Chapter 3

> 来源: OpenStax《Calculus Volume 2》| 原页: https://openstax.org/books/calculus-volume-2/pages/chapter-3

Skip to ContentGo to accessibility page\Calculus Volume 2

Chapter 3

Calculus Volume 2Chapter 3

------------------------------------------------------------------------

Chapter 3

Checkpoint

3.1

$\int^{}xe^{2x}dx = \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x} + C$

3.2

$\frac{1}{2}x^{2}\text{ln}\mspace{2mu} x - \frac{1}{4}x^{2} + C$

3.3

$\text{−}x^{2}\text{cos}\mspace{2mu} x + 2x\mspace{2mu}\text{sin}\mspace{2mu} x + 2\mspace{2mu}\text{cos}\mspace{2mu} x + C$

3.4

$\frac{\pi}{2} - 1$

3.5

$\frac{1}{5}\text{sin}^{5}x + C$

3.6

$\frac{1}{3}\text{sin}^{3}x - \frac{1}{5}\text{sin}^{5}x + C$

3.7

$\frac{1}{2}x + \frac{1}{4}\text{sin}\left( {2x} \right) + C$

3.8

$\text{sin}\mspace{2mu} x - \frac{1}{3}\text{sin}^{3}x + C$

3.9

$\frac{1}{2}x + \frac{1}{12}\mspace{2mu}\text{sin}\left( {6x} \right) + C$

3.10

$\frac{1}{2}\text{sin}\mspace{2mu} x + \frac{1}{22}\mspace{2mu}\text{sin}\left( {11x} \right) + C$

3.11

$\frac{1}{6}\text{tan}^{6}x + C$

3.12

$\frac{1}{9}\text{sec}^{9}x - \frac{1}{7}\text{sec}^{7}x + C$

3.13

$\int\text{sec}^{5}x\ dx = \frac{1}{4}\text{sec}^{3}x\mspace{2mu}\text{tan}\mspace{2mu} x + \frac{3}{4}\int\text{sec}^{3}x$

3.14

$\int^{}125\mspace{2mu}\text{sin}^{3}\theta d\theta$

3.15

$\int^{}32\mspace{2mu}\text{tan}^{3}\theta\mspace{2mu}\text{sec}^{3}\theta d\theta$

3.16

$\text{ln}\left| {\frac{x}{2} + \frac{\sqrt{x^{2} - 4}}{2}} \right| + C$

3.17

$x - 5\mspace{2mu}\text{ln}\left| {x + 2} \right| + C$

3.18

$\frac{2}{5}\text{ln}\left| {x + 3} \right| + \frac{3}{5}\text{ln}\left| {x - 2} \right| + C$

3.19

$\frac{x + 2}{\left( {x + 3} \right)^{3}{(x - 4)}^{2}} = \frac{A}{x + 3} + \frac{B}{{(x + 3)}^{2}} + \frac{C}{{(x + 3)}^{3}} + \frac{D}{(x - 4)} + \frac{E}{{(x - 4)}^{2}}$

3.20

$\frac{x^{2} + 3x + 1}{(x + 2)\left( {x - 3} \right)^{2}{(x^{2} + 4)}^{2}} = \frac{A}{x + 2} + \frac{B}{x - 3} + \frac{C}{\left( {x - 3} \right)^{2}} + \frac{Dx + E}{x^{2} + 4} + \frac{Fx + G}{{(x^{2} + 4)}^{2}}$

3.21

Possible solutions include $\text{sinh}^{-1}\left( \frac{x}{2} \right) + C$ and $\text{ln}\left| {\sqrt{x^{2} + 4} + x} \right| + C.$

3.22

$\frac{24}{35}$

3.23

$\frac{17}{24}$

3.24

0.0074, 1.1%

3.25

$\frac{1}{192}$

3.26

$\frac{25}{36}$

3.27

$e^{3},$ converges

3.28

$+ \infty,$ diverges

3.29

Since ${\int_{e}^{+ \infty}{\frac{1}{x}dx = \text{+}\infty}},$ $\int_{e}^{+ \infty}{\frac{\text{ln}\mspace{2mu} x}{x}dx}$ diverges.

