Compute the error (the difference between the exact value and the approximation) in the Simpson's Rule approximation using . Round your answer to 6 decimal places.

Question

Compute the error (the difference between the exact value and the approximation) in the Simpson's Rule approximation using  . Round your answer to 6 decimal places.

Quizplus · Accepted Answer

The error term for Simpson's Rule is proportional to the fourth derivative of the function evaluated at some point within the interval. In this case, the fourth derivative of the given function is 16cos(4x), which has a maximum absolute value of 16.

Therefore, the error in Simpson's Rule is bounded by (b-a)^5 / (180*n^4) * 16 for an n-segment approximation.

Plugging in the values for this problem, we have: 
Error = (π/16)^5 / (180*2^4) * 16 = 0.000058

Rounding to six decimal places gives us 0.000058, which is equivalent to choice C.

Compute the Error (The Difference Between the Exact Value and the Approximation)