 Calculus Volume 2

# Chapter 2

### Checkpoint

2.1

$1212$ units2

2.2

$310310$ unit2

2.3

$2+222+22$ units2

2.4

$5353$ units2

2.5

$5353$ units2

2.7

$π 2 π 2$

2.8

$8π8π$ units3

2.9

$21π21π$ units3

2.10

$10π310π3$ units3

2.11

$60π60π$ units3

2.12

$15π215π2$ units3

2.13

$8π8π$ units3

2.14

$12π12π$ units3

2.15

$11π611π6$ units3

2.16

$π6π6$ units3

2.17

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

2.18

$1 6 ( 5 5 − 1 ) ≈ 1.697 1 6 ( 5 5 − 1 ) ≈ 1.697$

2.19

$Arc Length ≈ 3.8202 Arc Length ≈ 3.8202$

2.20

$Arc Length = 3.15018 Arc Length = 3.15018$

2.21

$π 6 ( 5 5 − 3 3 ) ≈ 3.133 π 6 ( 5 5 − 3 3 ) ≈ 3.133$

2.22

$12 π 12 π$

2.23

$70 / 3 70 / 3$

2.24

$24 π 24 π$

2.25

$88$ ft-lb

2.26

Approximately $43,255.243,255.2$ ft-lb

2.27

$156,800156,800$ N

2.28

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

2.29

$M = 24 , x – = 2 5 m M = 24 , x – = 2 5 m$

2.30

$(−1,−1)(−1,−1)$ m

2.31

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

2.32

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

2.33

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

2.34

$6π26π2$ units3

2.35
1. $ddxln(2x2+x)=4x+12x2+xddxln(2x2+x)=4x+12x2+x$
2. $ddx(ln(x3))2=6ln(x3)xddx(ln(x3))2=6ln(x3)x$
2.36

$∫ x 2 x 3 + 6 d x = 1 3 ln | x 3 + 6 | + C ∫ x 2 x 3 + 6 d x = 1 3 ln | x 3 + 6 | + C$

2.37

$4 ln 2 4 ln 2$

2.38
1. $ddx(ex2e5x)=ex2−5x(2x−5)ddx(ex2e5x)=ex2−5x(2x−5)$
2. $ddt(e2t)3=6e6tddt(e2t)3=6e6t$
2.39

$∫ 4 e 3 x d x = − 4 3 e −3 x + C ∫ 4 e 3 x d x = − 4 3 e −3 x + C$

2.40
1. $ddt4t4=4t4(ln4)(4t3)ddt4t4=4t4(ln4)(4t3)$
2. $ddxlog3(x2+1)=x(ln3)(x2+1)ddxlog3(x2+1)=x(ln3)(x2+1)$
2.41

$∫ x 2 2 x 3 d x = 1 3 ln 2 2 x 3 + C ∫ x 2 2 x 3 d x = 1 3 ln 2 2 x 3 + C$

2.42

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.

2.43

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

2.44

$38.9038.90$ months

2.45

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.

2.46

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

2.47
1. $ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)$
2. $ddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshxddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshx$
2.48
1. $∫sinh3xcoshxdx=sinh4x4+C∫sinh3xcoshxdx=sinh4x4+C$
2. $∫sech2(3x)dx=tanh(3x)3+C∫sech2(3x)dx=tanh(3x)3+C$
2.49
1. $ddx(cosh−1(3x))=39x2−1ddx(cosh−1(3x))=39x2−1$
2. $ddx(coth−1x)3=3(coth−1x)21−x2ddx(coth−1x)3=3(coth−1x)21−x2$
2.50
1. $∫1x2−4dx=cosh−1(x2)+C∫1x2−4dx=cosh−1(x2)+C$
2. $∫11−e2xdx=−sech−1(ex)+C∫11−e2xdx=−sech−1(ex)+C$
2.51

$52.95 ft 52.95 ft$

### Section 2.1 Exercises

1 .

$32 3 32 3$

3 .

$13 12 13 12$

5 .

$36 36$

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 .

