15$

411.

$∬SF·dS=6.5759∬SF·dS=6.5759$

413.

$∬SF·dS=21∬SF·dS=21$

415.

$∬SF·dS=72∬SF·dS=72$

417.

$∬SF·dS=−33.5103∬SF·dS=−33.5103$

419.

$∬SF·dS=πa4b2∬SF·dS=πa4b2$

421.

$∬SF·dS=52π∬SF·dS=52π$

423.

$∬SF·dS=21π2∬SF·dS=21π2$

425.

$−(1−e−1)−(1−e−1)$

### Chapter Review Exercises

427.

False

429.

False

431. 433.

Conservative, $f(x,y)=xy−2eyf(x,y)=xy−2ey$

435.

Conservative, $f(x,y,z)=x2y+y2z+z2xf(x,y,z)=x2y+y2z+z2x$

437.

$−163−163$

439.

$3229(33−1)3229(33−1)$

441.

Divergence: $ex+xexy+xyexyz,ex+xexy+xyexyz,$ curl: $xzexyzi−yzexyzj+yexykxzexyzi−yzexyzj+yexyk$

443.

$−2π−2π$

445.

$−π−π$

447.

$31π/231π/2$

449.

$2(2+π)2(2+π)$

451.

$2π/32π/3$