Chapter 5
> 来源: OpenStax《Calculus Volume 3》| 原页: https://openstax.org/books/calculus-volume-3/pages/chapter-5
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Chapter 5
Calculus Volume 3Chapter 5
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Chapter 5
Checkpoint
5.1
$V = {\sum\limits_{i = 1}^{2}{\sum\limits_{j = 1}^{2}{f(x_{ij}^{*},y_{ij}^{*})\text{Δ}A}}} = 0$
5.2
a\. 26 b. Answers may vary.
5.3
$- \frac{1340}{3}$
5.4
$\frac{4 - \text{ln}\ 5}{\text{ln}\ 5}$
5.5
$\frac{\pi}{2}$
5.6
Answers to both parts a. and b. may vary.
5.7
Type I and Type II are expressed as $\left\{ {\left. \left( {x,y} \right) \right|0 \leq x \leq 2,x^{2} \leq y \leq 2x} \right\}$ and $\left\{ {\left. \left( {x,y} \right) \right|0 \leq y \leq 4,\frac{1}{2}y \leq x \leq \sqrt{y}} \right\},$ respectively.
5.8
$\pi\text{/}4$
5.9
$\left\{ \left. (x,y) \right|0 \leq y \leq \ln 2,1 \leq x \leq e^{y} \right\} \cup \left\{ \left. (x,y) \right|\ln 2 \leq y \leq e,1 \leq x \leq 2 \right\} \cup \left\{ \left. (x,y) \right|e \leq y \leq e^{2},\text{ln}\ y \leq x \leq 2 \right\}$
5.10
Same as in the example shown.
5.11
$\frac{216}{35}$
5.12
$\frac{e^{2}}{4} + 10e - \frac{49}{4}$ cubic units
5.13
$\frac{81}{4}$ square units
5.14
$\frac{3}{4}$
5.15
$\frac{\pi}{4}$
5.16
$\frac{11}{39} \approx 0.282$
5.17
$\frac{14}{3}$
5.18
$8\pi$
5.19
$\pi\text{/}4$
5.20
$V = {\int\limits_{0}^{2\pi}\ {\int\limits_{0}^{2\sqrt{2}}\left( {16 - 2r^{2}} \right)}}r\ dr\ d\theta = 64\pi$ cubic units
5.21
$A = 2{\int\limits_{\text{−}\pi\text{/}2}^{\pi\text{/}6}\ {\int\limits_{1 + \text{sin}\ \theta}^{3 - 3\ \text{sin}\ \theta}{r\ dr\ d\theta}}} = 8\pi + 9\sqrt{3}$
5.22
$\frac{\pi}{4}$
5.23
${\iiint\limits_{B}{z\ \text{sin}\ x\ \text{cos}\ y}}\ dV = 8$
5.24
${\iiint\limits_{E}1}dV = {\int_{x = -3}^{x = 3}{\int_{y = \text{−}\sqrt{9 - x^{2}}}^{y = \sqrt{9 - x^{2}}}{\int_{z = \text{−}\sqrt{9 - x^{2} - y^{2}}}^{z = \sqrt{9 - x^{2} - y^{2}}}{1dz\ dy\ dx = 36\pi.}}}}$
5.25
\(i\) ${\int\limits_{z = 0}^{z = 4}\ {\int\limits_{x = 0}^{x = \sqrt{4 - z}}\ {\int\limits_{y = x^{2}}^{y = 4 - z}{f\left( {x,y,z} \right)}}}}dy\ dx\ dz,$ (ii) ${\int\limits_{y = 0}^{y = 4}\ {\int\limits_{z = 0}^{z = 4 - y}\ {\int\limits_{x = 0}^{x = \sqrt{y}}{f\left( {x,y,z} \right)}}}}dx\ dz\ dy,$ (iii) ${\int\limits_{y = 0}^{y = 4}\ {\int\limits_{x = 0}^{x = \sqrt{y}}\ {\int\limits_{z = 0}^{z = 4 - y}{f\left( {x,y,z} \right)}}}}dz\ dx\ dy,$ (iv) ${\int\limits_{x = 0}^{x = 2}\ {\int\limits_{y = x^{2}}^{y = 4}\ {\int\limits_{z = 0}^{z = 4 - y}{f\left( {x,y,z} \right)}}}}dz\ dy\ dx,$ (v) ${\int\limits_{x = 0}^{x = 2}\ {\int\limits_{z = 0}^{z = 4 - x^{2}}\ {\int\limits_{y = x^{2}}^{y = 4 - z}{f\left( {x,y,z} \right)}}}}dy\ dz\ dx$
5.26
$f_{\text{ave}} = 8$
5.27
$\frac{16}{3}$
5.28
${\iiint\limits_{E}{f\left( {r,\theta,z} \right)}}r\ dz\ dr\ d\theta = {\int\limits_{\theta = 0}^{\theta = \pi}\ {\int\limits_{r = 0}^{r = 2\ \text{sin}\ \theta}\ {\int\limits_{z = 0}^{z = 4 - r\ \text{sin}\ \theta}{f\left( {r,\theta,z} \right)r\ dz\ dr\ d\theta}}}}.$
5.29
$E = \left\{ {\left. \left( {r,\theta,z} \right) \right|0 \leq \theta \leq 2\pi,0 \leq z \leq 1,z \leq r \leq 2 - z^{2}} \right\}$ and $V = {\int\limits_{r = 0}^{r = 1}\ {\int\limits_{z = r}^{z = 2 - r^{2}}\ {\int\limits_{\theta = 0}^{\theta = 2\pi}{r\ d\theta\ dz\ dr}}}}.$
5.30
$E_{2} = \left\{ {\left. \left( {r,\theta,z} \right) \right|0 \leq \theta \leq 2\pi,0 \leq r \leq 1,r \leq z \leq \sqrt{4 - r^{2}}} \right\}$ and $V = {\int\limits_{r = 0}^{r = 1}\ {\int\limits_{z = r}^{z = \sqrt{4 - r^{2}}}\ {\int\limits_{\theta = 0}^{\theta = 2\pi}{r\ d\theta\ dz\ dr}}}}.$
5.31
$V(E) = {\int\limits_{\theta = 0}^{\theta = 2\pi}\ {\int\limits_{\phi = 0}^{\varphi = \pi\text{/}3}\ {\int\limits_{\rho = 0}^{\rho = 2}{\rho^{2}\text{sin}\ \varphi\ d\rho\ d\varphi\ d\theta}}}}$
5.32
Rectangular: ${\int\limits_{x = -2}^{x = 2}\ {\int\limits_{y = \text{−}\sqrt{4 - x^{2}}}^{y = \sqrt{4 - x^{2}}}\ {\int\limits_{z = \text{−}\sqrt{4 - x^{2} - y^{2}}}^{z = \sqrt{4 - x^{2} - y^{2}}}{dz\ dy\ dx -}}}}{\int\limits_{x = -1}^{x = 1}\ {\int\limits_{y = \text{−}\sqrt{1 - x^{2}}}^{y = \sqrt{1 - x^{2}}}\ {\int\limits_{z = \text{−}\sqrt{4 - x^{2} - y^{2}}}^{z = \sqrt{4 - x^{2} - y^{2}}}{dz\ dy\ dx}}}}.$
Cylindrical: ${\int\limits_{\theta = 0}^{\theta = 2\pi}\ {\int\limits_{r = 1}^{r = 2}\ {\int\limits_{z = \text{−}\sqrt{4 - r^{2}}}^{z = \sqrt{4 - r^{2}}}{r\ dz\ dr\ d\theta}}}}.$
Spherical: ${\int\limits_{\varphi = \pi\text{/}6}^{\varphi = 5\pi\text{/}6}\ {\int\limits_{\theta = 0}^{\theta = 2\pi}\ {\int\limits_{\rho = \text{csc}\ \varphi}^{\rho = 2}{\rho^{2}\text{sin}\ \varphi\ d\rho\ d\theta\ d\varphi}}}}.$
5.33
$\frac{9\pi}{8}\ \text{kg}$
5.34
$M_{x} = \frac{81\pi}{64}$ and $M_{y} = \frac{81\pi}{64}$
5.35
$\overset{\text{−}}{x} = \frac{M_{y}}{m} = \frac{81\pi\text{/}64}{9\pi\text{/}8} = \frac{9}{8}$ and $\overset{\text{−}}{y} = \frac{M_{x}}{m} = \frac{81\pi\text{/}64}{9\pi\text{/}8} = \frac{9}{8}.$
5.36
$\overset{\text{−}}{x} = \frac{M_{y}}{m} = \frac{1\text{/}20}{1\text{/}12} = \frac{3}{5}$ and $\overset{\text{−}}{y} = \frac{M_{x}}{m} = \frac{1\text{/}24}{1\text{/}12} = \frac{1}{2}$
5.37
$x_{c} = \frac{M_{y}}{m} = \frac{1\text{/}15}{1\text{/}6} = \frac{2}{5}\ \text{and}\ y_{c} = \frac{M_{x}}{m} = \frac{1\text{/}12}{1\text{/}6} = \frac{1}{2}$
5.38
$I_{x} = {\int\limits_{x = 0}^{x = 2}\ {\int\limits_{y = 0}^{y = x}{y^{2}\sqrt{xy}\ dy\ dx}}} = \frac{64}{35}$ and $I_{y} = {\int\limits_{x = 0}^{x = 2}\ {\int\limits_{y = 0}^{y = x}{x^{2}\sqrt{xy}\ dy\ dx}}} = \frac{64}{35}.$ Also, $I_{0} = {\int\limits_{x = 0}^{x = 2}\ {\int\limits_{y = 0}^{y = x}{\left( {x^{2} + y^{2}} \right)\sqrt{xy}\ dy\ dx}}} = \frac{128}{35}.$
5.39
$R_{x} = \frac{6\sqrt{35}}{35},$ $R_{y} = \frac{6\sqrt{35}}{35},$ and $R_{0} = \frac{6\sqrt{70}}{35}.$
5.40
$\frac{54}{35} = 1.543$
5.41
$\left( {\frac{3}{2},\frac{9}{8},\frac{1}{2}} \right)$
5.42
The moments of inertia of the tetrahedron $Q$ about the $yz\text{-plane,}$ the $xz\text{-plane,}$ and the $xy\text{-plane}$ are $99\text{/}35,36\text{/}7,\text{and}\ 243\text{/}35,$ respectively.
