1. A closed, bounded region $R$ that has a boundary consisting of two smooth curves, $y=g_{1}(x)$ and $y=g_{2}(x),$ defined on $a\leq x\leq b,$ where $g_{1}(x)\leq g_{2}(x)$ on $[a,b]$ and possibly a portion of the vertical lines $x=a$ and $x=b,$ is called a(n) ________ region.

2. True or False The double integral of a function that is continuous on a closed, bounded, $y$-simple region $R$ can be expressed as an iterated integral.

3. True or False The region $R$ enclosed by the parabola $y=x^{2}+1$ and the line $y=3x+1$ is both $x$-simple and $y$-simple.

4. Explain why $\int_{0}^{1}\left[ \int_{0}^{x}f(x,y)\, {\it dy}\right]\! {\it dx}\neq \int_{0}^{1}\left[ \int_{0}^{y}f(x,y)\,{\it dx}\right]\! {\it dy}.$

9. $\displaystyle\int_{0}^{1}\left[ \int_{x^{2}}^{\sqrt{x}}\,{\it dy}\right]\! {\it dx}$

$\dfrac13$

10. $\displaystyle\int_{0}^{1}\left[ \int_{y^{2}}^{\sqrt{y}}x\,{\it dx}\right]\! {\it dy}$

11. $\displaystyle\int_{-1}^{2}\left[\int_{y^{2}}^{y+2}\,{\it dx}\right]\! {\it dy}$

$\dfrac92$

12. $\displaystyle\int_{0}^{1}\left[ \int_{x}^{2x}y\,{\it dy}\right]\! {\it dx}$

13. $\displaystyle\int_{0}^{1}\left[ \int_{x}^{3x}xy\,{\it dy}\right]\! {\it dx}$

1

14. $\displaystyle\int_{0}^{1}\left[ \int_{y}^{\sqrt{y}}x^{2}y\,{\it dx}\right]\! {\it dy}$

15. $\int_{0}^{1}\left[ \int_{1}^{e^{y}}\dfrac{y}{x}\,{\it dx}\right]\! {\it dy}$

$\dfrac13$

16. $\int_{0}^{1}\left[\int_{0}^{x^{2}}xe^{y}\,{\it dy}\right]\! {\it dx}$

17. $\int_{0}^{2}\left[ \int_{y}^{2y}xy\,{\it dx}\right]\! {\it dy}$

6

18. $\int_{2}^{4}\left[ \int_{1}^{y^{2}}\dfrac{y}{x^{2}}\,{\it dx}\right]\! {\it dy}$

19. $\int_{0}^{1}\left[ \int_{x^{2}}^{x}\sqrt{xy}\,{\it dy}\right]\! {\it dx}$

$\dfrac2{27}$

20. $\int_{0}^{1}\left[\int_{x^{2}}^{x}\sqrt{x}\,{\it dy}\right]\! {\it dx}$

21. $\int_{0}^{1}\left[ \int_{x}^{1}\dfrac{1}{x^{2}+1}\,{\it dy}\right]\! {\it dx}$

$\dfrac{\pi}4 - \dfrac{\ln 2}2$

22. $\int_{0}^{1}\left[ \int_{y}^{\sqrt{y}}(x^{2}+y^{2})\,{\it dx}\right]\! {\it dy}$

23. $\int_{1}^{2}\left[ \int_{0}^{\ln x}xe^{y}\,{\it dy}\right]\! {\it dx}$

$\dfrac56$

24. $\int_{2}^{3}\left[ \int_{0}^{1/y}\ln y\,{\it dx}\right]\! {\it dy}$

25. $\displaystyle\iint\limits_{\kern-3ptR}(x+y)\,{\it dA},$ where $R$ is the region enclosed by $y=x^{2}$ and $y^{2}=8x$

$\dfrac{36}5$

26. $\displaystyle\iint\limits_{\kern-3ptR}(x^{2}-y^{2})\,{\it dA},$ where $R$ is the region enclosed by $y=x$ and $y=x^{2}$

27. $\displaystyle\iint\limits_{\kern-3ptR}y^{2}\,{\it dA},$ where $R$ is the region enclosed by $y=2-x$ and $y=x^{2}$

$\dfrac{423}{28}$

28. $\displaystyle\iint\limits_{\kern-3ptR}xy\,{\it dA},$ where $R$ is the region enclosed by $y^{2}=x+1$ and $y=1-x$

