Gilbert Strang; Edwin “Jed” Herman

Checkpoint

5.1

$V = \sum_{i = 1}^{2} \sum_{j = 1}^{2} f (x_{i j}^{*}, y_{i j}^{*}) Δ A = 0$

5.2

a. 26 b. Answers may vary.

5.3

$- \frac{1340}{3}$

5.4

$\frac{4 - ln 5}{ln 5}$

5.5

$\frac{π}{2}$

5.6

Answers to both parts a. and b. may vary.

5.7

Type I and Type II are expressed as ${(x, y) | 0 \leq x \leq 2, x^{2} \leq y \leq 2 x}$ and ${(x, y) | 0 \leq y \leq 4, \frac{1}{2} y \leq x \leq \sqrt{y}},$ respectively.

5.8

$π / 4$

5.9

${(x, y) | 0 \leq y \leq \ln 2, 1 \leq x \leq e^{y}} \cup {(x, y) | \ln 2 \leq y \leq e, 1 \leq x \leq 2} \cup {(x, y) | e \leq y \leq e^{2}, ln y \leq x \leq 2}$

5.10

Same as in the example shown.

5.11

$\frac{216}{35}$

5.12

$\frac{e^{2}}{4} + 10 e - \frac{49}{4}$ cubic units

5.13

$\frac{81}{4}$ square units

5.14

$\frac{3}{4}$

5.15

$\frac{π}{4}$

5.16

$\frac{11}{39} \approx 0.282$

5.17

$\frac{14}{3}$

5.18

$8 π$

5.19

$π / 4$

5.20

$V = \int_{0}^{2 π} \int_{0}^{2 \sqrt{2}} (16 - 2 r^{2}) r d r d θ = 64 π$ cubic units

5.21

$A = 2 \int_{− π / 2}^{π / 6} \int_{1 + sin θ}^{3 - 3 sin θ} r d r d θ = 8 π + 9 \sqrt{3}$

5.22

$\frac{π}{4}$

5.23

$\underset{B}{∭} z sin x cos y d V = 8$

5.24

$\underset{E}{∭} 1 d V = \int_{x = −3}^{x = 3} \int_{y = − \sqrt{9 - x^{2}}}^{y = \sqrt{9 - x^{2}}} \int_{z = − \sqrt{9 - x^{2} - y^{2}}}^{z = \sqrt{9 - x^{2} - y^{2}}} 1 d z d y d x = 36 π .$

5.25

(i) $\int_{z = 0}^{z = 4} \int_{x = 0}^{x = \sqrt{4 - z}} \int_{y = x^{2}}^{y = 4 - z} f (x, y, z) d y d x d z,$ (ii) $\int_{y = 0}^{y = 4} \int_{z = 0}^{z = 4 - y} \int_{x = 0}^{x = \sqrt{y}} f (x, y, z) d x d z d y,$ (iii) $\int_{y = 0}^{y = 4} \int_{x = 0}^{x = \sqrt{y}} \int_{z = 0}^{z = 4 - y} f (x, y, z) d z d x d y,$ (iv) $\int_{x = 0}^{x = 2} \int_{y = x^{2}}^{y = 4} \int_{z = 0}^{z = 4 - y} f (x, y, z) d z d y d x,$ (v) $\int_{x = 0}^{x = 2} \int_{z = 0}^{z = 4 - x^{2}} \int_{y = x^{2}}^{y = 4 - z} f (x, y, z) d y d z d x$

5.26

$f_{ave} = 8$

5.27

$8$

5.28

$\underset{E}{∭} f (r, θ, z) r d z d r d θ = \int_{θ = 0}^{θ = π} \int_{r = 0}^{r = 2 sin θ} \int_{z = 0}^{z = 4 - r sin θ} f (r, θ, z) r d z d r d θ .$

5.29

$E = {(r, θ, z) | 0 \leq θ \leq 2 π, 0 \leq z \leq 1, z \leq r \leq 2 - z^{2}}$ and $V = \int_{r = 0}^{r = 1} \int_{z = r}^{z = 2 - r^{2}} \int_{θ = 0}^{θ = 2 π} r d θ d z d r .$

