Step 1 :

For rotation about the x - axis, the surface area formula : $\"\"$ .

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

Differentiate the curve with respect to x.

$\"\"$

Apply formula : $\"\"$ .

$\"\"$ .

Substitute corresponding values in $\"\"$ .

The surface area is $\"\"$ .

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

Evaluating boundaries :

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

Substitute corresponding values in $\"\"$

$\"\"$

Apply formula : $\"\"$ .

$\"\"$

The exact area of the surface by rotating the curve about the x - axis is 6.594 square units.