2. Use the slicing method to derive the formula for the volume of a cone.

3. Use the slicing method to derive the formula for the volume of a tetrahedron with side length [latex]a.[/latex]

4. Use the disk method to derive the formula for the volume of a trapezoidal cylinder.

5. Explain when you would use the disk method versus the washer method. When are they interchangeable?

6. A pyramid with height 6 units and square base of side 2 units, as pictured here.

7. A pyramid with height 4 units and a rectangular base with length 2 units and width 3 units, as pictured here.

8. A tetrahedron with a base side of 4 units, as seen here.

9. A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here.

10. A cone of radius [latex]r[/latex] and height [latex]h[/latex] has a smaller cone of radius [latex]r\text{/}2[/latex] and height [latex]h\text{/}2[/latex] removed from the top, as seen here. The resulting solid is called a frustum.

11. The base is a circle of radius [latex]a.[/latex] The slices perpendicular to the base are squares.

12. The base is a triangle with vertices [latex](0,0),(1,0),[/latex] and [latex](0,1).[/latex] Slices perpendicular to the xy-plane are semicircles.

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[latex]\frac{\pi }{24}[/latex] units³

13. The base is the region under the parabola [latex]y=1-{x}^{2}[/latex] in the first quadrant. Slices perpendicular to the xy-plane are squares.

14. The base is the region under the parabola [latex]y=1-{x}^{2}[/latex] and above the [latex]x\text{-axis}\text{.}[/latex] Slices perpendicular to the [latex]y\text{-axis}[/latex] are squares.

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2 units³

15. The base is the region enclosed by [latex]y={x}^{2}[/latex] and [latex]y=9.[/latex] Slices perpendicular to the [latex]x[/latex]-axis are right isosceles triangles.

16. The base is the area between [latex]y=x[/latex] and [latex]y={x}^{2}.[/latex] Slices perpendicular to the [latex]x[/latex]-axis are semicircles.

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[latex]\frac{\pi }{240}[/latex] units³

17. [latex]x+y=8,x=0,\text{ and }y=0[/latex]

18. [latex]y=2{x}^{2},x=0,x=4,\text{ and }y=0[/latex]

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[latex]\frac{4096\pi }{5}[/latex] units³

19. [latex]y={e}^{x}+1,x=0,x=1,\text{ and }y=0[/latex]

20. [latex]y={x}^{4},x=0,\text{ and }y=1[/latex]

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[latex]\frac{8\pi }{9}[/latex] units³

21. [latex]y=\sqrt{x},x=0,x=4,\text{ and }y=0[/latex]

22. [latex]y= \sin x,y= \cos x,\text{ and }x=0[/latex]

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[latex]\frac{\pi }{2}[/latex] units³

23. [latex]y=\frac{1}{x},x=2,\text{ and }y=3[/latex]

24. [latex]{x}^{2}-{y}^{2}=9\text{ and }x+y=9,y=0\text{ and }x=0[/latex]

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[latex]207\pi[/latex] units³

25. [latex]y=4-\frac{1}{2}x,x=0,\text{ and }y=0[/latex]

26. [latex]y=2{x}^{3},x=0,x=1,\text{ and }y=0[/latex]

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[latex]\frac{4\pi }{5}[/latex] units³

27. [latex]y=3{x}^{2},x=0,\text{ and }y=3[/latex]

28. [latex]y=\sqrt{4-{x}^{2}},y=0,\text{ and }x=0[/latex]

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[latex]\frac{16\pi }{3}[/latex] units³

29. [latex]y=\frac{1}{\sqrt{x+1}},x=0,\text{ and }x=3[/latex]

30. [latex]x= \sec (y)\text{ and }y=\frac{\pi }{4},y=0\text{ and }x=0[/latex]

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[latex]\pi[/latex] units³

31. [latex]y=\frac{1}{x+1},x=0,\text{ and }x=2[/latex]

32. [latex]y=4-x,y=x,\text{ and }x=0[/latex]