29. Enclosed by the graphs of $y=x^{3}$ and $y=x^{2}$

30. Enclosed by the graphs of $y=2 \sqrt{x}$ and $y=\dfrac{x^{2}}{4}$

31. Enclosed by the graphs of $y=x^{2}-9$ and $y=9-x^{2}$

32. Enclosed by the graphs of $x^{2}+y^{2}=16$ and $y^{2}=6x$

33. Enclosed by the graph $y=\dfrac{1}{\sqrt{x-1}}$, the $x$ -axis, and the lines $x=2$ and $x=5$

34. Enclosed by the graphs of $y=x^{3/2}$ and $y=x$

35. Enclosed by the line $x+y=3$ and the hyperbola $xy=2$

36. Enclosed by the hyperbola $xy=\sqrt{3}$ and the circle $x^{2}+y^{2}=4$, in the first quadrant only

37. $\int_{0}^{1}\left[ \int_{0}^{x}f(x,y)\,{\it dy}\right]\! {\it dx}$

38. $\int_{0}^{2}\left[ \int_{y^{2}}^{2y}f(x,y)\,{\it dx}\right]\! {\it dy}$

39. $\int_{0}^{a}\left[ \int_{0}^{\sqrt{a^{2}-y^{2}}}f(x,y)\,{\it dx}\right]\! {\it dy}$

40. $\int_{0}^{2\sqrt[3]{2}}\left[ \int_{x^{2}/4}^{\sqrt{x}}f(x,y)\,{\it dy}\right]\! {\it dx}$

41. $\int_{0}^{16}\left[\int^{y^{1/4}}_{y/8} f(x,y)\,{\it dx}\right]\! {\it dy}$

42. $\int_{2}^{5}\left[ \int_{x^{2}-6x+9}^{x-1}f(x,y)\,{\it dy}\right]\! {\it dx}$

43. $\int_{1}^{2}\left[ \int_{0}^{\ln y}f(x,y)\,{\it dx}\right]\! {\it dy}$

44. $\int_{1}^{e}\left[ \int_{\ln x}^{1}f(x,y)\,{\it dy} \right]\! {\it dx}$

45. $\displaystyle\iint\limits_{\kern-3ptR}[ f(x,y) -g(x,y) ] \, {\it dA}$

46. $\displaystyle\iint\limits_{\kern-3ptR}4f(x,y) \, {\it dA}$

47. $\displaystyle\iint\limits_{\kern-3ptR}[ 3f(x,y) +4g(x,y) ] \, {\it dA}$

48. $\displaystyle\iint\limits_{\kern-3ptR} [ 2f(x,y) +5g(x,y) ] \, {\it dA}$

49. Area Find the area of the region $R$ in the first quadrant enclosed by the graphs of $y=6x-x^{2}$ and $y=4x-8$ by partitioning $R$ with a horizontal line through the point $(4,8)$. Refer to Figure 20.

50. Area Use double integration to find the area in the first quadrant enclosed by the parabola $x^{2}=9y$, the $y$-axis, and the circle $x^{2}+y^{2}=10.$

51. The tetrahedron enclosed by the plane $x+2y+z=2$ and the coordinate planes

52. Enclosed by the elliptic paraboloid $4x^{2}+9y^{2}=36$ and the planes $z=0$ and $z=1$

53. Below the paraboloid $z=x^{2}+y^{2}$ and above the triangular region $R$ formed by the $x$- and $y$-axes and the line $x+y=1,$ as shown in the figure below.

    

54. Enclosed by the cylinder $z=9-y^{2}$ and the planes $x=0$, $x=4$, and $z=0$

55. Enclosed by the surface $z=4-x^{2}-y^{2}$ and $z=0$

56. Enclosed by the surfaces $x^{2}+y^{2}=1$, $z=x$, and $z=x^{2}$

57. The solid bounded from above by the graph of $z=x^{2}+y^{2}$, from below by the $xy$-plane, and on the sides by the cylinder $x^{2}+y^{2}=1$, as shown in the figure.

