The ellipse equation is where .

So that the major axis is along -axis and has length .

(a) Find the volume of the solid if the ellipse is rotated about its major axis.

Consider the function , .

The volume of the solid  raotated about its major axis is .

The parametric equations of the ellipse are and .

Consider .

Apply derivative on each side with respect to .

.

Substitute and and in .

Consider .

Apply derivative on each side with respect to .

.

If then .

If then  .

Substitute and .

.

The volume of the solid if the ellipse is rotated about its major axis is .

(b) Find the volume of the solid if the ellipse is rotated about its minor axis.

The ellipse equation is where .

So that the minor axis is along -axis and has length .

Consider the function , .

The volume of the solid  raotated about its minor axis is .

The parametric equations of the ellipse are and .

Consider .

Apply derivative on each side with respect to .

.

Substitute and and in .

.

Consider .

Apply derivative on each side with respect to .

.

If then .

If then  .

Substitute and in .

.

The volume of the solid if the ellipse is rotated about its minor axis is .

(a) The volume of the solid if the ellipse is rotated about its major axis is .

(b) The volume of the solid if the ellipse is rotated about its minor axis is .