Chapter 2

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Chapter 2

Calculus Volume 2Chapter 2

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Chapter 2

Checkpoint

2.1

$12$ units²

2.2

$\frac{3}{10}$ unit²

2.3

$2 + 2\sqrt{2}$ units²

2.4

$\frac{5}{3}$ units²

2.5

$\frac{5}{3}$ units²

2.7

$\frac{\pi}{2}$

2.8

$8\pi$ units³

2.9

$21\pi$ units³

2.10

$\frac{10\pi}{3}$ units³

2.11

$60\pi$ units³

2.12

$\frac{15\pi}{2}$ units³

2.13

$8\pi$ units³

2.14

$12\pi$ units³

2.15

$\frac{11\pi}{6}$ units³

2.16

$\frac{\pi}{6}$ units³

2.17

Use the method of washers; $V = {\int_{-1}^{1}\pi}\left\lbrack {\left( {2 - x^{2}} \right)^{2} - \left( x^{2} \right)^{2}} \right\rbrack dx$

2.18

$\frac{1}{6}\left( {5\sqrt{5} - 1} \right) \approx 1.697$

2.19

$\text{Arc Length} \approx 3.8202$

2.20

$\text{Arc Length} = 3.15018$

2.21

$\frac{\pi}{6}\left( {5\sqrt{5} - 3\sqrt{3}} \right) \approx 3.133$

2.22

$12\pi$

2.23

$70\text{/}3$

2.24

$24\pi$

2.25

$8$ ft-lb

2.26

Approximately $43,255.2$ ft-lb

2.27

$156,800$ N

2.28

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

2.29

$M = 24,\overset{–}{x} = \frac{2}{5}\ \text{m}$

2.30

$\left( {-1,-1} \right)$ m

2.31

The centroid of the region is $\left( {{3\text{/}2},{6\text{/}5}} \right).$

2.32

The centroid of the region is $\left( {1,{13\text{/}5}} \right).$

2.33

The centroid of the region is $\left( {0,{2\text{/}5}} \right).$

2.34

$6\pi^{2}$ units³

2.35

1. $\frac{d}{dx}\text{ln}\left( {2x^{2} + x} \right) = \frac{4x + 1}{2x^{2} + x}$

2. $\frac{d}{dx}\left( {\text{ln}\left( x^{3} \right)} \right)^{2} = \frac{6\ \text{ln}\left( x^{3} \right)}{x}$

2.36

${\int\frac{x^{2}}{x^{3} + 6}}dx = \frac{1}{3}\text{ln}\ \left| {x^{3} + 6} \right| + C$

2.37

$4\ \text{ln}\ 2$

2.38

1. $\frac{d}{dx}\left( \frac{e^{x^{2}}}{e^{5x}} \right) = e^{x^{2} - 5x}\left( {2x - 5} \right)$

2. $\frac{d}{dt}\left( e^{2t} \right)^{3} = 6e^{6t}$

2.39

${\int{\frac{4}{e^{3x}}dx}} = - \frac{4}{3}e^{-3x} + C$

2.40

1. $\frac{d}{dt}4^{t^{4}} = 4^{t^{4}}\left( {\text{ln}\ 4} \right)\left( {4t^{3}} \right)$

2. $\frac{d}{dx}\text{log}_{3}\left( \sqrt{x^{2} + 1} \right) = \frac{x}{\left( {\text{ln}\ 3} \right)\left( {x^{2} + 1} \right)}$

2.41

${\int{x^{2}2^{x^{3}}dx}} = \frac{1}{3\ \text{ln}\ 2}2^{x^{3}} + C$

2.42

There are $81,377,396$ bacteria in the population after $4$ hours. The population reaches $100$ million bacteria after $244.12$ minutes.

2.43

At $5\text{\%}$ interest, she must invest $\text{\$}223,130.16.$ At $6\text{\%}$ interest, she must invest $\text{\$}165,298.89.$

2.44

$38.90$ months

2.45

The coffee is first cool enough to serve about $3.5$ minutes after it is poured. The coffee is too cold to serve about $7$ minutes after it is poured.

2.46

A total of $94.13$ g of carbon remains. The artifact is approximately $13,300$ years old.

