Subtopic: Numerical integration – trapezoidal and Simpson's rule
Question Statement:
Simpson's $1/3$ rule of numerical integration gives the exact value of $\int_a^b f(x) \, dx$ when $f(x)$ is a polynomial of degree at most:
Options:
(A) 1
(B) 2
(C) 3
(D) 4
Answer Key:
C
Step-by-Step Solution & Explanation
Although Simpson's $1/3$ rule is derived by fitting a quadratic parabola ($2^{\text{nd}}$ degree polynomial) through three equidistant points, the symmetry of the truncation error term $E = -\frac{(b-a)h^4}{180} f^{(4)}(\xi)$ causes the third-order derivative term to vanish. Therefore, it is exact for all polynomials up to degree $3$ (cubics).