5.2. Volumes
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, and then he does the same volume problem using disks. KhanAcademySolid of Revolution(Part 6) (9:19).
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Together these videos show how both methods can be used to solve the same problem (though it’s not always done this easily!).
Review Questions In problems #1 - 4, find the volume of the solid generated by revolving the region bounded by the curves about the x−axis. p 1. y = 9 − x2 , y = 0 2. y = 3 + x, y = √ 1 + x2 3. y = sec x, y = 2, −π/4 ≤ x ≤ π/4 4. y = 1, y = x, x = 0 In problems #5–8, find the volume of the solid generated by revolving the region bounded by the curves about the y−axis. 5. y = x3 , x = 0, y = 1 6. x = y2 , y = x − 2 7. x = csc y, y √ = π/4, y = 3π/4, x = 0 8. y = 0, y = x, x = 4 In problems #9–12, use cylindrical shells to find the volume generated when the region bounded by the curves is revolved about the axis indicated. 9. y = 1x , y = 0, x = 1, x = 3, about the y−axis 10. y = x2 , x = 1, y = 0, about the x−axis 11. y = 2x − 1, y = −2x + 3, x = 2, about the y−axis 12. y2 = x, y = 1, x = 0, about the x−axis. 13. Use the cylindrical shells method to find the volume generated when the region is bounded by y = x3 , y = 1, x = 0 is revolved about the line y = 1. 256