5.43
$T^{-1}(x,y) = (u,v)$ where $u = \frac{3x - y}{3}$ and $v = \frac{y}{3}$
5.44
$J\left( {u,v} \right) = \frac{\partial\left( {x,y} \right)}{\partial\left( {u,v} \right)} = \left| \begin{array}{lll}
\frac{\partial x}{\partial u} & & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & & \frac{\partial y}{\partial v}
\end{array} \right| = \left| \begin{array}{lll}
1 & & 1 \\
0 & & 2
\end{array} \right| = 2$
5.45
$\int\limits_{0}^{\pi\text{/}2}\ {\int\limits_{0}^{1}{r^{3}dr\ d\theta}}$
5.46
$x = \frac{1}{2}\left( {v + u} \right)$ and $y = \frac{1}{2}\left( {v - u} \right)$ and ${\int\limits_{2}^{4}\ {\int\limits_{-u}^{u}{\frac{4}{u^{2}}\left( \frac{1}{2} \right)}}}dv\ du.$
5.47
$\frac{1}{2}\left( {\text{sin}\ 2 - 2} \right)$
5.48
${\int\limits_{0}^{3}\ {\int\limits_{0}^{2}\ {\int\limits_{1}^{2}\left( {\frac{v}{3} + \frac{vw}{3u}} \right)}}}du\ dv\ dw = 2 + \text{ln}\ 8$
Section 5.1 Exercises
1.
27\.
3.
0\.
5.
21.3.
7.
a\. 28 $\text{ft}^{3}$ b. 1.75 ft.
9.
a\. $0.112$ b. $f_{\text{ave}} \simeq 0.175;$ here $f(0.4,0.2) \simeq 0.1,$ $f(0.2,0.6) \simeq -0.2,$ $f(0.8,0.2) \simeq 0.6,$ and $f(0.8,0.6) \simeq 0.2.$
11.
$2\pi.$
13.
40\.
15.
$\frac{81}{2} + 39\sqrt[3]{2}.$
17.
$e - 1.$
19.
$15 - \frac{10\sqrt{2}}{9}.$
21.
0\.
23.
$(e - 1)(1 + \text{sin}\ 1 - \text{cos}\ 1).$
25.
$\frac{3}{4}\text{ln}\left( \frac{5}{3} \right) + 2\ \text{ln}^{2}2 - \text{ln}\ 2.$
27.
$\frac{1}{8}\left\lbrack {\left( {2\sqrt{3} - 3} \right)\pi + 6\ \text{ln}\ 2} \right\rbrack.$
29.
$\frac{1}{4}e^{4}\left( {e^{4} - 1} \right).$
31.
$4(e - 1)\left( {2 - \sqrt{e}} \right).$
33.
$- \frac{\pi}{4} + \text{ln}\left( \frac{5}{4} \right) - \frac{1}{2}\text{ln}\ 2 + \text{arctan}\ 2.$
35.
$\frac{1}{2}.$
37.
$\frac{1}{2}\left( {2\ \text{cosh}\ 1 + \text{cosh}\ 2 - 3} \right).$
49.
a\. $f(x,y) = \frac{1}{2}xy\left( {x^{2} + y^{2}} \right)$ b. $V = {\int\limits_{0}^{1}\ {\int\limits_{0}^{1}{f(x,y)dx\ dy}}} = \frac{1}{8}$ c. $f_{\text{ave}} = \frac{1}{8};$
d.
53.
a\. For $m = n = 2,$ $I = 4e^{-0.5} \approx 2.43$ b. $f_{\text{ave}} = e^{-0.5} \simeq 0.61;$
c.
55.
a\. $\frac{2}{n + 1} + \frac{1}{4}$ b. $\frac{1}{4}$
59.
$56.5\text{°}$ F; here $f(x_{1}^{*},y_{1}^{*}) = 71,$ $f(x_{2}^{*},y_{1}^{*}) = 72,$ $f(x_{1}^{*},y_{2}^{*}) = 40,$ $f(x_{2}^{*},y_{2}^{*}) = 43,$ where $x_{i}^{*}$ and $y_{j}^{*}$ are the midpoints of the subintervals of the partitions of $\lbrack a,b\rbrack$ and $\lbrack c,d\rbrack,$ respectively.
Section 5.2 Exercises
61.
$\frac{27}{20}$
63.
Type I but not Type II
65.
$\frac{\pi}{2}$
67.
$\frac{1}{6}\left( {8 + 3\pi} \right)$
69.
$\frac{1000}{3}$
71.
Type I and Type II
73.
The region $D$ is not of Type I: it does not lie between two vertical lines and the graphs of two continuous functions $g_{1}(x)$ and $g_{2}(x).$ The region $D$ is not of Type II: it does not lie between two horizontal lines and the graphs of two continuous functions $h_{1}(y)$ and $h_{2}(y).$
75.
$\frac{\pi}{2}$
77.
$0$
79.
$\frac{2}{3}$
81.
$\frac{41}{20}$
83.
$-63$
85.
$\pi$
87.
a\. Answers may vary; b. $\frac{2}{3}$
89.
a\. Answers may vary; b. $\frac{7}{3}$
91.
$\frac{8\pi}{3}$
93.
$e - \frac{3}{2}$
95.
$\frac{1}{3}$
97.
${\int\limits_{0}^{1}\ {\int\limits_{x - 1}^{1 - x}{x\ dy\ dx}}} = {\int\limits_{-1}^{0}\ {\int\limits_{0}^{y + 1}{x\ dx\ dy}}} + {\int\limits_{0}^{1}\ {\int\limits_{0}^{1 - y}{x\ dxd}}}y = \frac{1}{3}$
99.
