8.1 Challenging Integral Applications
Question #4 (Medium): Arc Length Approximation Using Simpson’s Rule
For functions where evaluating ∫ √ ( ) is challenging, Simpson’s Rule can be used
to approximate the length of the curve for the given interval. As a review, Simpson’s Rule follows:
∫ ( ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,
where is even and .
Using Simpson’s Rule with , estimate the length of the curve. Then compare the answer to the
value produced by the calculator.
First, taking the derivative of the function. √ . Thus, ∫ √ ( )
∫ √ ( ) ∫ √ .