$\"\"$

\

The curve is $\"\"$ and the interval is $\"\"$ .

\

The surface area formula for rotation about the $\"\"$ -axis is,

\

$\"\"$ .

\

$\"\"$

\

Differentiate the curve with respect to $\"\"$ .

\

$\"\"$

\

Apply formula: $\"\"$ .

\

$\"\"$ .

\

Substitute corresponding values in $\"\"$ .

\

The surface area is $\"\"$ .

\

Let $\"\"$ , then $\"\"$ .

\

Evaluating boundaries :

\

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

\

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

\

$\"\"$

\

Substitute corresponding values in $\"\"$ .

\

$\"\"$

\

Apply formula: $\"\"$ .

\

$\"\"$

\

Hyperbolic sine function: $\"\"$ .

\

$\"\"$

\

The exact area of the surface by rotating the curve about the $\"\"$ -axis is $\"\"$ square units.

\

$\"\"$

\

The exact area of the surface by rotating the curve $\"\"$ about the $\"\"$ -axis is $\"\"$ square units.