$\"\"$

\

The curves are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ and about $\"\"$ -axis.

\

Method of cylindrical shells:

\

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

\

$\"\"$ , where $\"\"$ .

\

Here $\"\"$ .

\

Integral limits are $\"\"$ and $\"\"$ .

\

Substitute corresponding values in volume formula.

\

$\"\"$ .

\

Consider $\"\"$ .

\

Differentiate on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

Change in integral limits :

\

If $\"\"$ , then $\"\"$ .

\

If $\"\"$ , then $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Substitute $\"\"$ , $\"\"$ and change in integral limits in the above integral.

\

$\"\"$

\

$\"\"$

\

Volume of the solid is $\"\"$ .

\

$\"\"$

\

Volume of the solid is $\"\"$ .