 Calculus Volume 1

# Chapter 6

### Checkpoint

6.1

$1212$ units2

6.2

$310310$ unit2

6.3

$2+222+22$ units2

6.4

$5353$ units2

6.5

$5353$ units2

6.7

$π2π2$

6.8

$8π8π$ units3

6.9

$21π21π$ units3

6.10

$10π310π3$ units3

6.11

$60π60π$ units3

6.12

$15π215π2$ units3

6.13

$8π8π$ units3

6.14

$12π12π$ units3

6.15

$11π611π6$ units3

6.16

$π6π6$ units3

6.17

Use the method of washers; $V=∫−11π[(2−x2)2−(x2)2]dxV=∫−11π[(2−x2)2−(x2)2]dx$

6.18

$16(55−1)≈1.69716(55−1)≈1.697$

6.19

$Arc Length≈3.8202Arc Length≈3.8202$

6.20

$Arc Length=3.15018Arc Length=3.15018$

6.21

$π6(55−33)≈3.133π6(55−33)≈3.133$

6.22

$12π12π$

6.23

$70/370/3$

6.24

$24π24π$

6.25

$88$ ft-lb

6.26

Approximately $43,255.243,255.2$ ft-lb

6.27

$156,800156,800$ N

6.28

Approximately 7,164,520,000 lb or 3,582,260 t

6.29

$M=24,x–=25mM=24,x–=25m$

6.30

$(−1,−1)(−1,−1)$ m

6.31

The centroid of the region is $(3/2,6/5).(3/2,6/5).$

6.32

The centroid of the region is $(1,13/5).(1,13/5).$

6.33

The centroid of the region is $(0,2/5).(0,2/5).$

6.34

$6π26π2$ units3

6.35
1. $ddxln(2x2+x)=4x+12x2+xddxln(2x2+x)=4x+12x2+x$
2. $ddx(ln(x3))2=6ln(x3)xddx(ln(x3))2=6ln(x3)x$
6.36

$∫x2x3+6dx=13ln|x3+6|+C∫x2x3+6dx=13ln|x3+6|+C$

6.37

$4ln24ln2$

6.38
1. $ddx(ex2e5x)=ex2−5x(2x−5)ddx(ex2e5x)=ex2−5x(2x−5)$
2. $ddt(e2t)3=6e6tddt(e2t)3=6e6t$
6.39

$∫4e3xdx=−43e−3x+C∫4e3xdx=−43e−3x+C$

6.40
1. $ddt4t4=4t4(ln4)(4t3)ddt4t4=4t4(ln4)(4t3)$
2. $ddxlog3(x2+1)=x(ln3)(x2+1)ddxlog3(x2+1)=x(ln3)(x2+1)$
6.41

$∫x22x3dx=13ln22x3+C∫x22x3dx=13ln22x3+C$

6.42

There are $81,377,39681,377,396$ bacteria in the population after $44$ hours. The population reaches $100100$ million bacteria after $244.12244.12$ minutes.

6.43

At $5%5%$ interest, she must invest $223,130.16.223,130.16.$ At $6%6%$ interest, she must invest $165,298.89.165,298.89.$

6.44

$38.9038.90$ months

6.45

The coffee is first cool enough to serve about $3.53.5$ minutes after it is poured. The coffee is too cold to serve about $77$ minutes after it is poured.

6.46

A total of $94.1394.13$ g of carbon remains. The artifact is approximately $13,30013,300$ years old.

6.47
1. $ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)$
2. $ddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshxddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshx$
6.48
1. $∫sinh3xcoshxdx=sinh4x4+C∫sinh3xcoshxdx=sinh4x4+C$
2. $∫sech2(3x)dx=tanh(3x)3+C∫sech2(3x)dx=tanh(3x)3+C$
6.49
1. $ddx(cosh−1(3x))=39x2−1ddx(cosh−1(3x))=39x2−1$
2. $ddx(coth−1x)3=3(coth−1x)21−x2ddx(coth−1x)3=3(coth−1x)21−x2$
6.50
1. $∫1x2−4dx=cosh−1(x2)+C∫1x2−4dx=cosh−1(x2)+C$
2. $∫11−e2xdx=−sech−1(ex)+C∫11−e2xdx=−sech−1(ex)+C$
6.51

$52.95ft52.95ft$

### Section 6.1 Exercises

1.

$323323$

3.

$13121312$

5.

$3636$

7. 243 square units

9. 4

11. $2(e−1)2e2(e−1)2e$

13. $1313$

15. $343343$

17. $5252$

19. $1212$

21. $9292$

23. $9292$

25. $332332$

27. $e−2e−2$

29. $274274$

31. $43−ln(3)43−ln(3)$

33. $1212$

35. $1212$

37. $−2(2−π)−2(2−π)$

39.