$\int\limits_{\text{−1}}^{1}\ \int\limits_{\text{−}\sqrt{1–y^{2}}}^{\sqrt{1–y^{2}}}y\ dx\ dy = \int\limits_{\text{−1}}^{1}\ \int\limits_{\text{−}\sqrt{1–x^{2}}}^{\sqrt{1–x^{2}}}y\ dy\ dx = 0$
101.
${\iint\limits_{D}{\left( {x^{2} - y^{2}} \right)dA}} = {\int\limits_{-1}^{1}\ {\int\limits_{y^{4} - 1}^{1 - y^{4}}{\left( {x^{2} - y^{2}} \right)dx\ dy}}} = \frac{464}{4095}$
103.
$\frac{4}{5}$
105.
$\frac{5\pi}{32}$
109.
$1$
111.
$2$
113.
a\. $\frac{1}{3};$ b. $\frac{1}{6};$ c. $\frac{1}{6}$
115.
a\. $\frac{4}{3};$ b. $2\pi;$ c. $\frac{6\pi - 4}{3}$
117.
$0\ \text{and}\ 0.865474;$ $A(D) = 0.621135$
119.
$P\left\lbrack {X + Y \leq 6} \right\rbrack = 1 + \frac{3}{2e^{2}} - \frac{5}{e^{6\text{/}5}} \approx 0.45;$ there is a $45\text{\%}$ chance that a customer will spend $6$ minutes in the drive-thru line.
Section 5.3 Exercises
123.
$D = \left\{ {\left. \left( {r,\theta} \right) \right|4 \leq r \leq 5,\frac{\pi}{2} \leq \theta \leq \pi} \right\}$
125.
$D = \left\{ {\left. \left( {r,\theta} \right) \right|0 \leq r \leq \sqrt{2},0 \leq \theta \leq \pi} \right\}$
127.
$D = \left\{ {\left. \left( {r,\theta} \right) \right|0 \leq r \leq 4\ \text{sin}\ \theta,0 \leq \theta \leq \pi} \right\}$
129.
$D = \left\{ {\left. \left( {r,\theta} \right) \right|3 \leq r \leq 5,\frac{\pi}{4} \leq \theta \leq \frac{\pi}{2}} \right\}$
131.
$D = \left\{ {\left. \left( {r,\theta} \right) \right|3 \leq r \leq 5,\frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}} \right\}$
133.
$D = \left\{ {\left. \left( {r,\theta} \right) \right|0 \leq r \leq \text{tan}\ \theta\ \text{sec}\ \theta,0 \leq \theta \leq \frac{\pi}{4}} \right\}$
135.
$0$
137.
$\frac{63\pi}{16}$
139.
$\frac{3367\pi}{18}$
141.
$\frac{35\pi^{2}}{576}$
143.
$\frac{7\pi^{2}}{576}\left( {21 - e^{2} + e^{4}} \right)$
145.
$\frac{5}{2}\text{ln}\left( {1 + \sqrt{2}} \right)$
147.
$\frac{1}{6}\left( {2 - \sqrt{2}} \right)$
149.
${\int\limits_{0}^{\pi}\ {\int\limits_{0}^{2}{r^{5}dr\ d\theta}}} = \frac{32\pi}{3}$
151.
${\int\limits_{\text{−}\pi\text{/}2}^{\pi\text{/}2}\ {\int\limits_{0}^{4}{r\ \text{sin}\left( r^{2} \right)dr\ d\theta}}} = \pi\ \text{sin}^{2}8$
153.
$\frac{3\pi}{4}$
155.
$\frac{\pi}{2}$
157.
$\frac{1}{3}\left( {4\pi - 3\sqrt{3}} \right)$
159.
$\frac{16}{3\pi}$
161.
$\frac{\pi}{18}$
163.
a\. $\frac{2\pi}{3};$ b. $\frac{\pi}{3};$ c. $\frac{\pi}{3}$
165.
$\frac{256\pi}{3}\ \text{cm}^{3}$
167.
$\frac{3\pi}{32}$
169.
$4\pi$
171.
$\frac{\pi}{4}$
173.
$\frac{1}{2}\pi e(e - 1)$
175.
$\sqrt{3} - \frac{\pi}{4}$
177.
$\frac{133\pi^{2}}{864}$
Section 5.4 Exercises
181.
$192$
183.
$0$
185.
${\int\limits_{1}^{2}\ {\int\limits_{2}^{3}\ {\int\limits_{0}^{1}\left( {x^{2} + \text{ln}\ y + z} \right)}}}dz\ dx\ dy = \frac{35}{6} + 2\ \text{ln}\ 2$
187.
${\int\limits_{1}^{3}\ {\int\limits_{0}^{4}\ {\int\limits_{-1}^{2}\left( {x^{2}z + \frac{1}{y}} \right)}}}dz\ dx\ dy = 64 + 12\ \text{ln}\ 3$
191.
$\frac{77}{12}$
193.
$2$
195.
$\frac{439}{120}$
197.
$0$
199.
$- \frac{64}{105}$
201.
$\frac{11}{26}$
203.
$\frac{113}{450}$
205.
$\frac{- 609 - 216\sqrt{3} - 80\pi}{5760} \approx - 0.21431$
207.
$\frac{3\pi}{2}$
209.
$1250$
211.
${\int\limits_{0}^{5}\ {\int\limits_{-3}^{3}\ {\int\limits_{0}^{\sqrt{9 - y^{2}}}{z\ dz\ dy\ dx}}}} = 90$
213.