2.47

1. $\frac{d}{dx}\left( {\text{tanh}\left( {x^{2} + 3x} \right)} \right) = \left( {\text{sech}^{2}\left( {x^{2} + 3x} \right)} \right)\left( {2x + 3} \right)$

2. $\frac{d}{dx}\left( \frac{1}{\left( {\text{sinh}\ x} \right)^{2}} \right) = \frac{d}{dx}\left( {\text{sinh}\ x} \right)^{-2} = -2\left( {\text{sinh}\ x} \right)^{-3}\text{cosh}\ x$

2.48

1. ${\int{\text{sinh}^{3}x\ \text{cosh}\ x\ d}}x = \frac{\text{sinh}^{4}x}{4} + C$

2. ${\int{\text{sech}^{2}\left( {3x} \right)d}}x = \frac{\text{tanh}\left( {3x} \right)}{3} + C$

2.49

1. $\frac{d}{dx}\left( {\text{cosh}^{-1}\left( {3x} \right)} \right) = \frac{3}{\sqrt{9x^{2} - 1}}$

2. $\frac{d}{dx}\left( {\text{coth}^{-1}x} \right)^{3} = \frac{3\left( {\text{coth}^{-1}x} \right)^{2}}{1 - x^{2}}$

2.50

1. ${\int{\frac{1}{\sqrt{x^{2} - 4}}d}}x = \text{cosh}^{-1}\left( \frac{x}{2} \right) + C$

2. ${\int{\frac{1}{\sqrt{1 - e^{2x}}}d}}x = \text{−}\text{sech}^{-1}\left( e^{x} \right) + C$

2.51

$52.95\ \text{ft}$

Section 2.1 Exercises

1.

$\frac{32}{3}$

3.

$\frac{13}{12}$

5.

$36$

7.

243 square units

9.

4

11.

$\frac{2\left( {e - 1} \right)^{2}}{e}$

13.

$\frac{1}{3}$

15.

$\frac{34}{3}$

17.

$\frac{5}{2}$

19.

$\frac{1}{2}$

21.

$\frac{9}{2}$

23.

$\frac{9}{2}$

25.

$\frac{3\sqrt{3}}{2}$

27.

$e-2$

29.

$\frac{27}{4}$

31.

$\frac{4}{3} - \text{ln}(3)$

33.

$\frac{1}{2}$

35.

$\frac{1}{2}$

37.

$-2\left( {\sqrt{2} - \pi} \right)$

39.

$1.067$

41.

$1.605$

43.

$7.523$

45.

$\frac{3\pi - 4}{12}$

47.

$1.429$

49.

$\text{\$}33,333.33$ total profit for $200$ cell phones sold

51.

$3.32$ mi represents how far ahead the hare is from the tortoise

53.

$\frac{343}{24}$

55.

$4\sqrt{3}$

57.

$\pi - \frac{32}{25}$

Section 2.2 Exercises

63.

8 units³

65.

$\frac{32}{3\sqrt{2}}$ units³

67.

$\frac{7}{24}\pi r^{2}h$ units³

69.

$\frac{\pi}{24}$ units³

71.

$2$ units³

73.

$\frac{\pi}{240}$ units³

75.

$\frac{4096\pi}{5}$ units³

77.

$\frac{8\pi}{9}$ units³

79.

$\frac{\pi}{2}$ units³

81.

$207\pi$ units³

83.

$\frac{4\pi}{5}$ units³

85.

$\frac{16\pi}{3}$ units³

87.

$\pi$ units³

89.

$\frac{16\pi}{3}$ units³

91.

$\frac{72\pi}{5}$ units³

93.

$\frac{108\pi}{5}$ units³

95.

$\frac{3\pi}{10}$ units³

97.

$2\sqrt{6}\pi$ units³

99.

$9\pi$ units³

101.

$\frac{\pi}{20}\left( {75 - 4\ \text{ln}^{5}(2)} \right)$ units³

103.

$\frac{m^{2}\pi}{3}\left( {b^{3} - a^{3}} \right)$ units³

105.

$\frac{4a^{2}b\pi}{3}$ units³

107.

$2\pi^{2}$ units³

109.

$\frac{2ab^{2}\pi}{3}$ units³

111.

$V = \frac{2\pi}{3}(2r + h)\left( {r - h} \right)^{2}$ units³

113.

$\frac{\pi}{3}\left( {h + R} \right)\left( {h - 2R} \right)^{2}$ units³

Section 2.3 Exercises

115.

$54\pi$ units³

117.

$81\pi$ units³

119.

$\frac{512\pi}{7}$ units³

121.

$2\pi$ units³

123.

$\frac{2\pi}{3}$ units³

125.

$2\pi$ units³

127.

$\frac{4\pi}{5}$ units³

129.

$\frac{64\pi}{3}$ units³

131.

$\frac{32\pi}{5}$ units³

133.

$\frac{7\pi}{6}$

135.

$48\pi$

137.

$\frac{308\pi}{5}$

139.

$\frac{512\pi}{7}$

141.

$\frac{8\pi}{5}$

143.

$\frac{28\pi}{15}$ units³

145.

$\frac{3\pi}{10}$ units³

147.

$\pi\left( {\frac{6 \cdot 2^{2/3}}{5} - \frac{11}{10}} \right) = \frac{\pi}{10}\left( {12 \cdot 2^{2/3} - 11} \right) \approx 2.5286$ units³

149.