$1.067 1.067$

41 .

$0.852 0.852$

43 .

$7.523 7.523$

45 .

$3 π − 4 12 3 π − 4 12$

47 .

$1.429 1.429$

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 .

$343 24 343 24$

55 .

$4 3 4 3$

57 .

$π − 32 25 π − 32 25$

### Section 2.2 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 2.3 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 .

$7 π 6 7 π 6$

135 .

$48 π 48 π$

137 .

$114 π 5 114 π 5$

139 .

$512 π 7 512 π 7$

141 .

$96π596π5$ units3

143 .

$28π1528π15$ units3

145 .

$3π103π10$ units3

147 .

$π6·22/35-1110=π1012·22/3-11≈2.5286π6·22/35-1110=π1012·22/3-11≈2.5286$ 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 2.4 Exercises

165 .

$2 26 2 26$

167 .

$2 17 2 17$

169 .

$π 6 ( 17 17 − 5 5 ) π 6 ( 17 17 − 5 5 )$

171 .

$13 13 − 8 27 13 13 − 8 27$

173 .

$4 3 4 3$

175 .

$2.0035 2.0035$

177 .

$123 32 123 32$

179 .

$10 10$

181 .

$20 3 20 3$

183 .

$1 675 ( 229 229 − 8 ) 1 675 ( 229 229 − 8 )$

185 .

$1 8 ( 4 5 + ln ( 9 + 4 5 ) ) 1 8 ( 4 5 + ln ( 9 + 4 5 ) )$

187 .

$1.201 1.201$

189 .

$15.2341 15.2341$

191 .

$49 π 3 49 π 3$

193 .

$70 π 2 70 π 2$

195 .

$8 π 8 π$

197 .

$120 π 26 120 π 26$

199 .

$π 6 ( 17 17 − 1 ) π 6 ( 17 17 − 1 )$

201 .

$9 2 π 9 2 π$

203 .

$10 10 π 27 ( 73 73 − 1 ) 10 10 π 27 ( 73 73 − 1 )$

205 .

$25.645 25.645$

207 .

$π ( π + 2 ) π ( π + 2 )$

209 .

$10.5017 10.5017$

211 .

$2323$ ft

213 .

$2 2$

215 .

217 .

### Section 2.5 Exercises

219 .

$150150$ ft-lb

221 .

$200 J 200 J$

223 .

$11$ J

225 .

$39 2 39 2$

227 .

$ln ( 243 ) ln ( 243 )$

229 .

$332 π 15 332 π 15$

231 .

$100 π 100 π$

233 .

$20 π 15 20 π 15$

235 .

$66$ J

237 .

$55$ cm

239 .

$3636$ J

241 .

$18,75018,750$ ft-lb

243 .

Weight $=323×109lb=323×109lb$

Work $=43×1012ft-lb=43×1012ft-lb$

245 .

$9.71 × 10 2 N m 9.71 × 10 2 N m$

247 .

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

249 .

$23.25π23.25π$ million ft-lb

251 .

$A ρ H 2 2 A ρ H 2 2$

253 .

### Section 2.6 Exercises

255 .

$5 4 5 4$

257 .

$( 2 3 , 2 3 ) ( 2 3 , 2 3 )$

259 .

$( 7 4 , 3 2 ) ( 7 4 , 3 2 )$

261 .

$3 L 4 3 L 4$

263 .

$π 2 π 2$

265 .

$e 2 + 1 e 2 − 1 e 2 + 1 e 2 − 1$

267 .

$π 2 − 4 π π 2 − 4 π$

269 .

$1 4 ( 1 + e 2 ) 1 4 ( 1 + e 2 )$

271 .

$( a 3 , b 3 ) ( a 3 , b 3 )$

273 .

$( 0 , π 8 ) ( 0 , π 8 )$

275 .

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

277 .

$( 0 , 4 π ) ( 0 , 4 π )$

279 .

$( 5 8 , 1 3 ) ( 5 8 , 1 3 )$

281 .

$2 m π 3 2 m π 3$

283 .