$1.0671.067$

41.

$0.8520.852$

43.

$7.5237.523$

45.

$3π−4123π−412$

47.

$1.4291.429$

49.

$33,333.3333,333.33$ total profit for $200200$ cell phones sold

51.

$3.2633.263$ mi represents how far ahead the hare is from the tortoise

53.

$3432434324$

55.

$4343$

57.

$π−3225π−3225$

### Section 6.2 Exercises

63.

8 units3

65.

$32323232$ units3

67.

$724πr2h724πr2h$ units3

69. $π24π24$ units3

71. $22$ units3

73. $π240π240$ units3

75. $4096π54096π5$ units3

77. $8π98π9$ units3

79. $π2π2$ units3

81. $207π207π$ units3

83. $4π54π5$ units3

85. $16π316π3$ units3

87. $ππ$ units3

89. $16π316π3$ units3

91. $72π572π5$ units3

93. $108π5108π5$ units3

95. $3π103π10$ units3

97. $26π26π$ units3

99. $9π9π$ units3

101. $π20(75−4ln5(2))π20(75−4ln5(2))$ units3

103.

$m2π3(b3−a3)m2π3(b3−a3)$ units3

105.

$4a2bπ34a2bπ3$ units3

107.

$2π22π2$ units3

109.

$2ab2π32ab2π3$ units3

111.

$π12(r+h)2(6r−h)π12(r+h)2(6r−h)$ units3

113.

$π3(h+R)(h−2R)2π3(h+R)(h−2R)2$ units3

### Section 6.3 Exercises

115. $54π54π$ units3

117. $81π81π$ units3

119. $512π7512π7$ units3

121.

$2π2π$ units3

123.

$2π32π3$ units3

125.

$2π2π$ units3

127.

$4π54π5$ units3

129.

$64π364π3$ units3

131.

$32π532π5$ units3

133.

$7π67π6$

135.

$48π48π$

137.

$97π597π5$

139.

$512π7512π7$

141.

$64π564π5$ units3

143.

$28π1528π15$ units3

145.

$3π103π10$ units3

147.

$52π552π5$ units3

149.

$0.98760.9876$ units3

151. $3232$ units3

153. $496π15496π15$ units3

155. $398π15398π15$ units3

157. $15.907415.9074$ units3

159.

$13πr2h13πr2h$ units3

161.

$πr2hπr2h$ units3

163.

$πa2πa2$ units3

### Section 6.4 Exercises

165.

$226226$

167.

$217217$

169.

$π6(1717−55)π6(1717−55)$

171.

$1313−8271313−827$

173.

$4343$

175.

$2.00352.0035$

177.

$1233212332$

179.

$1010$

181.

$203203$

183.

$1675(229229−8)1675(229229−8)$

185.

$18(45+ln(9+45))18(45+ln(9+45))$

187.

$1.2011.201$

189.

$15.234115.2341$

191.

$49π349π3$

193.

$70π270π2$

195.

$8π8π$

197.

$120π26120π26$

199.

$π6(1717−1)π6(1717−1)$

201.

$92π92π$

203.

$1010π27(7373−1)1010π27(7373−1)$

205.

$25.64525.645$

207.

$π(π+2)π(π+2)$

209.

$10.501710.5017$

211.

$2323$ ft

213.

$22$

215.

217.

### Section 6.5 Exercises

219.

$150150$ ft-lb

221.

$200J200J$

223.

$11$ J

225.

$392392$

227.

$ln(243)ln(243)$

229.

$332π15332π15$

231.

$100π100π$

233.

$20π1520π15$

235.

$66$ J

237.

$55$ cm

239.

$3636$ J

241.

$18,75018,750$ ft-lb

243.

$323×109ft-lb323×109ft-lb$

245.

$9.71×102N m9.71×102N m$

247.

a. $3,000,0003,000,000$ lb, b. $749,000749,000$ lb

249.

$23.25π23.25π$ million ft-lb

251.

$AρH22AρH22$

253.

### Section 6.6 Exercises

255.

$5454$

257.

$(23,23)(23,23)$

259.

$(74,32)(74,32)$

261.

$3L43L4$

263.

$π2π2$

265.

$e2+1e2−1e2+1e2−1$

267.

$π2−4ππ2−4π$

269.

$14(1+e2)14(1+e2)$

271.

$(a3,b3)(a3,b3)$

273.

$(0,π8)(0,π8)$

275.

$(0,3)(0,3)$

277.

$(0,4π)(0,4π)$

279.

