Exercises for using the shell method.

The region defined by the inequalities for is revolved around the -axis. Compute the volume of the resulting solid using the shell method.

  • When the slicing variable is , the radius of a shell is the horizontalvertical distance from an -slice to the axis of rotation. Thus
  • The height of an -slice is equal to
  • The volume is equal to the integral of , so (Note: to compute the integral, split it into two parts and make the substitution for one of them.)
The region in the plane bounded above by the graph , below by , and on the left by is revolved around the axis . Compute the volume of the resulting solid using the shell method.

  • When the slicing variable is , the radius of a shell is the horizontalvertical distance from an -slice to the axis . Thus
  • The height of an -slice is equal to
  • The volume is equal to the integral of , so
The region in the plane , , and is revolved around the -axis. Use the shell method to compute the volume.
The same region as above (bounded by , , and ) is revolved around the axis . Use the shell method to compute the volume.
The same region as above (bounded by , , and ) is revolved around the -axis. Use the shell method to compute the volume.
The “height” of a shell is in this case.
For the same region as above (bounded by , , and ), use the shell method to compute the volume when revolved around the axis .

Sample Quiz Questions

The region in the plane bounded below by the curve , above by the curve , on the right by the line , and on the left by the line is revolved around the axis . Compute the volume of the resulting solid. (Hints won’t reveal until after you choose a response.)

The region in the plane bounded on the left by the curve , on the right by the curve , and below by the line is revolved around the axis . Compute the volume of the resulting solid. (Hints won’t reveal until after you choose a response.)

The region in the plane between the -axis and the graph in the range is revolved around the axis . Compute the volume of the resulting solid. (Hints won’t reveal until after you choose a response.)