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Trapezoidal, midpoint and Simpson's rules

Problem 4.380 · medium

Approximate \( \displaystyle \int_{0}^{2} x^{2}\, dx \) using the Simpson's rule with \( \displaystyle n = 6 \).

\[ 1 \cdot \frac{1}{3} = \frac{1}{3} \]
Δx = (b − a)/n.✓ Proved
The Simpson's rule uses the points x = 0, 1/3, 2/3, 1, 4/3, 5/3, 2 with weights 1, 4, 2, 4, 2, 4, 1, all times 1/9.
\[ \frac{1 \cdot 0 + 4 \cdot 1 \cdot \frac{1}{9} + 2 \cdot 4 \cdot \frac{1}{9} + 2 \cdot 16 \cdot \frac{1}{9} + 4 \cdot 1 + 1 \cdot 4 + 4 \cdot 25 \cdot \frac{1}{9}}{9} = \frac{8}{3} \]
Weighted sum of the function values.✓ Proved
\[ \int\limits_{0}^{2} x^{2}\, dx = \frac{8}{3} \]
The exact value, ≈ 2.666667, for comparison: the error is about 0.00e+00.✓ Proved

Answer \( S_{6} = \frac{8}{3} \approx 2.666667 \)

Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the rule implemented separately in floating point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly states the multiplier for Simpson's Rule as 1/9. The correct multiplier is Δx/3, which equals (2/6)/3 = 1/9 only if Δx=2/3,

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly states the multiplier for Simpson's Rule as 1/9. The correct multiplier is Δx/3, which equals (2/6)/3 = 1/9 only if Δx=2/3,
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-05

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/numerical_integration, checked 2026-10-05 with SymPy 1.14.0.