$0.9876$ units³

151.

$3\sqrt{2}$ units³

153.

$\frac{496\pi}{15}$ units³

155.

$\frac{398\pi}{15}$ units³

157.

$15.9074$ units³

159.

$\frac{1}{3}\pi r^{2}h$ units³

161.

$\pi r^{2}h$ units³

163.

$\pi a^{2}$ units³

Section 2.4 Exercises

165.

$2\sqrt{26}$

167.

$2\sqrt{17}$

169.

$\frac{\pi}{6}\left( {17\sqrt{17} - 5\sqrt{5}} \right)$

171.

$\frac{13\sqrt{13} - 8}{27}$

173.

$\frac{4}{3}$

175.

$2.0035$

177.

$\frac{123}{32}$

179.

$10$

181.

$\frac{20}{3}$

183.

$\frac{1}{675}\left( {229\sqrt{229} - 8} \right)$

185.

$\frac{1}{8}\left( {4\sqrt{5} + \text{ln}\left( {9 + 4\sqrt{5}} \right)} \right)$

187.

$1.201$

189.

$15.2341$

191.

$\frac{49\pi}{3}$

193.

$70\pi\sqrt{2}$

195.

$8\pi$

197.

$120\pi\sqrt{26}$

199.

$\frac{\pi}{6}\left( {17\sqrt{17} - 1} \right)$

201.

$9\sqrt{2}\pi$

203.

$\frac{10\sqrt{10}\pi}{27}\left( {73\sqrt{73} - 1} \right)$

205.

$25.645$

207.

$\pi(2 + \pi)$

209.

$10.5017$

211.

$24$ ft

213.

$2$

215.

Answers may vary

217.

For more information, look up Gabriel’s Horn.

Section 2.5 Exercises

219.

$150$ ft-lb

221.

$200\ \text{J}$

223.

$1$ J

225.

$\frac{39}{2}$

227.

$\text{ln}(243)$

229.

$\frac{332\pi}{15}$

231.

$100\pi$

233.

$20\pi\sqrt{15}$

235.

$6$ J

237.

$5$ cm

239.

$36$ J

241.

$18,750$ ft-lb

243.

Weight $= {\frac{32}{3}\ \times \ 10^{9}\ \text{lb}}$

Work $= {\frac{4}{3}\ \times \ 10^{12}\ \text{ft-lb}}$

245.

$9.71\ \times \ 10^{8}\ \text{N m}$

247.

a\. $3,000,000$ lb, b. $750,000$ lbs

249.

$23.25\pi$ million ft-lb

251.

$\frac{A\rho H^{2}}{2}$

253.

Answers may vary

Section 2.6 Exercises

255.

$\frac{5}{4}$

257.

$\left( {\frac{2}{3},\frac{2}{3}} \right)$

259.

$\left( {\frac{7}{4},\frac{3}{2}} \right)$

261.

$\frac{3L}{4}$

263.

$\frac{\pi}{2}$

265.

$\frac{e^{2} + 1}{e^{2} - 1}$

267.

$\frac{\pi^{2} - 4}{\pi}$

269.

$\frac{1}{4}\left( {1 + e^{2}} \right)$

271.

$\left( {\frac{a}{3},\frac{b}{3}} \right)$

273.

$\left( {0,\frac{\pi}{8}} \right)$

275.

$(0,3)$

277.

$\left( {0,\frac{4}{\pi}} \right)$

279.

$\left( {\frac{5}{8},\frac{1}{3}} \right)$

281.

$\frac{2m\pi}{3}$

283.

$\pi a^{2}b$

285.

$\left( {\frac{4}{3\pi},\frac{4}{3\pi}} \right)$

287.

$\left( {\frac{1}{2},\frac{2}{5}} \right)$

289.

$\left( {0,\frac{28}{9\pi}} \right)$

291.

Center of mass: $\left( \frac{3a}{8},\frac{2a^{2}}{5} \right),$ volume: $\frac{\pi a^{4}}{2}$

293.

Volume: $2\pi^{2}a^{2}\left( {b + a} \right)$

Section 2.7 Exercises

295.

$\frac{1}{x}$

297.

$- \frac{1}{x{(\text{ln}\ x)}^{2}}$

299.

$\text{ln}\left( {x + 1} \right) + C$

301.

$\text{ln}(x) + 1$

303.

$\text{cot}(x)$

305.

$\frac{7}{x}$

307.

$\text{csc}(x)\text{sec}\ x$

309.

$-2\ \text{tan}\ x$

311.

$\frac{1}{2}\text{ln}\left( \frac{5}{3} \right)$

313.

$2 - \frac{1}{2}\text{ln}(5)$

315.

$\frac{1}{\text{ln}(2)} - 1$

317.

