Subtopic: Numerical integration – trapezoidal and Simpson's rule
Question Statement:
The highest order of polynomial integrand for which Simpson's 1/3 rule of integration is exact is
Options:
(A) first
(B) second
(C) third
(D) fourth
Official Answer Key:
C
Step-by-Step Solution & Explanation
Simpson's 1/3 rule fits a second-degree (quadratic) polynomial through three points, so by construction it integrates quadratics exactly. Because of an error-cancellation property (the cubic term's contribution vanishes by symmetry about the midpoint over the interval), the rule is in fact also exact for cubic (third-degree) polynomials — this is the well-known 'extra order of accuracy' of Simpson's rule. It is not exact for general fourth-degree polynomials, so the highest exact order is third, option (C).
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