1. Find \(\int_{0}^{1}\int_{1}^{e}\dfrac{y}{x}\,{\it dx}\,{\it dy}\).

2. Find \(\int_{1}^{3}\int_{3}^{6}(x^{2}+y^{2})\,{\it dy}\,{\it dx}\).

3. Find \(\int_{0}^{1}\int_{y}^{1}ye^{-x^{3}}\,{\it dx}\,{\it dy}\).

4. Find \(\int_{0}^{1}\int_{0}^{\sqrt{1+x^{2}}}\dfrac{1}{ x^{2}+y^{2}+1}\,{\it dy}\,{\it dx}\).

5. Find \(\int_{0}^{\pi /4}\int_{0}^{\tan x}\sec x\,{\it dy}\,{\it dx}\)

6. Find \(\int_{0}^{1}\int_{-x}^{x}e^{x+y}\,{\it dy}\,{\it dx}\)

7. Let \(R\) be the square region \(0\leq x\leq 4\), \(-2\leq y\leq 2\) in the \(xy\)-plane, and suppose that \(f(x,y)=x+2y\).

   1. (a) Partition \(R\) into four squares of equal area and evaluate \( f\) at the midpoint of each square. Find the corresponding Riemann sum of \(f\) over \(R\).
   2. (b) Using the same partition as in (a), but evaluating \(f\) at an appropriate point of each square, determine the largest possible Riemann sum and the smallest. What is the average of these values?
   3. (c) What is the actual value of \(\iint\limits_{\kern-3ptR}f(x, y)\,{\it dA}\)?

8. Find \(\iint\limits_{\kern-3ptR}xe^{y}\,{\it dA}\) if \(R\) is the closed rectangular region defined by \(0\leq x\leq 2\) and \(0\leq y\leq 1\).

9. Find \(\iint\limits_{\kern-3ptR}\sin x\cos y\,{\it dA}\) if \(R\) is the closed rectangular region defined by \(0\leq x\leq \dfrac{\pi }{2}\) and \(0\leq y\leq \dfrac{\pi }{3}\).

10. Volume Find the volume \(V\) under the surface \( z=f(x,y)=e^{x}\cos y\) over the rectangular region \(R\) defined by \(0\leq x\leq 1\) and \(0\leq y\leq \dfrac{\pi }{2}\).

11. Volume Find the volume \(V\) under the surface \( z=f(x, y)=2x^{2}+y^{2}\) over the rectangular region \(R\) defined by \( 0\leq x\leq 2\) and \(0\leq y\leq 1\).

12. The functions \(f\) and \(g\) are continuous on a closed, bounded region \(R,\) and \(\iint\limits_{\kern-3ptR}f(x,y)\, {\it dA}=-2,\) \(\iint\limits_{\kern-3ptR}g(x, y)\, {\it dA}=3\). Use the properties of double integrals to find each double integral.

    1. (a) \(\iint\limits_{\kern-3ptR}2[ f(x, y) +3g(x, y) ]\, {\it dA}\)
    2. (b) \(\iint\limits_{\kern-3ptR}[ g(x, y)-2f(x, y) ]\, {\it dA}\)

13. Find \(\iint\limits_{\kern-3ptR}x\sin (y^{3})\,{\it dA}\), where \(R\) is the region enclosed by the triangle with vertices \((0, 0)\), \((0, 2)\), and \((2, 2)\).

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

15. Volume Find the volume of the tetrahedron enclosed by the coordinate planes and the plane \(2x+y+3z=6\).

16. Volume Find the volume of the ellipsoid \(4x^{2}+y^{2}+\dfrac{ z^{2}}{4}=1\).

17. Volume Find the volume of the solid in the first octant enclosed by the coordinate planes, \(4y=3(4-x^{2})\) and \(4z=4-x^{2}\).

18. Area Use a double integral to find the area enclosed by the parabola \(y^{2}=16x\) and the line \(y=4x-8\).

19. Find \(\iint\limits_{\kern-3ptR}\sin\theta\,{\it dA}\), where \(R\) is the region enclosed by the rays \(\theta=0\) and \(\theta = \pi/6\) and the circle \(r=6\sin\theta\).