5.30

$E_{2} = {(r, θ, z) | 0 \leq θ \leq 2 π, 0 \leq r \leq 1, r \leq z \leq \sqrt{4 - r^{2}}}$ and $V = \int_{r = 0}^{r = 1} \int_{z = r}^{z = \sqrt{4 - r^{2}}} \int_{θ = 0}^{θ = 2 π} r d θ d z d r .$

5.31

$V (E) = \int_{θ = 0}^{θ = 2 π} \int_{ϕ = 0}^{φ = π / 3} \int_{ρ = 0}^{ρ = 2} ρ^{2} sin φ d ρ d φ d θ$

5.32

Rectangular: $\int_{x = −2}^{x = 2} \int_{y = − \sqrt{4 - x^{2}}}^{y = \sqrt{4 - x^{2}}} \int_{z = − \sqrt{4 - x^{2} - y^{2}}}^{z = \sqrt{4 - x^{2} - y^{2}}} d z d y d x - \int_{x = −1}^{x = 1} \int_{y = − \sqrt{1 - x^{2}}}^{y = \sqrt{1 - x^{2}}} \int_{z = − \sqrt{4 - x^{2} - y^{2}}}^{z = \sqrt{4 - x^{2} - y^{2}}} d z d y d x .$
Cylindrical: $\int_{θ = 0}^{θ = 2 π} \int_{r = 1}^{r = 2} \int_{z = − \sqrt{4 - r^{2}}}^{z = \sqrt{4 - r^{2}}} r d z d r d θ .$
Spherical: $\int_{φ = π / 6}^{φ = 5 π / 6} \int_{θ = 0}^{θ = 2 π} \int_{ρ = csc φ}^{ρ = 2} ρ^{2} sin φ d ρ d θ d φ .$

5.33

$\frac{9 π}{8} kg$

5.34

$M_{x} = \frac{81 π}{64}$ and $M_{y} = \frac{81 π}{64}$

5.35

$\overset{−}{x} = \frac{M_{y}}{m} = \frac{81 π / 64}{9 π / 8} = \frac{9}{8}$ and $\overset{−}{y} = \frac{M_{x}}{m} = \frac{81 π / 64}{9 π / 8} = \frac{9}{8} .$

5.36

$\overset{−}{x} = \frac{M_{y}}{m} = \frac{1 / 20}{1 / 12} = \frac{3}{5}$ and $\overset{−}{y} = \frac{M_{x}}{m} = \frac{1 / 24}{1 / 12} = \frac{1}{2}$

5.37

$x_{c} = \frac{M_{y}}{m} = \frac{1 / 15}{1 / 6} = \frac{2}{5} and y_{c} = \frac{M_{x}}{m} = \frac{1 / 12}{1 / 6} = \frac{1}{2}$

5.38

$I_{x} = \int_{x = 0}^{x = 2} \int_{y = 0}^{y = x} y^{2} \sqrt{x y} d y d x = \frac{64}{35}$ and $I_{y} = \int_{x = 0}^{x = 2} \int_{y = 0}^{y = x} x^{2} \sqrt{x y} d y d x = \frac{64}{35} .$ Also, $I_{0} = \int_{x = 0}^{x = 2} \int_{y = 0}^{y = x} (x^{2} + y^{2}) \sqrt{x y} d y d x = \frac{128}{35} .$

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

$(\frac{3}{2}, \frac{9}{8}, \frac{1}{2})$

5.42

The moments of inertia of the tetrahedron $Q$ about the $y z -plane,$ the $x z -plane,$ and the $x y -plane$ are $99 / 35, 36 / 7, and 243 / 35,$ respectively.

5.43

$T^{−1} (x, y) = (u, v)$ where $u = \frac{3 x - y}{3}$ and $v = \frac{y}{3}$

5.44

$J (u, v) = \frac{\partial (x, y)}{\partial (u, v)} = | \begin{array}{l} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array} | = | \begin{array}{l} 1 & 1 \\ 0 & 2 \end{array} | = 2$

5.45

$\int_{0}^{π / 2} \int_{0}^{1} r^{3} d r d θ$

5.46

$x = \frac{1}{2} (v + u)$ and $y = \frac{1}{2} (v - u)$ and $\int_{2}^{4} \int_{−u}^{u} \frac{4}{u^{2}} (\frac{1}{2}) d v d u .$

5.47

$\frac{1}{2} (sin 2 - 2)$

5.48

$\int_{0}^{3} \int_{0}^{2} \int_{1}^{2} (\frac{v}{3} + \frac{v w}{3 u}) d u d v d w = 2 + ln 8$

Section 5.1 Exercises

1.

