Subtopic: Numerical integration – trapezoidal and Simpson's rule
Question Statement:
Which of the following statements regarding Newton–Cotes numerical quadrature formulas is/are TRUE?
Options:
(A) The Trapezoidal rule integrates polynomials of degree up to $1$ with zero truncation error
(B) Simpson's $1/3$ rule integrates polynomials of degree up to $3$ exactly
(C) Simpson's $3/8$ rule requires the total number of subintervals $n$ to be a multiple of $3$
(D) Simpson's $1/3$ rule requires the total number of subintervals $n$ to be an even number
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. **Trapezoidal Precision (Option A)**: Linear interpolation makes the truncation error proportional to $f''(\xi)$, so it is exact for degree $\le 1$ (True).
2. **Simpson's 1/3 Precision (Option B)**: Simpson's 1/3 rule error is proportional to $f^{(4)}(\xi) h^5$, meaning odd symmetric cancellation makes it exact for cubic polynomials (degree $\le 3$) (True).
3. **Simpson's 3/8 Partition (Option C)**: Groups 3 subintervals per application, requiring $n$ to be a multiple of 3 (True).
4. **Simpson's 1/3 Partition (Option D)**: Pairs 2 subintervals per application, requiring $n$ to be an even integer (True).