$V = 5.33$
215.
${\int\limits_{0}^{1}\ {\int\limits_{1}^{3}\ {\int\limits_{2}^{4}\left( {y^{2}z^{2} + 1} \right)}}}dz\ dx\ dy;$ ${\int\limits_{0}^{1}\ {\int\limits_{1}^{3}\ {\int\limits_{2}^{4}\left( {x^{2}y^{2} + 1} \right)}}}dy\ dz\ dx$
217.
$\int\limits_{0}^{1}\ \int\limits_{\text{-z}}^{z}\ \int\limits_{0}^{1 - y^{4} - z^{4}}e^{x}\ dx\ dy\ dz;~\int\limits_{0}^{1}\ \int\limits_{\text{-x}}^{x}\ \int\limits_{0}^{1 - z^{4} - x^{4}}\ln~z\ dy\ dz\ dx$
219.
$V = \int\limits_{–a}^{a}\ \int\limits_{- \sqrt{a^{2} - z^{2}}}^{\sqrt{a^{2} - z^{2}}}\int\limits_{\sqrt{x^{2} + z^{2}}}^{a^{2}}dy\ dx\ dz$
221.
$\frac{9}{2}$
223.
$\frac{156}{5}$
225.
a\. Answers may vary; b. $\frac{128}{3}$
227.
a\. ${\int\limits_{0}^{r}\ {\int\limits_{0}^{\sqrt{r^{2} - x^{2}}}\ {\int\limits_{0}^{\sqrt{r^{2} - x^{2} - y^{2}}}{dz\ dy\ dx}}}};$ b. ${\int\limits_{0}^{r}\ {\int\limits_{0}^{\sqrt{r^{2} - y^{2}}}\ {\int\limits_{0}^{\sqrt{r^{2} - x^{2} - y^{2}}}{dz\ dx\ dy}}}},$ ${\int\limits_{0}^{r}\ {\int\limits_{0}^{\sqrt{r^{2} - z^{2}}}\ {\int\limits_{0}^{\sqrt{r^{2} - x^{2} - z^{2}}}{dy\ dx\ dz}}}},$ ${\int\limits_{0}^{r}\ {\int\limits_{0}^{\sqrt{r^{2} - x^{2}}}\ {\int\limits_{0}^{\sqrt{r^{2} - x^{2} - z^{2}}}{dy\ dz\ dx}}}},$ ${\int\limits_{0}^{r}\ {\int\limits_{0}^{\sqrt{r^{2} - z^{2}}}\ {\int\limits_{0}^{\sqrt{r^{2} - y^{2} - z^{2}}}{dx\ dy\ dz}}}},$ $\int\limits_{0}^{r}\ {\int\limits_{0}^{\sqrt{r^{2} - y^{2}}}\ {\int\limits_{0}^{\sqrt{r^{2} - y^{2} - z^{2}}}{dx\ dz\ dy}}}$
229.
$3$
231.
$\frac{250}{3}$
233.
$\frac{5}{16} \approx 0.313$
235.
$\frac{35}{2}$
Section 5.5 Exercises
241.
$\frac{9\pi}{8}$
243.
$\frac{1}{8}$
245.
$\frac{\pi e^{2}}{6}$
249.
a\. $E = \left\{ \left( {r,\theta,z} \right) \middle| 0 \leq \theta \leq \pi,0 \leq r \leq 4\ \text{sin}\ \theta,0 \leq z \leq \sqrt{16 - r^{2}} \right\};$ b. $\int\limits_{0}^{\pi}\ {\int\limits_{0}^{4\ \text{sin}\ \theta}\ {\int\limits_{0}^{\sqrt{16 - r^{2}}}{f\left( {r,\theta,z} \right)r\ dz\ dr\ d\theta}}}$
251.
a\. $E = \left\{ \left( {r,\theta,z} \right) \middle| 0 \leq \theta \leq \frac{\pi}{2},0 \leq r \leq \sqrt{3},9 - 3r^{2} \leq z \leq 20 - r\left( {\text{cos}\ \theta + \text{sin}\ \theta} \right) \right\};$ b. ${\int\limits_{0}^{\pi\text{/}2}\ {\int\limits_{0}^{\sqrt{3}}\ {\int\limits_{9 - 3r^{2}}^{20 - r{({\text{cos}\ \theta + \text{sin}\ \theta})}}{f\left( {r,\theta,z} \right)}}}}r\ dz\ dr\ d\theta$
253.
a\. $E = \left\{ \left( {r,\theta,z} \right) \middle| 0 \leq r \leq 3,0 \leq \theta \leq \frac{\pi}{2},0 \leq z \leq r\ \text{cos}\ \theta + 3 \right\},$ $f\left( {r,\theta,z} \right) = \frac{1}{r\ \text{cos}\ \theta + 3};$ b. $\int\limits_{0}^{3}\ {\int\limits_{0}^{\pi\text{/}2}\ {\int\limits_{0}^{r\ \text{cos}\ \theta + 3}{\frac{r}{r\ \text{cos}\ \theta + 3}dz\ d\theta\ dr = \frac{9\pi}{4}}}}$
255.
a\. $y = r\ \text{cos}\ \theta,z = r\ \text{sin}\ \theta,x = z,$ $E = \left\{ \left( {r,\theta,z} \right) \middle| 1 \leq r \leq 3,0 \leq \theta \leq 2\pi,0 \leq z \leq 9 - r^{2} \right\},f\left( {r,\theta,z} \right) = z;$ b. $\int\limits_{1}^{3}\ {\int\limits_{0}^{2\pi}\ {\int\limits_{0}^{9 - r^{2}}{zr\ dz\ d\theta\ dr = \frac{256\pi}{3}}}}$
257.
