Question

Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines.

Answer 1

(a)

Step 1:

The equations are and .

The volume of the solid generated revolving about the - axis.

Washer method:

The outer radius of revolution is .

The inner radius of revolution is .

Substitute and in .

Find intersection points of two line equations.

Apply zero product property.

and .

and .

Step 2:

Integrate between 0 and 2.

Apply power rule .

The volume of solid is  cubic units.

Solution:

The volume of solid is  cubic units.

Answer 2

(b)

Step 1:

The equations are and .

The volume of the solid generated revolving about the line .

Washer method:

The outer radius of revolution is .

The inner radius of revolution is .

Substitute and in .

Find intersection points of two line equations.

Apply zero product property.

and .

and .

Step 2:

Integrate between 0 and 2.

Apply power rule .

The volume of solid is cubic units.

Solution:

The volume of solid is cubic units.