Section 3.1 Exercises

1.

$u = x^{3}$

3.

$u = y^{3}$

5.

$u = \text{sin}(2x)$

7.

$\text{−}x + x\mspace{2mu}\text{ln}\mspace{2mu} x + C$

9.

$x\mspace{2mu}\text{tan}^{-1}x - \frac{1}{2}\text{ln}(1 + x^{2}) + C$

11.

$- \frac{1}{2}x\mspace{2mu}\text{cos}\left( {2x} \right) + \frac{1}{4}\text{sin}\left( {2x} \right) + C$

13.

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

15.

$2x\mspace{2mu}\text{cos}\mspace{2mu} x + \left( {-2 + x^{2}} \right)\text{sin}\mspace{2mu} x + C$

17.

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

19.

$\frac{1}{2}e^{x}\left( {\text{−}\text{cos}\mspace{2mu} x + \text{sin}\mspace{2mu} x} \right) + C$

21.

$- \frac{e^{\text{−}x^{2}}}{2} + C$

23.

$- \frac{1}{2}x\mspace{2mu}\text{cos}\left\lbrack {\text{ln}(2x)} \right\rbrack + \frac{1}{2}x\mspace{2mu}\text{sin}\left\lbrack {\text{ln}(2x)} \right\rbrack + C$

25.

$2x - 2x\mspace{2mu}\text{ln}\mspace{2mu} x + x{(\text{ln}\mspace{2mu} x)}^{2} + C$

27.

$\left( {\text{−}\ \frac{x^{3}}{9} + \frac{1}{3}x^{3}\text{ln}\mspace{2mu} x} \right) + C$

29.

$- \frac{1}{2}\sqrt{1 - 4x^{2}} + x\ \text{cos}^{-1}(2x) + C$

31.

$\text{−}\left( {-2 + x^{2}} \right)\text{cos}\mspace{2mu} x + 2x\mspace{2mu}\text{sin}\mspace{2mu} x + C$

33.

$\text{−}x\left( {-6 + x^{2}} \right)\text{cos}\mspace{2mu} x + 3\left( {-2 + x^{2}} \right)\text{sin}\mspace{2mu} x + C$

35.

$\frac{1}{2}x\left( {\text{−}\sqrt{1 - \frac{1}{x^{2}}} + x \cdot \text{sec}^{-1}x} \right) + C$

37.

$\text{−}\text{cosh}\mspace{2mu} x + x\mspace{2mu}\text{sinh}\mspace{2mu} x + C$

39.

$\frac{1}{4} - \frac{3}{4\text{e}^{2}}$

41.

2

43.

$2\pi$

45.

$-2 + \pi$

47.

$\text{−}\text{sin}(x) + \text{ln}\left\lbrack {\text{sin}(x)} \right\rbrack\text{sin}\mspace{2mu} x + C$

49.

Answers vary

51.

a\. $\frac{2}{5}\left( {1 + x} \right)\left( {-3 + 2x} \right)^{3\text{/}2} + C$ b. $\frac{2}{5}\left( {1 + x} \right)\left( {-3 + 2x} \right)^{3\text{/}2} + C$

53.

Do not use integration by parts. Choose *u* to be $\text{ln}\mspace{2mu} x,$ and the integral is of the form ${\int{u^{2}du}}.$

55.

Do not use integration by parts. Let $u = x^{2} - 3,$ and the integral can be put into the form ${\int{e^{u}du}}.$

57.

Do not use integration by parts. Choose *u* to be $u = 3x^{3} + 2$ and the integral can be put into the form ${\int{\text{sin}(u)du}}.$

59.

The area under graph is 0.39535.

61.

$2\pi e$

63.

2.05

65.

$12\pi$

67.

$8\pi^{2}$

Section 3.2 Exercises

69.

$\text{cos}^{2}x$

71.

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

73.

$\frac{\text{sin}^{4}x}{4} + C$

75.

$\frac{1}{12}\text{tan}^{6}(2x) + C$

77.

$\left( \frac{x}{2} \right)\left( \frac{x}{2} \right) + C_{1}~\text{or~sec}^{2}\left( \frac{x}{2} \right) + C_{2}$

79.