$\pi$
259.
$\frac{\pi}{3}$
261.
$\frac{\pi}{4}$
263.
$\frac{2\pi}{3}$
265.
$V = \frac{\pi}{12} \approx 0.2618$
267.
${\int\limits_{0}^{1}\ {\int\limits_{0}^{\pi}\ {\int\limits_{r^{2}}^{r}{zr^{2}\text{cos}\ \theta}}}}\ dz\ d\theta\ dr$
269.
$180\pi\sqrt{10}$
271.
$\frac{81\pi\left( {\pi - 2} \right)}{16}$
277.
a\. $f\left( {\rho,\theta,\varphi} \right) = \rho\ \text{sin}\ \varphi\left( {\text{cos}\ \theta + \text{sin}\ \theta} \right),$ $E = \left\{ \left( {\rho,\theta,\varphi} \right) \middle| 1 \leq \rho \leq 2,0 \leq \theta \leq \pi,0 \leq \varphi \leq \frac{\pi}{2} \right\};$ b. $\int\limits_{0}^{\pi}\ \int\limits_{0}^{\pi\text{/}2}\ \int\limits_{1}^{2}\rho^{3}\text{sin}^{2}\ \varphi\left( {\text{cos}\theta + \text{sin}\theta} \right) = \frac{15\pi}{8}$
279.
a\. $f\left( {\rho,\theta,\varphi} \right) = \rho\ \text{cos}\ \varphi;$ $E = \left\{ \left( {\rho,\theta,\varphi} \right) \middle| 0 \leq \rho \leq 2\ \text{cos}\ \varphi,0 \leq \theta \leq 2\pi,0 \leq \varphi \leq \frac{\pi}{4} \right\};$ b. ${\int\limits_{0}^{2\pi}\ {\int\limits_{0}^{\pi\text{/}4}\ {\int\limits_{0}^{2\ \text{cos}\ \varphi}{\rho^{3}\text{sin}\ \varphi}}}}\text{cos}\ \varphi\ d\rho\ d\varphi\ d\theta = \frac{7\pi}{6}$
281.
$\pi$
283.
$9\pi\left( {\sqrt{2} - 1} \right)$
285.
${\int\limits_{0}^{\pi\text{/}2}\ {\int\limits_{0}^{\pi}\ {\int\limits_{0}^{4}{\rho^{6}\text{sin}\ \varphi}}}}\ d\rho\ d\varphi\ d\theta$
287.
$V = \frac{4\pi\sqrt{3}}{3} \approx 7.255$
289.
$\frac{243\pi}{32}$
291.
${\int\limits_{0}^{2\pi}\ {\int\limits_{2}^{4}\ {\int\limits_{\text{−}\sqrt{16 - r^{2}}}^{\sqrt{16 - r^{2}}}{r\ dz\ dr\ d\theta}}}};$ $\int\limits_{\pi\text{/}6}^{5\pi\text{/}6}\ {\int\limits_{0}^{2\pi}\ {\int\limits_{2\ \text{csc}\ \varphi}^{4}{\rho^{2}\text{sin}\ \varphi\ d\rho\ d\theta\ d\varphi}}}$
293.
$P = \frac{64P_{0}\pi}{3}$ watts
295.
$Q = kr^{4}\pi\mu C$
Section 5.6 Exercises
297.
$\frac{27}{2}$
299.
$24\sqrt{2}$
301.
$76$
303.
$8\pi$
305.
$\frac{\pi}{2}$
307.
$2$
309.
a\. $M_{x} = \frac{81}{5},M_{y} = \frac{162}{5};$ b. $\overset{\text{−}}{x} = \frac{12}{5},\overset{\text{−}}{y} = \frac{6}{5};$
c.
311.
a\. $M_{x} = \frac{216\sqrt{2}}{5},M_{y} = \frac{432\sqrt{2}}{5};$ b. $\overset{\text{−}}{x} = \frac{18}{5},\overset{\text{−}}{y} = \frac{9}{5};$
c.
313.
a\. $M_{x} = \frac{368}{5},M_{y} = \frac{1552}{5};$ b. $\overset{\text{−}}{x} = \frac{388}{95},\overset{\text{−}}{y} = \frac{92}{95};$
c.
315.
a\. $M_{x} = 16\pi,M_{y} = 8\pi;$ b. $\overset{\text{−}}{x} = 1,\overset{\text{−}}{y} = 2;$
c.
317.
a\. $M_{x} = 0,M_{y} = 0;$ b. $\overset{\text{−}}{x} = 0,\overset{\text{−}}{y} = 0;$
c.
319.
a\. $M_{x} = 2,M_{y} = 0;$ b. $\overset{\text{−}}{x} = 0,\overset{\text{−}}{y} = 1;$
c.
