$\"\"$

\

The curves are $\"\"$ , $\"\"$ , $\"\"$ , $\"\"$ .

\

Find the volume about $\"\"$ .

\

Cylindrical shells method:

\

The volume of the solid obtained by rotating about the $\"\"$ -axis the region under the curve is

\

$\"\"$

\

$\"\"$ .

\

Integrate between $\"\"$ to $\"\"$ .

\

The distance from the center of the rectangle to the axis of revolution is

\

$\"\"$

\

$\"\"$ .

\

The shell height is $\"\"$ .

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

Consider $\"\"$ .

\

Apply parts of integration formula: $\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

Integrate on each side.

\

$\"\"$

\

$\"\"$ .

\

Substitute the corresponding values in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$ .

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ cubic units.

\

$\"\"$

\

$\"\"$ cubic units.