    

58. Enclosed by a portion of the parabolic cylinder $z=\dfrac{x^{2} }{2}$ and the four planes $y=0$, $y=x$, $x=2$, and $z=0$

59. Enclosed by the coordinate planes, the plane $y=3$, and the surface $z=y+1-x^{2}$

60. Enclosed by the surfaces $z=xe^{y}$, $z=0$, $y=0$, $x=1$, and $y=x^{2}$

61. $\int_{0}^{\sqrt{\pi /2}}\left[ \int_{y}^{\sqrt{\pi /2}}\sin x^{2}\,{\it dx}\right]\! {\it dy}$

62. $\int_{0}^{1/2}\left[ \int_{2x}^{1}e^{y^{2}} {\it dy}\right]\! {\it dx}$

63. $\int_{0}^{1}\left[ \int_{0}^{\sqrt{1-x^{2}}}\dfrac{1}{\sqrt{1-y^{2}}}\,{\it dy}\right]\! {\it dx}$

64. $\int_{0}^{1}\left[ \int_{y}^{1}\dfrac{\sin x}{x}\,{\it dx}\right]{\it dy}$

65. $\int_{0}^{1}\left[ \int_{y}^{1}\sqrt{2+x^{2}}\,{\it dx}\right]\! {\it dy}$

66. $\int_{0}^{1}\left[ \int_{\sqrt[3]{x}}^{1}\sqrt{1+y^{4}}~{\it dy}\right]\! {\it dx}$

67. $\int_{0}^{1}\left[ \int_{\sqrt{y}}^{1}e^{y/x}\,{\it dx}\right]\! {\it dy}$

68. $\int_{0}^{1}\left[ \int_{\sin ^{-1}y}^{\pi /2}e^{\cos x}\,{\it dx}\right]\! {\it dy}$

69. Properties of Double Integrals Let $R$ be the region enclosed by $y=1$, $y=-1$, $x=0$, and $x=1$; let $R_{1}$ and $R_{2}$ be the subregions of $R$ in the first and fourth quadrants, respectively. Suppose $f$ is continuous on $R$ and $$\begin{eqnarray*} &&\displaystyle\iint\limits_{\kern-3ptR}3f(x,y)\,{\it dA}-2\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=\displaystyle\iint \limits_{\kern-3ptR_{2}}f(x,y)\,{\it dA}, \\[3pt] &&\displaystyle\iint\limits_{\kern-3ptR_{2}}5f(x,y)\,{\it dA}-2\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=18 \end{eqnarray*}$$

    Find $\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA}$.

70. Properties of Double Integrals Let $R$ be the region enclosed by $y=0$, $y=2$, $x=-2$, and $x=1$; let $R_{1}$ and $R_{2}$ be the subregions of $R$ in the first and second quadrants, respectively. Suppose $f$ is continuous on $R$ and $$\begin{eqnarray*} &&\displaystyle\iint\limits_{\kern-3ptR_{2}}3f(x,y)\,{\it dA}-\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=2\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA},\\[4pt] &&\displaystyle\iint\limits_{\kern-3ptR_{1}}4f(x,y)\,{\it dA}+\displaystyle\iint\limits_{\kern-3ptR_{2}}2f(x,y)\,{\it dA}=7 \end{eqnarray*}$$

    Find $\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA}$.

922

71. Properties of Double Integrals Let $R$ be the region enclosed by $y=x$, $y=-1$, and $x=2$; let $R_{1}$ and $R_{2}$ be the subregions of $R$ above the $x$-axis and below the $x$-axis, respectively. Suppose $f$ is continuous on $R$ and $$\begin{eqnarray*} &&2\displaystyle\iint\limits_{\kern-3ptR_2}f(x,y)\,{\it dA}-7\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=17\\[4pt] &&\displaystyle\iint\limits_{\kern-3ptR_{2}}f(x,y)\,{\it dA}-2\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=7 \end{eqnarray*}$$

    Find $\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA}$.

72. Properties of Double Integrals Let $R$ be the region enclosed by $y=x^{2}+1$, $y=0$, $x=-1$, and $x=2$; let $R_{1}$ and $R_{2}$ be the subregions of $R$ in the first and second quadrants, respectively. Suppose $f$ is continuous on $R$ and $$\begin{eqnarray*} &&\displaystyle\iint\limits_{\kern-3ptR_2}6f(x,y)\,{\it dA}-3\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=-12\\[4pt] &&\displaystyle\iint\limits_{\kern-3ptR_{2}}4f(x,y)\,{\it dA}+2= \displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA} \end{eqnarray*}$$

    Find $\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA}$.

