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.

 .


2. about the -axis.

 .


3. about the -axis.

 .


4. about the -axis.

 .


5. about the -axis.

 .


6. about the -axis.

 .


7. about the -axis.

 .


9. about the -axis.

 .


10. about the -axis.

 .


11. about .

 .


12. about .

 .


13. about .

 

.


14. about .

 .


15. about .

 .

16. about .

 .


17. about .

 .


18. about .

 .


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 .

 .


20. about .

 .


21. about .

 .


22. about .

 .


23. about .

 .


24. about .

 .  


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.

 .


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

 .


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.      

 .


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.

  .


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 .

.


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)

        

        

        

        .