321.
a\. $I_{x} = \frac{243}{10},I_{y} = \frac{486}{5},\text{and}\ I_{0} = \frac{243}{2};$ b. $R_{x} = \frac{3\sqrt{5}}{5},R_{y} = \frac{6\sqrt{5}}{5},\text{and}\ R_{0} = 3$
323.
a\. $I_{x} = \frac{648\sqrt{2}}{7},I_{y} = \frac{2592\sqrt{2}}{7},\text{and}\ I_{0} = \frac{3240\sqrt{2}}{7};$ b. $R_{y} = \frac{3\sqrt{21}}{7},R_{x} = \frac{6\sqrt{21}}{7},\text{and}\ R_{0} = \frac{3\sqrt{105}}{7}$
325.
a\. $I_{x} = 88,I_{y} = 1560,\text{and}\ I_{0} = 1648;$ b. $R_{x} = \frac{\sqrt{418}}{19},R_{y} = \frac{\sqrt{7410}}{19},$ and $R_{0} = \frac{2\sqrt{1957}}{19}$
327.
a\. $I_{x} = \frac{128\pi}{3},I_{y} = \frac{56\pi}{3},\text{and}\ I_{0} = \frac{184\pi}{3};$ b. $R_{x} = \frac{4\sqrt{3}}{3},R_{y} = \frac{\sqrt{21}}{3},$ and $R_{0} = \frac{\sqrt{69}}{3}$
329.
a\. $I_{x} = \frac{\pi}{32},I_{y} = \frac{\pi}{8},\text{and}\ I_{0} = \frac{5\pi}{32};$ b. $R_{x} = \frac{1}{4},R_{y} = \frac{1}{2},\text{and}\ R_{0} = \frac{\sqrt{5}}{4}$
331.
a\. $I_{x} = \frac{7}{3},I_{y} = \frac{1}{3},\text{and}\ I_{0} = \frac{8}{3};$ b. $R_{x} = \frac{\sqrt{42}}{6},R_{y} = \frac{\sqrt{6}}{6},\text{and}\ R_{0} = \frac{2\sqrt{3}}{3}$
333.
$m = \frac{1}{3}$
337.
a\. $m = \frac{9\pi}{8};$ b. $M_{xy} = \frac{3\pi}{4},M_{xz} = \frac{9}{2},M_{yz} = \frac{9}{2};$ c. $\overset{\text{−}}{x} = \frac{4}{\pi},\overset{\text{−}}{y} = \frac{4}{\pi},\overset{\text{−}}{z} = \frac{2}{3};$ d. the solid $Q$ and its center of mass are shown in the following figure.
339.
a\. $\overset{\text{−}}{x} = \frac{3\sqrt{2}}{2\pi},\overset{\text{−}}{y} = \frac{3\left( {2 - \sqrt{2}} \right)}{2\pi},\overset{\text{−}}{z} = 0;$ b. the solid $Q$ and its center of mass are shown in the following figure.
343.
$n = -1$
349.
a\. $\rho\left( {x,y,z} \right) = x^{2} + y^{2};$ b. $\frac{16\pi}{7}$
351.
$M_{xy} = \pi\left( {f(0) - f(a) + af^{\prime}(a)} \right)$
355.
$I_{x} = I_{y} = I_{z} \simeq 0.84$
Section 5.7 Exercises
357.
a\. $T\left( {u,v} \right) = \left( {g\left( {u,v} \right),h\left( {u,v} \right)} \right),x = g\left( {u,v} \right) = \frac{u}{2}$ and $y = h\left( {u,v} \right) = \frac{v}{3}.$ The functions $g$ and $h$ are continuous and differentiable, and the partial derivatives $g_{u}\left( {u,v} \right) = \frac{1}{2},$ $g_{v}\left( {u,v} \right) = 0,h_{u}\left( {u,v} \right) = 0\ \text{and}\ h_{v}\left( {u,v} \right) = \frac{1}{3}$ are continuous on $S;$ b. $T\left( {0,0} \right) = \left( {0,0} \right),$ $T\left( {1,0} \right) = \left( {\frac{1}{2},0} \right),T\left( {0,1} \right) = \left( {0,\frac{1}{3}} \right),$ and $T\left( {1,1} \right) = \left( {\frac{1}{2},\frac{1}{3}} \right);$ c. $R$ is the rectangle of vertices $\left( {0,0} \right),\left( {\frac{1}{2},0} \right),\left( {\frac{1}{2},\frac{1}{3}} \right),\text{and}\ \left( {0,\frac{1}{3}} \right)$ in the $xy\text{-plane;}$ the following figure.
359.
a\. $T\left( {u,v} \right) = \left( {g\left( {u,v} \right),h\left( {u,v} \right)} \right),x = g\left( {u,v} \right) = 2u - v,$ and $y = h\left( {u,v} \right) = u + 2v.$ The functions $g$ and $h$ are continuous and differentiable, and the partial derivatives $g_{u}\left( {u,v} \right) = 2,$ $g_{v}\left( {u,v} \right) = -1,$ $h_{u}\left( {u,v} \right) = 1,$ and $h_{v}\left( {u,v} \right) = 2$ are continuous on $S;$ b. $T\left( {0,0} \right) = \left( {0,0} \right),$ $T\left( {1,0} \right) = \left( {2,1} \right),$ $T\left( {0,1} \right) = \left( {-1,2} \right),$ and $T\left( {1,1} \right) = \left( {1,3} \right);$ c. $R$ is the sqaure of vertices $\left( {0,0} \right),\left( {2,1} \right),\left( {1,3} \right),\text{and}\ \left( {-1,2} \right)$ in the $xy\text{-plane;}$ see the following figure.