27.

3.

0.

5.

21.3.

7.

a. 28 ${ft}^{3}$ b. 1.75 ft.

9.

a. $0.112$ b. $f_{ave} ≃ 0.175;$ here $f (0.4, 0.2) ≃ 0.1,$ $f (0.2, 0.6) ≃ −0.2,$ $f (0.8, 0.2) ≃ 0.6,$ and $f (0.8, 0.6) ≃ 0.2 .$

11.

$2 π .$

13.

40.

15.

$\frac{81}{2} + 39 \sqrt[3]{2} .$

17.

$e - 1 .$

19.

$15 - \frac{10 \sqrt{2}}{9} .$

21.

0.

23.

$(e - 1) (1 + sin 1 - cos 1) .$

25.

$\frac{3}{4} ln (\frac{5}{3}) + 2 {ln}^{2} 2 - ln 2 .$

27.

$\frac{1}{8} [(2 \sqrt{3} - 3) π + 6 ln 2] .$

29.

$\frac{1}{4} e^{4} (e^{4} - 1) .$

31.

$4 (e - 1) (2 - \sqrt{e}) .$

33.

$- \frac{π}{4} + ln (\frac{5}{4}) - \frac{1}{2} ln 2 + arctan 2 .$

35.

$\frac{1}{2} .$

37.

$\frac{1}{2} (2 cosh 1 + cosh 2 - 3) .$

49.

a. $f (x, y) = \frac{1}{2} x y (x^{2} + y^{2})$ b. $V = \int_{0}^{1} \int_{0}^{1} f (x, y) d x d y = \frac{1}{8}$ c. $f_{ave} = \frac{1}{8};$
d.

In xyz space, a plane is formed at z = 1/8, and there is another shape that starts at the origin, increases through the plane in a line roughly running from (1, 0.25, 0.125) to (0.25, 1, 0.125), and then rapidly increases to (1, 1, 1).

53.

a. For $m = n = 2,$ $I = 4 e^{−0.5} \approx 2.43$ b. $f_{ave} = e^{−0.5} ≃ 0.61;$
c.

In xyz space, a plane is formed at z = 0.61, and there is another shape with maximum roughly at (0, 0, 0.92), which decreases along all the sides to the points (plus or minus 1, plus or minus 1, 0.12).

55.

a. $\frac{2}{n + 1} + \frac{1}{4}$ b. $\frac{1}{4}$

59.

$56.5 °$ 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 $[a, b]$ and $[c, d],$ respectively.

Section 5.2 Exercises

61.

$\frac{27}{20}$

63.

Type I but not Type II

65.

$\frac{π}{2}$

67.

$\frac{1}{6} (8 + 3 π)$

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.

$\frac{π}{2}$

77.

$0$

79.

$\frac{2}{3}$

81.

$\frac{41}{20}$

83.

$−63$

85.

$π$

87.

a. Answers may vary; b. $\frac{2}{3}$

89.

a. Answers may vary; b. $\frac{7}{3}$

91.

$\frac{8 π}{3}$

93.

$e - \frac{3}{2}$

95.

$\frac{1}{3}$

97.

$\int_{0}^{1} \int_{x - 1}^{1 - x} x d y d x = \int_{−1}^{0} \int_{0}^{y + 1} x d x d y + \int_{0}^{1} \int_{0}^{1 - y} x d x d y = \frac{1}{3}$

99.

$\int_{−1}^{1} \int_{− \sqrt{1 - y^{2}}}^{\sqrt{1 - y^{2}}} y d x d y = \int_{−1}^{1} \int_{− \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y d y d x = 0$

101.

$\iint_{D} (x^{2} - y^{2}) d A = \int_{−1}^{1} \int_{y^{4} - 1}^{1 - y^{4}} (x^{2} - y^{2}) d x d y = \frac{464}{4095}$

103.