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[latex]\frac{16\pi }{3}[/latex] units³

33. [latex]y=x+2,y=x+6,x=0,\text{ and }x=5[/latex]

34. [latex]y={x}^{2}\text{ and }y=x+2[/latex]

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[latex]\frac{72\pi }{5}[/latex] units³

35. [latex]{x}^{2}={y}^{3}\text{ and }{x}^{3}={y}^{2}[/latex]

36. [latex]y=4-{x}^{2}\text{ and }y=2-x[/latex]

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[latex]\frac{108\pi }{5}[/latex] units³

37. [T] [latex]y= \cos x,y={e}^{\text{−}x},x=0,\text{ and }x=1.2927[/latex]

38. [latex]y=\sqrt{x}\text{ and }y={x}^{2}[/latex]

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[latex]\frac{3\pi }{10}[/latex] units³

39. [latex]y= \sin x\text{,}y=5 \sin x,x=0\text{ and }x=\pi[/latex]

40. [latex]y=\sqrt{1+{x}^{2}}\text{ and }y=\sqrt{4-{x}^{2}}[/latex]

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[latex]2\sqrt{6}\pi[/latex] units³

41. [latex]y=\sqrt{x},x=4,\text{ and }y=0[/latex]

42. [latex]y=x+2,y=2x-1,\text{ and }x=0[/latex]

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[latex]9\pi[/latex] units³

43. [latex]y=\sqrt[3]{x}\text{ and }y={x}^{3}[/latex]

44. [latex]x={e}^{2y},x={y}^{2},y=0,\text{ and }y=\text{ln}(2)[/latex]

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[latex]\frac{\pi }{20}(75-4{\text{ln}}^{5}(2))[/latex] units³

45. [latex]x=\sqrt{9-{y}^{2}},x={e}^{\text{−}y},y=0,\text{ and }y=3[/latex]

46. Yogurt containers can be shaped like frustums. Rotate the line [latex]y=\frac{1}{m}x[/latex] around the [latex]y[/latex]-axis to find the volume between [latex]y=a\text{ and }y=b.[/latex]

47. Rotate the ellipse [latex]({x}^{2}\text{/}{a}^{2})+({y}^{2}\text{/}{b}^{2})=1[/latex] around the [latex]x[/latex]-axis to approximate the volume of a football, as seen here.

48. Rotate the ellipse [latex]({x}^{2}\text{/}{a}^{2})+({y}^{2}\text{/}{b}^{2})=1[/latex] around the [latex]y[/latex]-axis to approximate the volume of a football.

49. A better approximation of the volume of a football is given by the solid that comes from rotating [latex]y= \sin x[/latex] around the [latex]x[/latex]-axis from [latex]x=0[/latex] to [latex]x=\pi .[/latex] What is the volume of this football approximation, as seen here?

50. What is the volume of the Bundt cake that comes from rotating [latex]y= \sin x[/latex] around the [latex]y[/latex]-axis from [latex]x=0[/latex] to [latex]x=\pi ?[/latex]

51. The base is the region between [latex]y=x[/latex] and [latex]y={x}^{2}.[/latex] Slices perpendicular to the [latex]x[/latex]-axis are semicircles.

52. The base is the region enclosed by the generic ellipse [latex]({x}^{2}\text{/}{a}^{2})+({y}^{2}\text{/}{b}^{2})=1.[/latex] Slices perpendicular to the [latex]x[/latex]-axis are semicircles.

53. Bore a hole of radius [latex]a[/latex] down the axis of a right cone and through the base of radius [latex]b,[/latex] as seen here.

54. Find the volume common to two spheres of radius [latex]r[/latex] with centers that are [latex]2h[/latex] apart, as shown here.

55. Find the volume of a spherical cap of height [latex]h[/latex] and radius [latex]r[/latex] where [latex]h

56. Find the volume of a sphere of radius [latex]R[/latex] with a cap of height [latex]h[/latex] removed from the top, as seen here.