$(58,13)(58,13)$

281.

$2mπ32mπ3$

283.

$πa2bπa2b$

285.

$(43π,43π)(43π,43π)$

287.

$(12,25)(12,25)$

289.

$(0,289π)(0,289π)$

291.

Center of mass: $(a6,4a25),(a6,4a25),$ volume: $2πa492πa49$

293.

Volume: $2π2a2(b+a)2π2a2(b+a)$

### Section 6.7 Exercises

295.

$1x1x$

297.

$−1x(lnx)2−1x(lnx)2$

299.

$ln(x+1)+Cln(x+1)+C$

301.

$ln(x)+1ln(x)+1$

303.

$cot(x)cot(x)$

305.

$7x7x$

307.

$csc(x)secxcsc(x)secx$

309.

$−2tanx−2tanx$

311.

$12ln(53)12ln(53)$

313.

$2−12ln(5)2−12ln(5)$

315.

$1ln(2)−11ln(2)−1$

317.

$12ln(2)12ln(2)$

319.

$13(lnx)313(lnx)3$

321.

$2x3x2+1x2−12x3x2+1x2−1$

323.

$x−2−(1/x)(lnx−1)x−2−(1/x)(lnx−1)$

325.

$exe−1exe−1$

327.

$11$

329.

$−1x2−1x2$

331.

$π−ln(2)π−ln(2)$

333.

$1x1x$

335.

$e5−6units2e5−6units2$

337.

$ln(4)−1units2ln(4)−1units2$

339.

$2.86562.8656$

341.

$3.15023.1502$

### Section 6.8 Exercises

349.

True

351.

False; $k=ln(2)tk=ln(2)t$

353.

$2020$ hours

355.

No. The relic is approximately $871871$ years old.

357.

$71.9271.92$ years

359.

$55$ days $66$ hours $2727$ minutes

361.

$1212$

363.

$8.618%8.618%$

365.

$6766.766766.76$

367.

$99$ hours $1313$ minutes

369.

$239,179239,179$ years

371.

$P′(t)=43e0.01604t.P′(t)=43e0.01604t.$ The population is always increasing.

373.

The population reaches $1010$ billion people in $2027.2027.$

375.

$P′(t)=2.259e0.06407t.P′(t)=2.259e0.06407t.$ The population is always increasing.

### Section 6.9 Exercises

377.

$exande−xexande−x$

379.

381.

383.

385.

$3sinh(3x+1)3sinh(3x+1)$

387.

$−tanh(x)sech(x)−tanh(x)sech(x)$

389.

$4cosh(x)sinh(x)4cosh(x)sinh(x)$

391.

$xsech2(x2+1)x2+1xsech2(x2+1)x2+1$

393.

$6sinh5(x)cosh(x)6sinh5(x)cosh(x)$

395.

$12sinh(2x+1)+C12sinh(2x+1)+C$

397.

$12sinh2(x2)+C12sinh2(x2)+C$

399.

$13cosh3(x)+C13cosh3(x)+C$

401.

$ln(1+cosh(x))+Cln(1+cosh(x))+C$

403.

$cosh(x)+sinh(x)+Ccosh(x)+sinh(x)+C$

405.

$41−16x241−16x2$

407.

$sinh(x)cosh2(x)+1sinh(x)cosh2(x)+1$

409.

$−csc(x)−csc(x)$

411.

$−1(x2−1)tanh−1(x)−1(x2−1)tanh−1(x)$

413.

$1atanh−1(xa)+C1atanh−1(xa)+C$

415.

$x2+1+Cx2+1+C$

417.

$cosh−1(ex)+Ccosh−1(ex)+C$

419.

421.

$37.3037.30$

423.

$y=1ccosh(cx)y=1ccosh(cx)$

425.

$−0.521095−0.521095$

427.

$1010$

### Review Exercises

435.

False

437.

False

439.

$323323$

441.

$162π5162π5$

443.

a. $4,4,$ b. $128π7,128π7,$ c. $64π564π5$

445.

a. $1.949,1.949,$ b. $21.952,21.952,$ c. $17.09917.099$

447.

a. $316,316,$ b. $452π15,452π15,$ c. $31π631π6$

449.

$245.282245.282$

451.

Mass: $12,12,$ center of mass: $(1835,922)(1835,922)$

453.

$17+18ln(33+817)17+18ln(33+817)$

455.

Volume: $3π4,3π4,$ surface area: $π(2−sinh−1(1)+sinh−1(16)−25716)π(2−sinh−1(1)+sinh−1(16)−25716)$

457.

11:02 a.m.

459.

$π(1+sinh(1)cosh(1))π(1+sinh(1)cosh(1))$

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