$\frac{4}{5}$

105.

$\frac{5 π}{32}$

109.

$1$

111.

$2$

113.

a. $\frac{1}{3};$ b. $\frac{1}{6};$ c. $\frac{1}{6}$

115.

a. $\frac{4}{3};$ b. $2 π;$ c. $\frac{6 π - 4}{3}$

117.

$0 and 0.865474;$ $A (D) = 0.621135$

119.

$P [X + Y \leq 6] = 1 + \frac{3}{2 e^{2}} - \frac{5}{e^{6 / 5}} \approx 0.45;$ there is a $45 %$ chance that a customer will spend $6$ minutes in the drive-thru line.

Section 5.3 Exercises

123.

$D = {(r, θ) | 4 \leq r \leq 5, \frac{π}{2} \leq θ \leq π}$

125.

$D = {(r, θ) | 0 \leq r \leq \sqrt{2}, 0 \leq θ \leq π}$

127.

$D = {(r, θ) | 0 \leq r \leq 4 sin θ, 0 \leq θ \leq π}$

129.

$D = {(r, θ) | 3 \leq r \leq 5, \frac{π}{4} \leq θ \leq \frac{π}{2}}$

131.

$D = {(r, θ) | 3 \leq r \leq 5, \frac{3 π}{4} \leq θ \leq \frac{5 π}{4}}$

133.

$D = {(r, θ) | 0 \leq r \leq tan θ sec θ, 0 \leq θ \leq \frac{π}{4}}$

135.

$0$

137.

$\frac{63 π}{16}$

139.

$\frac{3367 π}{18}$

141.

$\frac{35 π^{2}}{576}$

143.

$\frac{7 π^{2}}{576} (21 - e^{2} + e^{4})$

145.

$\frac{5}{2} ln (1 + \sqrt{2})$

147.

$\frac{1}{6} (2 - \sqrt{2})$

149.

$\int_{0}^{π} \int_{0}^{2} r^{5} d r d θ = \frac{32 π}{3}$

151.

$\int_{− π / 2}^{π / 2} \int_{0}^{4} r sin (r^{2}) d r d θ = π {sin}^{2} 8$

153.

$\frac{3 π}{4}$

155.

$\frac{π}{2}$

157.

$\frac{1}{3} (4 π - 3 \sqrt{3})$

159.

$\frac{16}{3 π}$

161.

$\frac{π}{18}$

163.

a. $\frac{2 π}{3};$ b. $\frac{π}{3};$ c. $\frac{π}{3}$

165.

$\frac{256 π}{3} {cm}^{3}$

167.

$\frac{3 π}{32}$

169.

$4 π$

171.

$\frac{π}{4}$

173.

$\frac{1}{2} π e (e - 1)$

175.

$\sqrt{3} - \frac{π}{4}$

177.

$\frac{133 π^{2}}{864}$

Section 5.4 Exercises

181.

$192$

183.

$0$

185.

$\int_{1}^{2} \int_{2}^{3} \int_{0}^{1} (x^{2} + ln y + z) d z d x d y = \frac{35}{6} + 2 ln 2$

187.

$\int_{1}^{3} \int_{0}^{4} \int_{−1}^{2} (x^{2} z + \frac{1}{y}) d z d x d y = 64 + 12 ln 3$

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.

$\frac{- 609 - 216 \sqrt{3} - 80 π}{5760} \approx - 0.21431$

207.

$\frac{3 π}{2}$

209.

$1250$

211.

$\int_{0}^{5} \int_{−3}^{3} \int_{0}^{\sqrt{9 - y^{2}}} z d z d y d x = 90$

213.

$V = 5.33$

A complex shape that starts at the origin and reaches its maximum at (negative 2, negative 2, 8). The shape is truncated by the x = y plane, the x = 0 plane, the y = negative 2 plane, the z = 0 plane, and a complex triangular-like shape with curved edges and sides (negative 2, negative 2, 8), (0, 0, 0), and (0, negative 2, 4).

215.

$\int_{0}^{1} \int_{1}^{3} \int_{2}^{4} (y^{2} z^{2} + 1) d z d x d y;$ $\int_{0}^{1} \int_{1}^{3} \int_{2}^{4} (x^{2} y^{2} + 1) d y d z d x$

217.

