Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

x = y^2,   x = 1 - y^2;      about x = 3

Answer 1

Step 1:

The curve equations are and about .

Definition of volume:

The volume of the solid  is , where is the cross sectional area of the solid .

Find the intersection points of the two curves and .

Equate both the curves.

Graph:

Step 2:

Area of the region bounded by the curve and is

Area of the region bounded by the curve and is

Cross sectional area of the solid is .

Step 3:

Find the volume of the solid over the interval  and .

Volume of the solid is .

Substitute ,and .

Volume of the solid is .

Solution:

Volume of the solid is .