The equations is , , and .

(a) Find the volume of the region about the -axis.

Disk method:

The volume of the solid is .

Substitute , and .

.

The equations is , , and .

(b) Find the volume of the region about the -axis.

Shell method:

Vertical axis of revolution.

The volume of the solid is .

The distance from the center of the rectangle to the axis of revolution is .

The height of the rectangle is .

Substitute , and and in .

The graph is symmetrical about the -axis.

Apply integral substitution : , where .

Here then .

Apply the power rule of integral : .

Susbtitute .

.

The equations are , , and .

(c) Find the volume of the region about the line .

Shell method:

The volume of the solid is .

The distance from the center of the rectangle to the axis of revolution is .

The height of the rectangle is .

Substitute , and and in .

The graph is symmetrical about the -axis.

Apply integral substitution : , where .

Here then .

Apply the power rule of integral : .

Susbtitute .

.

Because of the symmetry we can calculate the volume of the revolution around the y-axis of the area above the x-axis and multiply by .

.

(a) The volume of the solid about  the -axis is .

(b) The volume of the solid about -axis is .

(c) The volume of the solid about is .