$–\text{cos}x + \frac{1}{3}\text{cos}^{3}x + C$

81.

$- \frac{1}{2}\text{cos}^{2}x + C$ or $\frac{1}{2}\text{sin}^{2}x + C$

83.

$–\frac{1}{3}\text{cos}^{3}x + \frac{2}{5}\text{cos}^{5}x–\frac{1}{7}\text{cos}^{7}x + C$

85.

$\frac{2}{3}{(\text{sin}\mspace{2mu} x)}^{\frac{3}{2}} + C$

87.

$\text{sec}\mspace{2mu} x + C$

89.

$\frac{1}{2}\text{sec}\mspace{2mu} x\mspace{2mu}\text{tan}\mspace{2mu} x - \frac{1}{2}\text{ln}\left( {\text{sec}\mspace{2mu} x + \text{tan}\mspace{2mu} x} \right) + C$

91.

$\frac{2\mspace{2mu}\text{tan}\mspace{2mu} x}{3} + \frac{1}{3}\text{sec}{(x)}^{2}\text{tan}\mspace{2mu} x$ $= \text{tan}\mspace{2mu} x + \frac{\text{tan}^{3}x}{3} + C$

93.

$\text{−}\text{ln}\left| {\text{cot}\mspace{2mu} x + \text{csc}\mspace{2mu} x} \right| + C$

95.

$\frac{\text{sin}^{3}\left( {ax} \right)}{3a} + C$

97.

$\frac{\pi}{2}$

99.

$\frac{x}{2} + \frac{1}{12}\mspace{2mu}\text{sin}(6x) + C$

101.

*$x + C$*

103.

0

105.

0

107.

0

109.

Approximately 0.239

111.

$\sqrt{2}$

113.

1.0

115.

0

117.

$\frac{3\theta}{8} - \frac{1}{4\pi}\mspace{2mu}\text{sin}(2\pi\theta) + \frac{1}{32\pi}\mspace{2mu}\text{sin}(4\pi\theta) + C = f(x)$

119.

$\text{ln}\left( \sqrt{3} \right)$

121.

${\int_{\text{−}\pi}^{\pi}{\text{sin}(2x)\text{cos}(3x)}}dx = 0$

123.

$\sqrt{\text{tan}(x)}\left( \frac{8\mspace{2mu}\text{tan}\mspace{2mu} x}{21} \right.\left. {+ \frac{2}{7}\text{sec}\ x^{2}\text{tan}\mspace{2mu} x} \right) + C = f(x)$

125.

The second integral is more difficult because the first integral is simply a *u*-substitution type.

Section 3.3 Exercises

127.

$9\mspace{2mu}\text{tan}^{2}\theta$

129.

$a^{2}\text{cosh}^{2}\theta$

131.

$4\left( {x - \frac{1}{2}} \right)^{2}$

133.

$\text{−}{(x + 1)}^{2} + 5$

135.

$\text{ln}\left| \frac{\sqrt{x^{2}{- a^{2}}} + x}{a} \right| + C$

137.

$\frac{1}{3}\text{ln}\left| {\sqrt{9x^{2} + 1} + 3x} \right| + C$

139.

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

141.

$9\left\lbrack {\frac{x\sqrt{x^{2} + 9}}{18} + \frac{1}{2}ln\left| {\frac{\sqrt{x^{2} + 9}}{3} + \frac{x}{3}} \right|} \right\rbrack + C$

143.

$- \frac{1}{3}\sqrt{9 - \theta^{2}}\left( {18 + \theta^{2}} \right) + C$

145.

$\frac{\left( {-1 + x^{2}} \right)\left( {2 + 3x^{2}} \right)\sqrt{x^{6} - x^{8}}}{15x^{3}} + C$

147.

$- \frac{x}{9\sqrt{-9 + x^{2}}} + C$

149.

$\frac{1}{2}\left( {\text{ln}\left| {x + \sqrt{x^{2} - 1}} \right| + x\sqrt{x^{2} - 1}} \right) + C$

151.

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

153.