$\frac{1}{2}\text{ln}(2)$

319.

$\frac{1}{3}\left( {\text{ln}\ x} \right)^{3}$

321.

$\frac{2x^{3}}{\sqrt{x^{2} + 1}\sqrt{x^{2} - 1}}$

323.

$x^{-2 - (1\text{/}x)}\left( {\text{ln}\ x - 1} \right)$

325.

$ex^{e - 1}$

327.

$1$

329.

$- \frac{1}{x^{2}}$

331.

$\pi - \text{ln}(2)$

333.

$\frac{1}{x}$

335.

$e^{5} - 6\ \text{units}^{2}$

337.

$\text{ln}(4) - 1\ \text{units}^{2}$

339.

$2.8656$

341.

$3.1502$

Section 2.8 Exercises

349.

True

351.

True; $k = \frac{\text{ln}\ (2)}{t}$

353.

$20$ hours

355.

No. The relic is approximately $871$ years old.

357.

$73$ years

359.

$5$ days $6$ hours $27$ minutes

361.

$12$

363.

$8.618\text{\%}$

365.

$\text{\$}6766.76$

367.

$9$ hours $13$ minutes

369.

$239,179$ years

371.

$P\prime(t) = 43e^{0.01604t}.$ The population is always increasing.

373.

The population reaches $10$ billion people in $2032.$

375.

$P\prime(t) = 2.259e^{0.06407t}.$ The population is always increasing.

Section 2.9 Exercises

377.

$e^{x}\ \text{and}\ e^{\text{−}x}$

379.

Answers may vary

381.

Answers may vary

383.

Answers may vary

385.

$3\ \text{sinh}\left( {3x + 1} \right)$

387.

$\text{−}\text{tanh}(x)\text{sech}(x)$

389.

$4\ \text{cosh}(x)\text{sinh}(x)$

391.

$\frac{x\ \text{sech}^{2}\left( \sqrt{x^{2} + 1} \right)}{\sqrt{x^{2} + 1}}$

393.

$6\ \text{sinh}^{5}(x)\text{cosh}(x)$

395.

$\frac{1}{2}\text{sinh}\left( {2x + 1} \right) + C$

397.

$\frac{1}{2}\text{sinh}\left( x^{2} \right) + C$

399.

$\frac{1}{3}\text{cosh}^{3}(x) + C$

401.

$\text{ln}\left( {1 + \text{cosh}(x)} \right) + C$

403.

$\text{cosh}(x) + \text{sinh}(x) + C$

405.

$\frac{4}{1 - 16x^{2}}$

407.

$\frac{\text{sinh}(x)}{\sqrt{\text{cosh}^{2}(x) + 1}}$

409.

$\text{−}\text{csc}(x)$

411.

$- \frac{1}{\left( {x^{2} - 1} \right)\text{tanh}^{-1}(x)}$

413.

$\frac{1}{a}\text{tanh}^{-1}\left( \frac{x}{a} \right) + C$

415.

$\sqrt{x^{2} + 1} + C$

417.

$\text{cosh}^{-1}\left( e^{x} \right) + C$

419.

Answers may vary

421.

$37.30$

423.

$y = \frac{1}{c}\text{cosh}(cx)$

425.

$-0.521095$

427.

$10$

Review Exercises

435.

False

437.

False

439.

$16\sqrt{3}~{units}^{3}$

441.

$\frac{162\pi}{5}$

443.

a\. $4,$ b. $\frac{128\pi}{7},$ c. $\frac{64\pi}{5}$

445.

a\. $1.949,$ b. $21.952,$ c. $17.099$

447.

a\. $\frac{31}{6},$ b. $\frac{452\pi}{15},$ c. $\frac{31\pi}{6}$

449.

$\pi\text{ln}17$

451.

Mass: $\frac{1}{2},$ center of mass: $\left( {\frac{18}{35},\frac{9}{22}} \right)$

453.

$\sqrt{17} + \frac{1}{8}\text{ln}(33 + 8\sqrt{17})$

455.

Volume: $\frac{3\pi}{4},$ surface area: $\pi\left( {\sqrt{2} - \text{sinh}^{-1}(1) + \text{sinh}^{-1}(16) - \frac{\sqrt{257}}{16}} \right)$

457.

11:02 a.m.

459.

$\pi(1 + \text{sinh}(1)\text{cosh}(1))$

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• Authors: Gilbert Strang, Edwin “Jed” Herman
• Publisher/website: OpenStax
• Book title: Calculus Volume 2
• Publication date: Mar 30, 2016
• Location: Houston, Texas
• Book URL: https://openstax.org/books/calculus-volume-2/pages/1-introduction
• Section URL: https://openstax.org/books/calculus-volume-2/pages/chapter-2

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