Step 1:

The equations are , -axis, and .

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

Waher method:

Volume of the solid: . \ \

Where is the outer radius, is the inner radius.

Graph the functions , , and .

Observe the graph:

The outer radius is .

the inner radius is .

The integral limits are and .

Step 2:

Find the volume of the solid.

Substitute , , and in .

\ \

Apply power rule of integration: .

.

The volume of the solid is .

Solution: \ \

The volume of the solid is .

Step 1:

The equations are , -axis, and .

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

Waher method:

Volume of the solid: . \ \

Where is the outer radius, is the inner radius.

Graph the functions , , and .

Observe the graph:

The outer radius is .

the inner radius is .

The integral limits are and .

Step 2:

Find the volume of the solid.

Substitute , , and in .

Apply power rule of integration: .

.

The volume of the solid is .

Solution: \ \

The volume of the solid is .

The function is  on interval .

Average value of the function  on is defined as .

In this case  and .

Substitute , and in .

The average value is .

Apply formula: .

. \ \

The average value is .

Solution:

The average value is .