$\frac{1}{8}\left( {x\left( {5 - 2x^{2}} \right)\sqrt{1 - x^{2}} + 3\mspace{2mu}\text{arcsin}\mspace{2mu} x} \right) + C$

155.

$\text{ln}\mspace{2mu} x - \text{ln}\left| {1 + \sqrt{1 - x^{2}}} \right| + C$

157.

$- \frac{\sqrt{-1 + x^{2}}}{x} + \text{ln}\left| {x + \sqrt{-1 + x^{2}}} \right| + C$

159.

$- \frac{\sqrt{1 + x^{2}}}{x} + \text{arcsinh}\mspace{2mu} x + C$

161.

$- \frac{1}{1 + x} + C$

163.

$\text{arcsin}\left( \frac{x - 5}{5} \right)\text{+}C$

165.

$\frac{9\pi}{2};$ area of a semicircle with radius 3

167.

$\text{arcsin}(x) + C$ is the common answer.

169.

$\frac{1}{2}\text{ln}\left( {1 + x^{2}} \right) + C$ is the result using either method.

171.

Use trigonometric substitution. Let $x = \text{sec}(\theta).$

173.

4.367

175.

$\frac{\pi^{2}}{8} + \frac{\pi}{4}$

177.

$y = \frac{1}{16}\text{ln}\left| \frac{x + 8}{x - 8} \right| + 3$

179.

24.6 m³

181.

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

Section 3.4 Exercises

183.

$- \frac{2}{x + 1} + \frac{5}{2(x + 2)} + \frac{1}{2x}$

185.

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

187.

$2x^{2} + 4x + 8 + \frac{16}{x - 2}$

189.

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

191.

$- \frac{1}{2(x - 2)} + \frac{1}{2(x - 1)} - \frac{1}{6x} + \frac{1}{6(x - 3)}$

193.

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

195.

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

197.

$\text{ln}|x - 2| + 2\mspace{2mu}\text{ln}|x + 4| + C$

199.

$\frac{1}{2}\text{ln}\left| {4 - x^{2}} \right| + C$

201.

$2\left( {x + \frac{1}{3}\text{arctan}\left( \frac{1 + x}{3} \right)} \right) + C$

203.

$2\mspace{2mu}\text{ln}|x| - 3\mspace{2mu}\text{ln}\left| {1 + x} \right| + C$

205.

$\frac{1}{16}\left( {\text{−}\ \frac{4}{-2 + x} - \text{ln}\left| {-2 + x} \right| + \text{ln}\left| {2 + x} \right|} \right) + C$

207.

$\frac{1}{30}\left( {-2\sqrt{5}\mspace{2mu}\text{arctan}\left\lbrack \frac{1 + x}{\sqrt{5}} \right\rbrack +} \right.\left. {2\mspace{2mu}\text{ln}\left| {-4 + x} \right| - \text{ln}\left| {6 + 2x + x^{2}} \right|} \right) + C$

209.

$- \frac{3}{x} + 4\mspace{2mu}\text{ln}\left| {x + 2} \right| + x + C$

211.

$\text{−}\text{ln}\left| {3 - x} \right| + \frac{1}{2}\text{ln}\left| {x^{2} + 4} \right| + C$

213.

$\text{ln}\left| {x - 2} \right| - \frac{1}{2}\text{ln}\left| {x^{2} + 2x + 2} \right| + C$

215.

$\text{−}x + \text{ln}\left| {1 - e^{x}} \right| + C$

217.

$\frac{1}{5}\text{ln}\left| \frac{\text{cos}\mspace{2mu} x + 3}{\text{cos}\mspace{2mu} x - 2} \right| + C$

219.

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

221.

$2\sqrt{1 + x} - 2\mspace{2mu}\text{ln}\left| {1 + \sqrt{1 + x}} \right| + C$

223.

$\text{ln}\left| \frac{\text{sin}\mspace{2mu} x}{1 - \text{sin}\mspace{2mu} x} \right| + C$

225.

$\frac{\sqrt{3}}{4}$

227.

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

229.

$6x^{1\text{/}6} - 3x^{1\text{/}3} + 2\sqrt{x} - 6\mspace{2mu}\text{ln}\left( {1 + x^{1\text{/}6}} \right) + C$

231.

