8.2 Challenging Integral Applications
Surface Area of Revolution
Question #5 (Medium): Surface Area Approximation Using Simpson’s Rule
Surface area is given by: ∫ √ ( )
Simpson’s rule states: ∫ ( ) [ ( ) ( ) ( ) ( ) ( )
( ) ( )] , where is even and .
Using Simpson’s Rule with approximate the surface area obtained by rotating the function about
the x-axis. Then compare the answer to the value from the calculator.
First, the integral expression needs to be established, which requires . Thus: . Plugging into
the surface area formula: ∫ √ ( ) ∫ ( )√ ( )
( ) ( √
∫ ( ) √ ( ) ∫ ( ) . To apply Simpson’s rule, let ( )
( √ [ ( )