\

Step 1:

\

The function is and .

\

Definition of surface area:

\

If the curve is described as ,  then the surface area of the curve obtained by rotating about the -axis is

\

\

The curve is .

\

Differentiate on each side.

\

\ \ Area of the surface obtained by rotating the curve about the -axis is

\

 

\

\

Step 2:

\

Simpson\\'s Rule:

\

Let  be continuous on . The Midpoint Rule for approximating  is given by

\

,

\

where  and

\

\

\

\ \

\

Using Simpson Rule,

\

Step 3:

\

Area of the surface obtained by rotating the curve about the -axis is

\

 

\

\

Using calculator, the value of the integral is .

\

Solution:

\

Area of the surface obtained by rotating the curve about the -axis is .

\