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Mathematical Expression Editor

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 axis is parallel to the direction of slices using the integration variable , which
indicates the shell method. The region lies to the right of the axis, which must be the
case because the interval lies to the right of the axis .

The integral to compute
equals

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 axis is parallel to the direction of slices using the integration variable , which
indicates the shell method. The lower endpoint of integration will be ; the upper
endpoint can be determined by setting and choosing the solution which is greater
than . This gives the range . The region lies above the axis, which must be the case
because the interval lies above the axis .

The integral to compute equals

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.)

If the variable is used for slicing, then slices are parallel to the axis of rotation,
which indicates the shell method should be used. The radius of a shell is . The height
of a shell is exactly .

The volume of the region is therefore given by

To compute
the integral we can use the substitution which implies the equality for
the differentials. This gives the equality Reversing the substitution gives