20. Area Use a double integral to find the first-quadrant area enclosed by \(y^{2}=x^{3}\) and \(y=x\).

21. Area Use a double integral and polar coordinates to find the area of the circle \(r=2\cos \theta\).

22. Area Find the first-quadrant area outside \(r=2a\) and inside \( r=4a\cos \theta \) by using polar coordinates.

23. Mass Find the mass of a lamina in the shape of a right triangle of height 6 and base 4 if the mass density is proportional to the square of the distance from the vertex at the right angle.

24. Center of Mass Find the center of mass of the homogeneous lamina enclosed by \(3x^{2}=y\) and \(y=3x\).

25. Center of Mass Find the center of mass of the homogeneous lamina in quadrant \(I\), enclosed by \( 4x=y^{2}\), \(y=0\), and \(x=4\).

26. Moments of Inertia Use double integration to find the moments of inertia for a homogeneous lamina in the shape of the ellipse \( 2x^{2}+9y^{2}=18.\)

27. Surface Area Find the area of the first-octant portion of the plane \(\dfrac{x}{2}+y+\dfrac{z}{3}=1\).

28. Surface Area Find the surface area of the paraboloid \( z=x^{2}+y^{2}\) below the plane \(z=1\).

29. Surface Area Find the surface area of the cone \( x^{2}+y^{2}=3z^{2}\) that lies inside the cylinder \(x^{2}+y^{2}=4y\).

30. Find \(\iiint\limits_{\kern-3ptE}e^{z}\cos x\,{\it dV},\) where \(E\) is the solid given by \(0\leq x\leq \dfrac{\pi }{2}\), \(0\leq y\leq 2\), \(0\leq z\leq 1\).

31. Find \(\int_{0}^{1}\int_{0}^{2-3x} \int_{0}^{2y}x^{2}\,{\it dz}\,{\it dy}\,{\it dx}\).

32. Find \(\iiint\limits_{\kern-3ptE}ye^{x}\,{\it dV},\) where \(E\) is the solid given by \( 0\leq x\leq 1\), \(0\leq y\leq 1\), \(\ y^{2}\leq z\leq y\).

33. Find \(\iiint\limits_{\kern-3ptE}xyz~{\it dV},\) where \(E\) is the solid enclosed by \(x=0\) and \(x=2-y-z\) whose projection onto the \(yz\)-plane is the region enclosed by the rectangle \( 0\leq y\leq 1\) and \(0\leq z\leq 1\).

34. Center of Mass Find the center of mass and the moments of inertia \(I_x\) and \(I_y\) of the homogeneous hemispherical shell \(0\leq a\leq r\leq b\), \(0\leq \phi \leq \dfrac{\pi }{2}\).

35. Volume Find the volume in the first octant of the paraboloid \( x^{2}+y^{2}+z=9\).

36. Volume Find the volume in the first octant enclosed by \(y=0\), \( z=0\), \(x+y=2\), \(x+2y=6\), and \(y^{2}+z^{2}=4\).

37. \((3, 0, 4)\)

38. \(( -2\sqrt{2},2\sqrt{2},3)\)

39. \((-1, 1, -2)\)

40. \((1, \sqrt{3}, 4)\)

971

41. Volume Use spherical coordinates with triple integrals to find the volume between the spheres \(x^{2}+y^{2}+z^{2}=4\) and \(x^{2}+y^{2}+z^{2}=1 \).

42. Volume Find the volume cut from the sphere \(\rho =4\) by the cone \(\phi =\dfrac{\pi }{4}\).

43. Use cylindrical coordinates to find \[ \displaystyle\int_{0}^{2}\int_{0}^{\sqrt{4-x^{2}}}\int_{0}^{\sqrt{4-x^{2}-y^{2}}}\dfrac{z}{ \sqrt{x^{2}+y^{2}}}\,{\it dz}\,{\it dy}\,{\it dx} \]

44. Volume Use cylindrical coordinates to find the volume in the first octant inside the cylinder \(r=1\) and below the plane \(3x+2y+6z=6\).