361.
a\. $T\left( {u,v} \right) = \left( {g\left( {u,v} \right),h\left( {u,v} \right)} \right),x = g\left( {u,v} \right) = u^{3},$ and $y = h\left( {u,v} \right) = v^{3}.$ The functions $g$ and $h$ are continuous and differentiable, and the partial derivatives $g_{u}\left( {u,v} \right) = 3u^{2},$ $g_{v}\left( {u,v} \right) = 0,$ $h_{u}\left( {u,v} \right) = 0,$ and $h_{v}\left( {u,v} \right) = 3v^{2}$ are continuous on $S;$ b. $T\left( {0,0} \right) = \left( {0,0} \right),$ $T\left( {1,0} \right) = \left( {1,0} \right),$ $T\left( {0,1} \right) = \left( {0,1} \right),$ and $T\left( {1,1} \right) = \left( {1,1} \right);$ c. $R$ is the unit square in the $xy\text{-plane;}$ see the following figure.
363.
$T$ is not one-to-one: two points of $S$ have the same image. Indeed, $T\left( {-2,0} \right) = T\left( {2,0} \right) = \left( {16,4} \right).$
365.
$T$ is one-to-one: We argue by contradiction. $T\left( {u_{1},v_{1}} \right) = T\left( {u_{2},v_{2}} \right)$ implies $2u_{1} - v_{1} = 2u_{2} - v_{2}$ and $u_{1} = u_{2}.$ Thus, $u_{1} = u_{2}$ and $v_{1} = v_{2}.$
367.
$T$ is not one-to-one: $T\left( {1,v,w} \right) = \left( {-1,v,w} \right)$
369.
$u = \frac{x - 2y}{3},v = \frac{x + y}{3}$
371.
$u = e^{x},v = e^{\text{−}x + y}$
373.
$u = \frac{x - y + z}{2},v = \frac{x + y - z}{2},w = \frac{\text{−}x + y + z}{2}$
375.
$S = \left\{ {\left. \left( {u,v} \right) \right|u^{2} + v^{2} \leq 1} \right\}$
377.
$R = \left\{ {\left. \left( {u,v,w} \right) \right|u^{2} - v^{2} - w^{2} \leq 1,w > 0} \right\}$
379.
$\frac{3}{2}$
381.
$-1$
383.
$2uv$
385.
$\frac{2vw}{u^{2}}\text{or}\frac{2w\left( {u + w^{2}} \right)}{u^{2}}$
387.
$2$
389.
a\. $T\left( {u,v} \right) = \left( {2u + v,3v} \right);$ b. The area of $R$ is
$A(R) = {\int\limits_{0}^{3}\ {\int\limits_{y\text{/}3}^{{({6 - y})}\text{/}3}{dx\ dy =}}}{\int\limits_{0}^{1}\ {\int\limits_{0}^{1 - u}{\left| \frac{\partial\left( {x,y} \right)}{\partial\left( {u,v} \right)} \right|dv\ du =}}}{\int\limits_{0}^{1}\ {\int\limits_{0}^{1 - u}{6dv\ du =}}}3.$
391.
$- \frac{1}{4}$
393.
$-1 + \text{cos}\ 2$
395.
$\frac{\pi}{15}$
397.
$\frac{31}{5}$
399.
$T\left( {r,\theta,z} \right) = \left( {r\ \text{cos}\ \theta,r\ \text{sin}\ \theta,z} \right);S = \left\lbrack {0,3} \right\rbrack\ \times \ \left\lbrack {0,\frac{\pi}{2}} \right\rbrack\ \times \ \left\lbrack {0,1} \right\rbrack$ in the $r\theta z\text{-space}$
403.
The area of the region is $~\frac{\ln 2}{2}.~$ A graph of the region is:
405.
$8$
409.
a\. $R = \left\{ \left( {x,y} \right) \middle| y^{2} + x^{2} - 2y - 4x + 1 \leq 0 \right\};$ b. $R$ is graphed in the following figure;
c. $3.16$
411.
a\. $T_{0,2} \circ T_{3,0}\left( {u,v} \right) = \left( {u + 3v,2u + 7v} \right);$ b. The image $R$ is the quadrilateral of vertices $\left( {0,0} \right),\left( {3,7} \right),\left( {2,4} \right),\text{and}\ \left( {4,9} \right);$ c. $S$ is graphed in the following figure;
d. $\frac{3}{2}$
413.
$\frac{2662}{3\pi} \simeq 282.45{\ \text{in}}^{3}$
415.
$A(R) = 2,177,216~\text{yd}^{2}$
Review Exercises
417.
True.
419.
False.
421.
0
423.
$\frac{1}{4}$
425.
1.475
427.
$\frac{52}{3}\pi$
429.
$\frac{\pi}{16}$
431.
$16\pi \approx 50.265$
433.
$\left( {\frac{8}{15},\frac{8}{15}} \right)$
435.
$\left( {0,0,\frac{4}{5}} \right)$
437.
$1.452\pi\ \times \ 10^{15}$ ft-lb
439.
$y = -1.238\ \times \ 10^{-7}x^{3} + 0.001196x^{2} - 3.666x + 7208;$ average temperature approximately $2800^{\text{°}}C$
441.
$\frac{\pi}{3}$
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- Book title: Calculus Volume 3
- Publication date: Mar 30, 2016
- Location: Houston, Texas
- Book URL: https://openstax.org/books/calculus-volume-3/pages/1-introduction
- Section URL: https://openstax.org/books/calculus-volume-3/pages/chapter-5
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