The surface area formula, for rotation about the - axis is .

The function is  and the interval is .

Differentiate the function with respect to .

Substitute corresponding values in .

The surface area :

Let , then .

Evaluating boundaries :

If , then .

If , then .

Substitute corresponding values in .

Therefore, the exact area of the surface by rotating the curve about the - axis is .

The exact area of the surface by rotating the curve about the - axis is .