$\int_{0}^{1} \int_{-z}^{z} \int_{0}^{1 - y^{4} - z^{4}} e^{x} d x d y d z; \int_{0}^{1} \int_{-x}^{x} \int_{0}^{1 - z^{4} - x^{4}} \ln z d y d z d x$

219.

$V = \int_{a^{2}}^{- a^{2}} \int_{- \sqrt{a^{2} - z^{2}}}^{\sqrt{a^{2} - z^{2}}} \int_{\sqrt{x^{2} + z^{2}}}^{a^{2}} d y d x d z$

221.

$\frac{9}{2}$

223.

$\frac{156}{5}$

225.

a. Answers may vary; b. $\frac{128}{3}$

227.

a. $\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - x^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - y^{2}}} d z d y d x;$ b. $\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - y^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - y^{2}}} d z d x d y,$ $\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - z^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - z^{2}}} d y d x d z,$ $\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - x^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - z^{2}}} d y d z d x,$ $\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - z^{2}}} \int_{0}^{\sqrt{r^{2} - y^{2} - z^{2}}} d x d y d z,$ $\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - y^{2}}} \int_{0}^{\sqrt{r^{2} - y^{2} - z^{2}}} d x d z d y$

229.

$3$

231.

$\frac{250}{3}$

233.

$\frac{5}{16} \approx 0.313$

235.

$\frac{35}{2}$

Section 5.5 Exercises

241.

$\frac{9 π}{8}$

243.

$\frac{1}{8}$

245.

$\frac{π e^{2}}{6}$

249.

a. $E = {(r, θ, z) | 0 \leq θ \leq π, 0 \leq r \leq 4 sin θ, 0 \leq z \leq \sqrt{16 - r^{2}}};$ b. $\int_{0}^{π} \int_{0}^{4 sin θ} \int_{0}^{\sqrt{16 - r^{2}}} f (r, θ, z) r d z d r d θ$

251.

a. $E = {(r, θ, z) | 0 \leq θ \leq \frac{π}{2}, 0 \leq r \leq \sqrt{3}, 9 - 3 r^{2} \leq z \leq 20 - r (cos θ + sin θ)};$ b. $\int_{0}^{π / 2} \int_{0}^{\sqrt{3}} \int_{9 - 3 r^{2}}^{20 - r (cos θ + sin θ)} f (r, θ, z) r d z d r d θ$

253.

a. $E = {(r, θ, z) | 0 \leq r \leq 3, 0 \leq θ \leq \frac{π}{2}, 0 \leq z \leq r cos θ + 3},$ $f (r, θ, z) = \frac{1}{r cos θ + 3};$ b. $\int_{0}^{3} \int_{0}^{π / 2} \int_{0}^{r cos θ + 3} \frac{r}{r cos θ + 3} d z d θ d r = \frac{9 π}{4}$

255.

a. $y = r cos θ, z = r sin θ, x = z,$ $E = {(r, θ, z) | 1 \leq r \leq 3, 0 \leq θ \leq 2 π, 0 \leq z \leq 1 - r^{2}}, f (r, θ, z) = z;$ b. $\int_{1}^{3} \int_{0}^{2 π} \int_{0}^{1 - r^{2}} z r d z d θ d r = \frac{256 π}{3}$

257.

$π$

259.

$\frac{π}{3}$

261.

$\frac{π}{4}$

263.

$\frac{2 π}{3}$

265.

$V = \frac{π}{12} \approx 0.2618$

A quarter section of an ellipsoid with width 2, height 1, and depth 1.

267.

$\int_{0}^{1} \int_{0}^{π} \int_{r^{2}}^{r} z r^{2} cos θ d z d θ d r$

269.

$180 π \sqrt{10}$

271.