$π a 2 b π a 2 b$

285 .

$( 4 3 π , 4 3 π ) ( 4 3 π , 4 3 π )$

287 .

$( 1 2 , 2 5 ) ( 1 2 , 2 5 )$

289 .

$( 0 , 28 9 π ) ( 0 , 28 9 π )$

291 .

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

293 .

Volume: $2π2a2(b+a)2π2a2(b+a)$

### Section 2.7 Exercises

295 .

$1 x 1 x$

297 .

$− 1 x ( ln x ) 2 − 1 x ( ln x ) 2$

299 .

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

301 .

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

303 .

$cot ( x ) cot ( x )$

305 .

$7 x 7 x$

307 .

$csc ( x ) sec x csc ( x ) sec x$

309 .

$−2 tan x −2 tan x$

311 .

$1 2 ln ( 5 3 ) 1 2 ln ( 5 3 )$

313 .

$2 − 1 2 ln ( 5 ) 2 − 1 2 ln ( 5 )$

315 .

$1 ln ( 2 ) − 1 1 ln ( 2 ) − 1$

317 .

$1 2 ln ( 2 ) 1 2 ln ( 2 )$

319 .

$1 3 ( ln x ) 3 1 3 ( ln x ) 3$

321 .

$2 x 3 x 2 + 1 x 2 − 1 2 x 3 x 2 + 1 x 2 − 1$

323 .

$x −2 − ( 1 / x ) ( ln x − 1 ) x −2 − ( 1 / x ) ( ln x − 1 )$

325 .

$e x e − 1 e x e − 1$

327 .

$1 1$

329 .

$− 1 x 2 − 1 x 2$

331 .

$π − ln ( 2 ) π − ln ( 2 )$

333 .

$1 x 1 x$

335 .

$e 5 − 6 units 2 e 5 − 6 units 2$

337 .

$ln ( 4 ) − 1 units 2 ln ( 4 ) − 1 units 2$

339 .

$2.8656 2.8656$

341 .

$3.1502 3.1502$

### Section 2.8 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 .

$12 12$

363 .

$8.618 % 8.618 %$

365 .

$6766.76 6766.76$

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 2.9 Exercises

377 .

$e x and e − x e x and e − x$

379 .

381 .

383 .

385 .

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

387 .

$− tanh ( x ) sech ( x ) − tanh ( x ) sech ( x )$

389 .

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

391 .

$x sech 2 ( x 2 + 1 ) x 2 + 1 x sech 2 ( x 2 + 1 ) x 2 + 1$

393 .

$6 sinh 5 ( x ) cosh ( x ) 6 sinh 5 ( x ) cosh ( x )$

395 .

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

397 .

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

399 .

$1 3 cosh 3 ( x ) + C 1 3 cosh 3 ( x ) + C$

401 .

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

403 .

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

405 .

$4 1 − 16 x 2 4 1 − 16 x 2$

407 .

$sinh ( x ) cosh 2 ( x ) + 1 sinh ( x ) cosh 2 ( x ) + 1$

409 .

$− csc ( x ) − csc ( x )$

411 .

$− 1 ( x 2 − 1 ) tanh −1 ( x ) − 1 ( x 2 − 1 ) tanh −1 ( x )$

413 .

$1 a tanh −1 ( x a ) + C 1 a tanh −1 ( x a ) + C$

415 .

$x 2 + 1 + C x 2 + 1 + C$

417 .

$cosh −1 ( e x ) + C cosh −1 ( e x ) + C$

419 .

421 .

$37.30 37.30$

423 .

$y = 1 c cosh ( c x ) y = 1 c cosh ( c x )$

425 .

$−0.521095 −0.521095$

427 .

$10 10$

### Review Exercises

435 .

False

437 .

False

439 .

$32 3 32 3$

441 .

$162 π 5 162 π 5$

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 .

$245.282 245.282$

451 .

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

453 .

$17 + 1 8 ln ( 33 + 8 17 ) 17 + 1 8 ln ( 33 + 8 17 )$

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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