1. Derive the formula for the volume of a sphere using the slicing method.
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 a.a.
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?
For the following exercises (6-10), draw a typical slice and find the volume using the slicing method for the given volume.
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 rr and height hh has a smaller cone of radius r/2r/2 and height h/2h/2 removed from the top, as seen here. The resulting solid is called a frustum.
For the following exercises (11-16), draw an outline of the solid and find the volume using the slicing method.
11. The base is a circle of radius a.a. The slices perpendicular to the base are squares.
12. The base is a triangle with vertices (0,0),(1,0),(0,0),(1,0), and (0,1).(0,1). Slices perpendicular to the xy-plane are semicircles.
13. The base is the region under the parabola y=1−x2y=1−x2 in the first quadrant. Slices perpendicular to the xy-plane are squares.
14. The base is the region under the parabola y=1−x2y=1−x2 and above the x-axis.x-axis. Slices perpendicular to the y-axisy-axis are squares.
15. The base is the region enclosed by y=x2y=x2 and y=9.y=9. Slices perpendicular to the xx-axis are right isosceles triangles.
16. The base is the area between y=xy=x and y=x2.y=x2. Slices perpendicular to the xx-axis are semicircles.
For the following exercises (17-24), draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the xx-axis.
17. x+y=8,x=0, and y=0x+y=8,x=0, and y=0
18. y=2x2,x=0,x=4, and y=0y=2x2,x=0,x=4, and y=0
19. y=ex+1,x=0,x=1, and y=0y=ex+1,x=0,x=1, and y=0
20. y=x4,x=0, and y=1
21. y=√x,x=0,x=4, and y=0
22. y=sinx,y=cosx, and x=0
23. y=1x,x=2, and y=3
24. x2−y2=9 and x+y=9,y=0 and x=0
For the following exercises (25-32), draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.
25. y=4−12x,x=0, and y=0
26. y=2x3,x=0,x=1, and y=0
27. y=3x2,x=0, and y=3
28. y=√4−x2,y=0, and x=0
29. y=1√x+1,x=0, and x=3
30. x=sec(y) and y=π4,y=0 and x=0
31. y=1x+1,x=0, and x=2
32. y=4−x,y=x, and x=0
For the following exercises (33-40), draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.
33. y=x+2,y=x+6,x=0, and x=5
34. y=x2 and y=x+2
35. x2=y3 and x3=y2
36. y=4−x2 and y=2−x
37. [T] y=cosx,y=e−x,x=0, and x=1.2927
38. y=√x and y=x2
39. y=sinx,y=5sinx,x=0 and x=π
40. y=√1+x2 and y=√4−x2
For the following exercises (41-45), draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.
41. y=√x,x=4, and y=0
42. y=x+2,y=2x−1, and x=0
43. y=3√x and y=x3
44. x=e2y,x=y2,y=0, and y=ln(2)
45. x=√9−y2,x=e−y,y=0, and y=3
46. Yogurt containers can be shaped like frustums. Rotate the line y=1mx around the y-axis to find the volume between y=a and y=b.
47. Rotate the ellipse (x2/a2)+(y2/b2)=1 around the x-axis to approximate the volume of a football, as seen here.
48. Rotate the ellipse (x2/a2)+(y2/b2)=1 around the y-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 y=sinx around the x-axis from x=0 to x=π. What is the volume of this football approximation, as seen here?
50. What is the volume of the Bundt cake that comes from rotating y=sinx around the y-axis from x=0 to x=π?
For the following exercises (51-56), find the volume of the solid described.
51. The base is the region between y=x and y=x2. Slices perpendicular to the x-axis are semicircles.
52. The base is the region enclosed by the generic ellipse (x2/a2)+(y2/b2)=1. Slices perpendicular to the x-axis are semicircles.
53. Bore a hole of radius a down the axis of a right cone and through the base of radius b, as seen here.
54. Find the volume common to two spheres of radius r with centers that are 2h apart, as shown here.
55. Find the volume of a spherical cap of height h and radius r where [latex]h
56. Find the volume of a sphere of radius R with a cap of height h removed from the top, as seen here.
Candela Citations
- Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-2/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction