What is the order of the polynomial that is applied for the Simpson's 1/3 rule?

Question

BlackTom AI · Accepted Answer

Simpson's 1/3 rule uses a quadratic (degree 2) interpolating polynomial fitted through three points, so the order of the polynomial applied is 2.

1) Relevant knowledge points and concepts
- Numerical integration by Newton–Cotes methods approximates f(x) by a low‑degree polynomial through sample points and integrates that polynomial exactly.
- Simpson's 1/3 rule is the Newton–Cotes rule that fits a degree‑2 (quadratic) interpolant through three equally spaced nodes and integrates it on the subinterval.

2) Complete reasoning process, step by step
- Take an interval [a,b] and split it into two equal subintervals by h = (b − a)/2. The three nodes are x0 = a, x1 = a + h, x2 = a + 2h = b.
- Form the Lagrange quadratic interpolant p(x) of degree 2 that satisfies p(xi) = f(xi) for i = 0,1,2.
- Integrate p(x) exactly from a to b. The integration of the Lagrange basis over [a,b] yields weights 1, 4, 1 (after factoring).

3) Intermediate calculation steps
- For the single-panel Simpson formula the integral of the quadratic interpolant gives
  ∫_a^b f(x) dx ≈ ∫_a^b p(x) dx = (h/3)[f(a) + 4 f(a+h) + f(a+2h)].
- Substitute h = (b − a)/2:
  (h/3) = ((b − a)/2) / 3 = (b − a)/6,
  so ∫_a^b f(x) dx ≈ (b − a)/6 [f(a) + 4 f((a+b)/2) + f(b)].

4) Why the correct answer is correct
- The derivation above shows Simpson's 1/3 rule is obtained by integrating a quadratic interpolant (degree 2). Therefore the polynomial order applied is 2.

5) Why the other choices are incorrect
- a. 1: A degree‑1 polynomial (linear) leads to the trapezoidal rule, not Simpson’s 1/3.
- c. 3 and d. 4: Those are higher-degree polynomials; Simpson’s 1/3 specifically uses degree 2. (Note: Simpson’s rule has algebraic degree of precision 3—it integrates all cubics exactly—but the interpolant used is quadratic.)

6) Conclusion
- The polynomial order applied in Simpson’s 1/3 rule is 2, confirming the correct choice b. 2.

What is the order of the polynomial that is applied for the Simpson's 1/3 rule?单项选择题

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