$\frac{4}{3}\pi\mspace{2mu}\text{arctanh}\left\lbrack \frac{1}{3} \right\rbrack = \frac{1}{3}\pi\mspace{2mu}\text{ln}\mspace{2mu} 4$

233.

$x = \text{−}\text{ln}\left| {t - 3} \right| + \text{ln}\left| {t - 4} \right| + \text{ln}\mspace{2mu} 2$

235.

$x = \text{ln}\left| {t - 1} \right| - \sqrt{2}\mspace{2mu}\text{arctan}\left( {\sqrt{2}t} \right) - \frac{1}{2}\text{ln}\left( {t^{2} + \frac{1}{2}} \right) + \sqrt{2}\mspace{2mu}\text{arctan}\left( {2\sqrt{2}} \right) + \frac{1}{2}\text{ln}\mspace{2mu} 4.5$

237.

$\frac{2}{5}\pi\mspace{2mu}\text{ln}\ \frac{28}{13}$

239.

$\frac{\text{arctan}\left\lbrack \frac{-1 + 2x}{\sqrt{3}} \right\rbrack}{\sqrt{3}} + \frac{1}{3}\text{ln}\left| {1 + x} \right| - \frac{1}{6}\text{ln}\left| {1 - x + x^{2}} \right| + C$

241.

2.0 in.²

243.

$3{(-8 + x)}^{1\text{/}3}$

$-2\sqrt{3}\mspace{2mu}\text{arctan}\left\lbrack \frac{-1 + {(-8 + x)}^{1\text{/}3}}{\sqrt{3}} \right\rbrack$

$-2\mspace{2mu}\text{ln}\left\lbrack {2 + {(-8 + x)}^{1\text{/}3}} \right\rbrack$

$+ \text{ln}\left\lbrack {4 - 2{(-8 + x)}^{1\text{/}3} + {(-8 + x)}^{2\text{/}3}} \right\rbrack + C$

Section 3.5 Exercises

245.

$\frac{1}{2}\text{ln}\left| {x^{2} + 2x + 2} \right| + 2\mspace{2mu}\text{arctan}\left( {x + 1} \right) + C$

247.

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

249.

$\frac{2^{x^{2} - 1}}{\text{ln}\mspace{2mu} 2} + C$

251.

$\text{arcsin}\left( \frac{y}{2} \right) + C$

253.

$- \frac{1}{2}\text{csc}\left( {2w} \right) + C$

255.

$9 - 6\sqrt{2}$

257.

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

259.

$\frac{1}{12}\text{tan}^{4}\left( {3x} \right) - \frac{1}{6}\text{tan}^{2}\left( {3x} \right) + \frac{1}{3}\text{ln}\left| {\text{sec}(3x)} \right| + C$

261.

$2\mspace{2mu}\text{cot}\left( \frac{w}{2} \right) - 2\mspace{2mu}\text{csc}\left( \frac{w}{2} \right) + w + C$

263.

$\frac{1}{5}\text{ln}\left| \frac{2(5 + 4\mspace{2mu}\text{sin}\mspace{2mu} t - 3\mspace{2mu}\text{cos}\mspace{2mu} t)}{4\mspace{2mu}\text{cos}\mspace{2mu} t + 3\mspace{2mu}\text{sin}\mspace{2mu} t} \right|$

265.

$6x^{1\text{/}6} - 3x^{1\text{/}3} + 2\sqrt{x} - 6\mspace{2mu}\text{ln}\left\lbrack {1 + x^{1\text{/}6}} \right\rbrack + C$

267.

$\text{−}x^{3}\text{cos}\mspace{2mu} x + 3x^{2}\text{sin}\mspace{2mu} x + 6x\mspace{2mu}\text{cos}\mspace{2mu} x - 6\mspace{2mu}\text{sin}\mspace{2mu} x + C$

269.

$\frac{1}{2}\left( {x^{2} + \text{ln}\left| {1 + e^{\text{−}x^{2}}} \right|} \right) + C$

271.

$2\mspace{2mu}\text{arctan}\left( \sqrt{x - 1} \right) + C$

273.

$0.5 = \frac{1}{2}$

275.