45. Use cylindrical coordinates to find \[ \int_{0}^{a}\int_{0}^{\sqrt{a^{2}-x^{2}}}\int_{0}^{\sqrt{a^{2}-x^{2}-y^{2}} }\,{\it dz}\,{\it dy}\,{\it dx} \]

46. Use spherical coordinates to find the iterated integral given in Problem 45.

47. \(\int_{-2}^{2}\int_{-\sqrt{4-y^{2}}}^{\sqrt{4-y^{2}} }\int_{0}^{\sqrt{16-x^{2}-y^{2}}}\,{\it dz}\,{\it dx}\,{\it dy}\)

48. \(\int_{0}^{1}\int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}\int_{\sqrt{x^{2}+y^{2} }}^{\sqrt{2-x^{2}-y^{2}}}\,{\it dz}\,{\it dy}\,{\it dx}\)

49. \(\int_{-\sqrt{2}}^{\sqrt{2}}\int_{-\sqrt{2-x^{2}}}^{\sqrt{ 2-x^{2}}}\int_{0}^{4}(x^{2}+y^{2})~z\,{\it dz}\,{\it dy}\,{\it dx}\)

50. \(\int_{0}^{1}\int_{0}^{1}\int_{0}^{\sqrt{1-x^{2}}}\,{\it dy}\,{\it dx}\,{\it dz}\)

51. \(\int_{0}^{a}\int_{0}^{\sqrt{a^{2}-x^{2}}}\int_{\dfrac{h}{a}\sqrt{ x^{2}+y^{2}}}^{\sqrt{a^{2}-x^{2}-y^{2}}}\,{\it dz}\,{\it dy}\,{\it dx}\)

52. \(\int_{0}^{1}\int_{0}^{\sqrt{1-x^{2}}}(x^{2}+y^{2})^{3/2}\,{\it dy} \,{\it dx}\)

53. \(\int_{-1}^{1}\int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}}\int_{\sqrt{x^{2}+y^{2}}}^{\sqrt{2-x^{2}-y^{2}}}\,{\it dz}\,{\it dx}\,{\it dy}\)

54. \(\int_{0}^{1}\int_{y}^{1}(x^{2}+1)^{2/3}\,{\it dx}\,{\it dy}\)

55. \(x=e^{u+v},\quad y=e^{v-u}\)

56. \(x=u+3v,\quad y=2u-v\)

57. \(x=u^{2},\quad y=3v,\quad and z=u+w\)

58. Find \(\iint\limits_{\kern-3ptR}y^{2}\,{\it dx}\,{\it dy}\), where the region \(R\) is enclosed by \(x-2y=1\), \(x-2y=3\), \(2x+3y=1\), and \(2x+3y=2\), using the change of variables \(u=x-2y\) and \(v=2x+3y.\)

59. Find \(\iint\limits_{\kern-3ptR}(x+y)^{3}\,{\it dx}\,{\it dy}\), where the region \(R\) is enclosed by the lines \(x+y=2\), \(x+y=5\), \(x-2y=-1\), and \(x-2y=3\), using the change of variables \(u=x+y\) and \(v=x-2y\).

60. Find \(\iint\limits_{\kern-3ptR}xy\,{\it dx}\,{\it dy}\), where \(R\) is the region enclosed by \(2x+y=0\), \(2x+y=3\), \(x-y=0\), and \(x-y=2\).

61. Find \(\iiint\limits_{\kern-3ptE}xz^{2}\,{\it dx}\,{\it dy}\,{\it dz}\), where \(E\) is the solid enclosed by the ellipsoid \(\dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}+z^{2}=1,\) using the change of variables \(u=\dfrac{x}{2}\), \(v=\dfrac{y}{3}\), and \(w=z.\)

62. Mass A solid in the first octant enclosed by the surface \( z=xy,\) the \(xy\)-plane, and the cylinder whose intersection with the \(xy\)-plane forms a triangle with vertices \((0, 0, 0),\) \(( 2, 0, 0),\) and \((0, 1, 0)\) has mass density \(\rho=x+y.\) Find the mass of the solid.