73. $\int_{0}^{1}\left[ \int_{0}^{\sqrt{1-x^{2}}}\sqrt{1-x^{2}-y^2}\,{\it dy}\right]\! {\it dx}$

74. $\int_{0}^{2}\left[ \int_{0}^{\sqrt{4-y^{2}}}(4-x^2-y^2)\,{\it dx}\right]\! {\it dy}$

75. $\int_{0}^{4}\left[ \int_{0}^{1}3\,{\it dy}\right]\! {\it dx}$

76. $\int_{0}^{1}\left[ \int_{0}^{2}2\,{\it dy}\right]\! {\it dx}$

77. Find:

    1. (a) $\displaystyle\iint\limits_{\kern-3ptR}x\,{\it dA},$ where $R$ is the region enclosed by the circle of radius 1 centered at the origin.
    2. (b) $4\displaystyle\iint\limits_{\kern-3ptR_{1}}x\,{\it dA},$ where $R_{1}$ is the region in the first quadrant enclosed by the circle of radius 1 centered at the origin (This shows that although the region over which we are integrating is symmetric about the $x$-axis and the $y$-axis, we must also consider properties of the integrand on this region before we use symmetry.)

78. 1. (a) Find $$\int_{1}^{e}\left[ \int_{0}^{\ln x}\dfrac{{\it dy}}{x}\right]\! {\it dx}.$$
    2. (b) Reverse the order of integration and find the resulting iterated integral.

79. 1. (a) Find $$\int_{0}^{\pi /2}\left[ \int_{0}^{\cos x}\sin x\,{\it dy}\right]\! {\it dx}.$$
    2. (b) Reverse the order of integration and find the resulting iterated integral.

80. Volume of a Liquid A tank in the shape of a half-cylinder is lying on its flat side. The radius of the tank is 1 m and its length is 4 m. If the tank is filled with liquid to a depth of $a$ meters, what is the volume of the liquid? (Hint: Position the half cylinder as shown in the figure below.)

    

81. Volume of a Liquid Repeat Problem 80 if the tank is buried so its rounded base is 1 m under the ground and its flat side is at ground level. See the figure below.

    

82. Damming a Valley A dam is built in a V-shaped valley. The top of the dam is $200$ m wide and the lowest point in the V is $150$ m below the top, as shown in the figure. The pressure $P$ due to the water at a depth $d$ meters below the surface is $P=\rho gd$, where $\rho =1000 \text{kg}/ \text{m}^{3}$ and $g=9.8 \text{m}/ {\text s}^{2}$. What is the total outward force $F$ from the water that this dam must be able to withstand? Express your answer in both newtons and pounds using the fact that $1 {\text N} \approx 0.2248 \text{lb}$. (Recall that pressure is $P=\dfrac{F}{A},$ so the force $F=PA.)$

    

83. Explain if the following equality is true or false without finding the exact value of the two integrals: $$\int_{-3}^{3}\int_{-\sqrt{ 9-y^{2}}}^{\sqrt{9-y^{2}}} ( 7-x)\,{\it dx}\, {\it dy}=4\int_{0}^{3}\int_{0}^{ \sqrt{9-y^{2}}}( 7-x)\,{\it dx}\, {\it dy}$$

84. Volume Find the volume of the tetrahedron enclosed by the coordinate planes and the plane $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$. (Assume that $a,b$, and $c$ are positive.)

85. Find $\displaystyle\iint\limits_{\kern-3ptR}xy\,{\it dA}$, where $R$ is the region enclosed by $y=4-x^{2}$ and $y=x^{2}$ to the right of the $y$-axis.

86. Find $\displaystyle\iint\limits_{\kern-3ptR}xy\,{\it dA}$, where $R$ is the region enclosed by $x=4-y^{2}$ and $x=y^{2}$ above the $x$-axis.

87. Find $\displaystyle\iint\limits_{\kern-3ptR}e^{\sqrt{x}}\,{\it dA}$, where $R$ is the dregion enclosed by $y=x$, $x=1$, $y=0$.

923

88. 1. (a) Graph the region $R$ defined by $0\leq a\leq y\leq b,$ $0\leq a\leq x\leq y.$
    2. (b) If $z=f(x)$ is continuous on $R,$ show that $\displaystyle\iint\limits_{\kern-3ptR}f(x) \,{\it dA}=\int_{a}^{b}(b-x)f(x)\,{\it dx}$.

89. Properties of Double Integrals Using properties of double integrals, show that if the functions $f$ and $g$ are integrable on $R$ and $f(x,y)\leq g(x,y)$ for all $(x,y)$ in $R$, then $\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA} \leq \displaystyle\iint\limits_{\kern-3ptR}g(x,y)\,{\it dA}$.

90. Volume Find the volume of the solid under the graph of the elliptic paraboloid $z=8-2x^{2}-y^{2}$ and above the first quadrant of the $xy$-plane. (Hint: Use Wallis’s formula from Section 7.1, Problem 69.)