$\"\"$

\

The equations of the graphs are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

\

(a)

\

Find the volume of the solid generated by revolving the region about the $\"\"$ -axis.

\

The volume of the solid generated revolving about the $\"\"$ - axis.

\

Formula for the volume of the solid with the Washer method,

\

$\"\"$ .

\

The outer radius of revolution is $\"\"$ .

\

The inner radius of revolution is $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Consider $\"\"$ .

\

Solve the integral using integration by parts.

\

Formula for integration by parts: $\"\"$ .

\

Here $\"\"$ and $\"\"$ .

\

Consider $\"\"$ .

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Consider $\"\"$ .

\

Apply integral on each side.

\

$\"\"$

\

$\"\"$ .

\

Substitute corresponding values in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

The volume of the solid generated by revolving the region about the $\"\"$ -axis is $\"\"$ .

\

$\"\"$

\

(b)

\

Find the volume of the solid generated by revolving the region about the $\"\"$ -axis.

\

The volume of the solid generated revolving about the $\"\"$ - axis is $\"\"$ .

\

Here $\"\"$ and $\"\"$ .

\

$\"\"$ .

\

Solve the integral using integration by parts.

\

Formula for integration by parts: $\"\"$ .

\

Here $\"\"$ and $\"\"$ .

\

Consider $\"\"$ .

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Consider $\"\"$ .

\

Apply integral on each side.

\

$\"\"$

\

$\"\"$ .

\

Substitute corresponding values in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

The volume of the solid generated by revolving the region about the $\"\"$ -axis is $\"\"$ .

\

$\"\"$

\

(c)

\

Find the centroid of the region.

\

Moments and center of mass of a planar lamina:

\

Let $\"\"$ and $\"\"$ be continuous functions such that $\"\"$ on $\"\"$ , and consider the planar lamina of uniform density $\"\"$ bounded by the graphs of $\"\"$ and $\"\"$ .

\

The moments about the $\"\"$ -and $\"\"$ -axes are $\"\"$ . $\"\"$ .

\

The center of mass $\"\"$ is $\"\"$ and $\"\"$ , where $\"\"$ is the mass of the lamina.

\

Find $\"\"$ .

\

Substitute $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Find $\"\"$ .

\

Substitute $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Substitute $\"\"$ .

\

$\"\"$ .

\

Find $\"\"$ .

\

Substitute $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

Substitute $\"\"$ .

\

$\"\"$ .

\

Find the centroid.

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

The centroid of the region is $\"\"$ .

\

$\"\"$

\

(a) The volume of the solid generated by revolving the region about the $\"\"$ -axis is $\"\"$ .

\

(b) The volume of the solid generated by revolving the region about the $\"\"$ -axis is $\"\"$ .

\

(c) The centroid of the region is $\"\"$ .