Subtopic: Numerical integration – trapezoidal and Simpson's rule
Question Statement:
Which of the following statements regarding Simpson's $1/3$ rule for numerical integration are CORRECT?
Options:
(A) The total number of sub-intervals must be an EVEN integer.
(B) The rule approximates the integrand by a quadratic parabola over each pair of intervals.
(C) It gives the EXACT value of the integral for any polynomial up to degree 3 (cubic polynomials).
(D) The truncation error is proportional to the first derivative of the integrand.
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
Simpson's $1/3$ rule requires an even number of sub-intervals ($n$ even). It fits a second-degree polynomial across successive point triplets and yields exact results for polynomials up to degree 3 because third-order error terms cancel out. Statement D is false: error is proportional to $h^4 f^{(4)}(\xi)$.