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.

y^2 = x,   x = 2y; about the y - axis

Answer 1

Step 1:

The curve equations are and about -axis.

Definition of volume:

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

Now we find the intersection points of the two curves and .

Equate both the curves.

Now we find the volume of the region over the interval  0 and 2.

Graph:

Step 2:

Area of the region bounded by the curve and -axis is

Area of the region bounded by the curve and -axis is

Cross sectional area of the solid is .

Step 3:

Volume of the solid is .

Substitute , and .

Volume of the solid is .

Solution:

Volume of the solid is .