$\frac{81 π (π - 2)}{16}$

277.

a. $f (ρ, θ, φ) = ρ sin φ (cos θ + sin θ),$ $E = {(ρ, θ, φ) | 1 \leq ρ \leq 2, 0 \leq θ \leq π, 0 \leq φ \leq \frac{π}{2}};$ b. $\int_{0}^{π} \int_{0}^{π / 2} \int_{1}^{2} ρ^{3} {sin}^{2} φ (cos θ + sin θ) = \frac{15 π}{8}$

279.

a. $f (ρ, θ, φ) = ρ cos φ;$ $E = {(ρ, θ, φ) | 0 \leq ρ \leq 2 cos φ, 0 \leq θ \leq 2 π, 0 \leq φ \leq \frac{π}{4}};$ b. $\int_{0}^{2 π} \int_{0}^{π / 4} \int_{0}^{2 cos φ} ρ^{3} sin φ cos φ d ρ d φ d θ = \frac{7 π}{6}$

281.

$π$

283.

$9 π (\sqrt{2} - 1)$

285.

$\int_{0}^{π / 2} \int_{0}^{π} \int_{0}^{4} ρ^{6} sin φ d ρ d φ d θ$

287.

$V = \frac{4 π \sqrt{3}}{3} \approx 7.255$

A quarter section of an ellipsoid with width 2, height 2, and depth 1

289.

$\frac{243 π}{32}$

291.

$\int_{0}^{2 π} \int_{2}^{4} \int_{− \sqrt{16 - r^{2}}}^{\sqrt{16 - r^{2}}} r d z d r d θ;$ $\int_{π / 6}^{5 π / 6} \int_{0}^{2 π} \int_{2 csc φ}^{4} ρ^{2} sin ρ d ρ d θ d φ$

293.

$P = \frac{64 P_{0} π}{3}$ watts

295.

$Q = k r^{4} π μ C$

Section 5.6 Exercises

297.

$\frac{27}{2}$

299.

$24 \sqrt{2}$

301.

$76$

303.

$8 π$

305.

$\frac{π}{2}$

307.

$2$

309.

a. $M_{x} = \frac{81}{5}, M_{y} = \frac{162}{5};$ b. $\overset{−}{x} = \frac{12}{5}, \overset{−}{y} = \frac{6}{5};$
c.

A triangular region R bounded by the x and y axes and the line y = negative x/2 + 3, with a point marked at (12/5, 6/5).

311.

a. $M_{x} = \frac{216 \sqrt{2}}{5}, M_{y} = \frac{432 \sqrt{2}}{5};$ b. $\overset{−}{x} = \frac{18}{5}, \overset{−}{y} = \frac{9}{5};$
c.

A rectangle R bounded by the x and y axes and the lines x = 6 and y = 3 with point marked (18/5, 9/5).

313.

a. $M_{x} = \frac{368}{5}, M_{y} = \frac{1552}{5};$ b. $\overset{−}{x} = \frac{388}{95}, \overset{−}{y} = \frac{92}{95};$
c.

A trapezoid R bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5 with the point marked (92/95, 388/95).

315.

a. $M_{x} = 16 π, M_{y} = 8 π;$ b. $\overset{−}{x} = 1, \overset{−}{y} = 2;$
c.

A circle with radius 2 centered at (1, 2), which is tangent to the x axis at (1, 0) and has pointed marked at the center (1, 2).

317.

a. $M_{x} = 0, M_{y} = 0;$ b. $\overset{−}{x} = 0, \overset{−}{y} = 0;$
c.

An ellipse R with center the origin, major axis 2, and minor axis 0.5, with point marked at the origin.

319.

a. $M_{x} = 2, M_{y} = 0;$ b. $\overset{−}{x} = 0, \overset{−}{y} = 1;$
c.

A square R with side length square root of 2 rotated 45 degrees, with corners at the origin, (2, 0), (1, 1), and (negative 1, 1). A point is marked at (0, 1).

321.

a. $I_{x} = \frac{243}{10}, I_{y} = \frac{486}{5}, and I_{0} = \frac{243}{2};$ b. $R_{x} = \frac{3 \sqrt{5}}{5}, R_{y} = \frac{6 \sqrt{5}}{5}, and R_{0} = 3$

323.

a. $I_{x} = \frac{648 \sqrt{2}}{7}, I_{y} = \frac{2592 \sqrt{2}}{7}, and I_{0} = \frac{3240 \sqrt{2}}{7};$ b. $R_{y} = \frac{3 \sqrt{21}}{7}, R_{x} = \frac{6 \sqrt{21}}{7}, and R_{0} = \frac{3 \sqrt{105}}{7}$

