Which of the following statements regarding numerical integration rules for approximating $I = \int_a^b f(x) \, dx$ are **CORRECT**?

(A) The single-segment Trapezoidal rule integrates polynomials of degree up to 1 (linear functions) exactly with zero truncation error

(B) Simpson's $1/3$ rule approximates the integrand using quadratic parabolas and integrates polynomials of degree up to 3 (cubics) exactly

(C) Simpson's $3/8$ rule requires the total number of subintervals $n$ to be a multiple of 3

(D) For a given fixed step size $h$, the global truncation error of the composite Trapezoidal rule is of order $O(h^4)$

Answer Key:

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### 1. Strategy & Theoretical Principles 1. **Trapezoidal Rule:** - Degree of precision = 1. Integrates linear polynomials $f(x) = c_0 + c_1 x$ exactly. - Composite Trapezoidal global error $E_T = -\frac{b-a}{12} h^2 f''(\xi) = O(h^2)$. 2. **Simpson's 1/3 Rule:** - Derived by passing a 2nd-degree parabola through 3 points ($n=2$). - Due to odd-symmetry cancellation of the cubic error term, its degree of precision is 3 (integrates up to cubics $a_3 x^3 + a_2 x^2 + a_1 x + a_0$ exactly). - Composite error $E_S = O(h^4)$. 3. **Simpson's 3/8 Rule:** - Passes a cubic polynomial through 4 points across 3 intervals. Consequently, the composite form strictly requires the number of subintervals $n$ to be a multiple of 3. ### 2. Statement Evaluation - **Statement A:** **CORRECT.** Degree of precision of Trapezoidal rule is 1. - **Statement B:** **CORRECT.** Simpson's 1/3 rule has degree of precision 3. - **Statement C:** **CORRECT.** Simpson's 3/8 rule requires $n$ to be a multiple of 3. - **Statement D:** **INCORRECT.** The global error of the composite Trapezoidal rule is $O(h^2)$, not $O(h^4)$ ($O(h^4)$ applies to Simpson's rule). ### 3. Exam Trap & Standard Reference - **Trap Alert:** Do not confuse local truncation error with composite global error. Trapezoidal global error is $O(h^2)$. - **Standard Reference:** Chapra, S.C. & Canale, R.P., *Numerical Methods for Engineers*, 7th Edition, McGraw-Hill, Chapter 21.

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