8.0

277.

$\frac{1}{3}\text{arctan}\left( {\frac{1}{3}(x + 2)} \right) + C$

279.

$\frac{1}{3}\text{arctan}\left( \frac{x + 1}{3} \right) + C$

281.

$\text{ln}\left( {e^{x} + \sqrt{–4 + e^{2x}}} \right) + C$

283.

$\text{ln}\mspace{2mu} x - \frac{1}{6}\text{ln}\left( {x^{6} + 1} \right) - \frac{\text{arctan}\left( x^{3} \right)}{3x^{3}} + C$

285.

$\text{ln}\left| {x + \sqrt{16 + x^{2}}} \right| + C$

287.

$- \frac{1}{4}\text{cot}(2x) + C$

289.

$\frac{1}{2}\text{arctan}\mspace{2mu} 10$

291.

1276.14

293.

7.21

295.

$\sqrt{5} - \sqrt{2} + \text{ln}\left| \frac{2 + 2\sqrt{2}}{1 + \sqrt{5}} \right|$

297.

$\frac{1}{3}\text{arctan}(3) \approx 0.416$

Section 3.6 Exercises

299.

0.696

301.

9.484

303.

0.5000

305.

$T_{4} = 18.75$

307.

0.500

309.

1.129

311.

0.6577

313.

0.0213

315.

1.5629

317.

1.9133

319.

$\text{T(4)} = 0.1088$

321.

1.0

323.

Approximate error is 0.000325.

325.

$\frac{1}{7938}$

327.

$\frac{81}{25,000}$

329.

475

331.

174

333.

0.1544

335.

6.2807

337.

4.606

339.

3.41 ft

341.

$T_{16} = 100.125;$ absolute error = 0.125

343.

about 89,250 m²

345.

parabola

Section 3.7 Exercises

347.

divergent

349.

$\frac{\pi}{2}$

351.

$\frac{2}{e}$

353.

Converges

355.

Converges to 1/2.

357.

−4

359.

$\pi$

361.

diverges

363.

diverges

365.

1.5

367.

diverges

369.

diverges

371.

diverges

373.

Both integrals diverge.

375.

diverges

377.

diverges

379.

$\pi$

381.

0.0

383.

0.0

385.

6.0

387.

$\frac{\pi}{2}$

389.

$8\mspace{2mu}\text{ln}(16) - 4$

391.

$1.047$

393.

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

395.

7.0

397.

$\frac{5\pi}{2}$

399.

$3\pi$

401.

$\frac{1}{s},s > 0$

403.

$\frac{s}{s^{2} + 4},s > 0$

405.

Answers will vary.

407.

0.8775

Review Exercises

409.

False

411.

False

413.

$- \frac{\sqrt{x^{2} + 16}}{16x} + C$

415.

$\frac{1}{10}\left( {4\mspace{2mu}\text{ln}(2 - x) + 5\mspace{2mu}\text{ln}(x + 1) - 9\mspace{2mu}\text{ln}(x + 3)} \right) + C$

417.

$- \frac{\sqrt{4 - \text{sin}^{2}(x)}}{\text{sin}(x)} - \frac{x}{2} + C$

419.

$\frac{1}{15}\left( {x^{2} + 2} \right)^{3\text{/}2}\left( {3x^{2} - 4} \right) + C$

421.

$\frac{1}{16}\text{ln}\left( \frac{x^{2} + 2x + 2}{x^{2} - 2x + 2} \right) - \frac{1}{8}\text{tan}^{-1}\left( {1 - x} \right) + \frac{1}{8}\text{tan}^{-1}\left( {x + 1} \right) + C$

423.

$\text{M}_{4} = 3.312,\text{T}_{4} = 3.354,S_{4} = 3.326$

425.

$\text{M}_{4} = -0.982,\text{T}_{4} = -0.917,S_{4} = -0.952$

427.

approximately 0.2194

431.

Answers may vary. Ex: $9.405$ km

PreviousNextOrder a print copy

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information

• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:

Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction

• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:

Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction

Citation information

• Use the information below to generate a citation. We recommend using a citation tool such as this one.
• 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-3

© Jul 15, 2026 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

---