6.2 Volumes

1-18. Calculate the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

1. about the -axis.

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2. about the -axis.

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3. about the -axis.

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4. about the -axis.

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5. about the -axis.

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6. about the -axis.

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7. about the -axis.

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9. about the -axis.

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10. about the -axis.

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11. about .

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12. about .

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13. about .

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14. about .

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15. about .

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16. about .

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17. about .

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18. about .

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19-24. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

19. about .

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20. about .

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21. about .

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22. about .

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23. about .

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24. about .

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27-30. Describe the solid of revolution whose volume is represented by the integral.

27. .

; about -axis.

28. .

; about -axis.

29. .

; about -axis.

30. .

; about -axis.

31-41. Find the volume of the described solid S.

31. A right circular cone with height h and base radius r.

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32. A frustum of a right circular cone with height h, lower base radius R, and top radius r.

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37. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 6cm, 8cm and 10cm.

, .

39. The base of S is an elliptical region with boundary curve . Cross-sections perpendicular to the -axis are isosceles right triangles with the hypotenuse in the base.

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44. A bowl is shaped like a hemisphere with diameter 40 cm. A ball with diameter 20 cm is placed in the bowl and water is poured into the bowl to a depth of centimeters. Find the volume of water in the bowl.

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45-48. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

45. , ; about the -axis.

46. , ; about -axis

47. , ; about .

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48. , ; about .

49. Find the volume of the solids obtained by rotating the region bounded by the curves and about the following lines:

(a) the -axis.

(b) the -axis.

(c) .

(a)

(b)

(c)

50. Let be the region in the first quadrant bounded by the curves and . Calculate the following quantities.

(a) The area of .

(b) The volume obtained by rotating about the -axis.

(c) The volume obtained by rotating about the -axis.

(a)

(b)

(c)

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