325.

a. $I_{x} = 88, I_{y} = 1560, and I_{0} = 1648;$ b. $R_{x} = \frac{\sqrt{418}}{19}, R_{y} = \frac{\sqrt{7410}}{19},$ and $R_{0} = \frac{2 \sqrt{1957}}{19}$

327.

a. $I_{x} = \frac{128 π}{3}, I_{y} = \frac{56 π}{3}, and I_{0} = \frac{184 π}{3};$ b. $R_{x} = \frac{4 \sqrt{3}}{3}, R_{y} = \frac{\sqrt{21}}{3},$ and $R_{0} = \frac{\sqrt{69}}{3}$

329.

a. $I_{x} = \frac{π}{32}, I_{y} = \frac{π}{8}, and I_{0} = \frac{5 π}{32};$ b. $R_{x} = \frac{1}{4}, R_{y} = \frac{1}{2}, and R_{0} = \frac{\sqrt{5}}{4}$

331.

a. $I_{x} = \frac{7}{3}, I_{y} = \frac{1}{3}, and I_{0} = \frac{8}{3};$ b. $R_{x} = \frac{\sqrt{42}}{6}, R_{y} = \frac{\sqrt{6}}{6}, and R_{0} = \frac{2 \sqrt{3}}{3}$

333.

$m = \frac{1}{3}$

337.

a. $m = \frac{9 π}{8};$ b. $M_{x y} = \frac{3 π}{4}, M_{x z} = \frac{9}{2}, M_{y z} = \frac{9}{2};$ c. $\overset{−}{x} = \frac{4}{π}, \overset{−}{y} = \frac{4}{π}, \overset{−}{z} = \frac{2}{3};$ d. the solid $Q$ and its center of mass are shown in the following figure.

A quarter cylinder in the first quadrant with height 1 and radius 3. A point is marked at (9/(2 pi), 9/(2 pi), 2/3).

339.

a. $\overset{−}{x} = \frac{3 \sqrt{2}}{2 π}, \overset{−}{y} = \frac{3 (2 - \sqrt{2})}{2 π}, \overset{−}{z} = 0;$ b. the solid $Q$ and its center of mass are shown in the following figure.

A wedge from a cylinder in the first quadrant with height 2, radius 1, and angle roughly 45 degrees. A point is marked at (3 times the square root of 2/(2 pi), 3 times (2 minus the square root of 2)/(2 pi), 0).

343.

$n = −1$

349.

a. $ρ (x, y, z) = x^{2} + y^{2};$ b. $\frac{16 π}{7}$

351.

$M_{x y} = π (f (0) - f (a) + a f^{'} (a))$

355.

$I_{x} = I_{y} = I_{z} ≃ 0.84$

Section 5.7 Exercises

357.

a. $T (u, v) = (g (u, v), h (u, v)), x = g (u, v) = \frac{u}{2}$ and $y = h (u, v) = \frac{v}{3} .$ The functions $g$ and $h$ are continuous and differentiable, and the partial derivatives $g_{u} (u, v) = \frac{1}{2},$ $g_{v} (u, v) = 0, h_{u} (u, v) = 0 and h_{v} (u, v) = \frac{1}{3}$ are continuous on $S;$ b. $T (0, 0) = (0, 0),$ $T (1, 0) = (\frac{1}{2}, 0), T (0, 1) = (0, \frac{1}{3}),$ and $T (1, 1) = (\frac{1}{2}, \frac{1}{3});$ c. $R$ is the rectangle of vertices $(0, 0), (\frac{1}{2}, 0), (\frac{1}{2}, \frac{1}{3}), and (0, \frac{1}{3})$ in the $x y -plane;$ the following figure.

A rectangle with one corner at the origin, horizontal length 0.5, and vertical height 0.34.

359.

a. $T (u, v) = (g (u, v), h (u, v)), x = g (u, v) = 2 u - v,$ and $y = h (u, v) = u + 2 v .$ The functions $g$ and $h$ are continuous and differentiable, and the partial derivatives $g_{u} (u, v) = 2,$ $g_{v} (u, v) = −1,$ $h_{u} (u, v) = 1,$ and $h_{v} (u, v) = 2$ are continuous on $S;$ b. $T (0, 0) = (0, 0),$ $T (1, 0) = (2, 1),$ $T (0, 1) = (−1, 2),$ and $T (1, 1) = (1, 3);$ c. $R$ is the sqaure of vertices $(0, 0), (2, 1), (1, 3), and (−1, 2)$ in the $x y -plane;$ see the following figure.

A square of side length square root of 5 with one corner at the origin and another at (2, 1).

361.

a. $T (u, v) = (g (u, v), h (u, v)), x = g (u, v) = u^{3},$ and $y = h (u, v) = v^{3} .$ The functions $g$ and $h$ are continuous and differentiable, and the partial derivatives $g_{u} (u, v) = 3 u^{2},$ $g_{v} (u, v) = 0,$ $h_{u} (u, v) = 0,$ and $h_{v} (u, v) = 3 v^{2}$ are continuous on $S;$ b. $T (0, 0) = (0, 0),$ $T (1, 0) = (1, 0),$ $T (0, 1) = (0, 1),$ and $T (1, 1) = (1, 1);$ c. $R$ is the unit square in the $x y -plane;$ see the following figure.

A Cartesian coordinate graph with axes ranging from 0 to 1 on both x (horizontal) and y (vertical) axes, displaying an empty plot area with no data or functions presented.

363.

$T$ is not one-to-one: two points of $S$ have the same image. Indeed, $T (−2, 0) = T (2, 0) = (16, 4) .$

365.

$T$ is one-to-one: We argue by contradiction. $T (u_{1}, v_{1}) = T (u_{2}, v_{2})$ implies $2 u_{1} - v_{1} = 2 u_{2} - v_{2}$ and $u_{1} = u_{2} .$ Thus, $u_{1} = u_{2}$ and $v_{1} = v_{2} .$

367.

$T$ is not one-to-one: $T (1, v, w) = (−1, v, w)$

369.

$u = \frac{x - 2 y}{3}, v = \frac{x + y}{3}$

371.

$u = e^{x}, v = e^{− x + y}$

373.

$u = \frac{x - y + z}{2}, v = \frac{x + y - z}{2}, w = \frac{− x + y + z}{2}$

375.

$S = {(u, v) | u^{2} + v^{2} \leq 1}$

377.

$R = {(u, v, w) | u^{2} - v^{2} - w^{2} \leq 1, w > 0}$

379.

$\frac{3}{2}$

381.

$−1$

383.

$2 u v$

385.

$2 v w$

387.

$2$

389.

a. $T (u, v) = (2 u + v, 3 v);$ b. The area of $R$ is
$A (R) = \int_{0}^{3} \int_{y / 3}^{(6 - y) / 3} d x d y = \int_{0}^{1} \int_{0}^{1 - u} | \frac{\partial (x, y)}{\partial (u, v)} | d v d u = \int_{0}^{1} \int_{0}^{1 - u} 6 d v d u = 3 .$

391.

$- \frac{1}{4}$

393.

$−1 + cos 2$

395.

$\frac{π}{15}$

397.

$\frac{31}{5}$

399.

$T (r, θ, z) = (r cos θ, r sin θ, z); S = [0, 3] \times [0, \frac{π}{2}] \times [0, 1]$ in the $r θ z -space$

403.

The area of $R$ is $10 - 4 \sqrt{6};$ the boundary curves of $R$ are graphed in the following figure.

Four lines are drawn, namely, y = 3, y = 2, y = 3/(x squared), and y = 2/(x squared). The lines y = 3 and y = 2 are parallel to each other. The lines y = 3/(x squared) and y = 2/(x squared) are curves that run somewhat parallel to each other.

405.

$8$

409.

a. $R = {(x, y) | y^{2} + x^{2} - 2 y - 4 x + 1 \leq 0};$ b. $R$ is graphed in the following figure;

A circle with radius 2 and center (2, 1).

c. $3.16$

411.

a. $T_{0, 2} \circ T_{3, 0} (u, v) = (u + 3 v, 2 u + 7 v);$ b. The image $R$ is the quadrilateral of vertices $(0, 0), (3, 7), (2, 4), and (4, 9);$ c. $S$ is graphed in the following figure;