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Trapezoidal, midpoint and Simpson's rules

Problem 4.677 · medium

Approximate \( \displaystyle \int_{0}^{1} \frac{4}{x^{2} + 1}\, dx \) using the trapezoidal rule with \( \displaystyle n = 4 \).

\[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]
Δx = (b − a)/n.✓ Proved
The trapezoidal rule uses the points x = 0, 1/4, 1/2, 3/4, 1 with weights 1, 2, 2, 2, 1, all times 1/8.
Reviewed
\[ \frac{1 \cdot 2 + 1 \cdot 4 + 2 \cdot 64 \cdot \frac{1}{25} + 2 \cdot 16 \cdot \frac{1}{5} + 2 \cdot 64 \cdot \frac{1}{17}}{8} = \frac{5323}{1700} \]
Weighted sum of the function values.✓ Proved
\[ \int\limits_{0}^{1} \frac{4}{x^{2} + 1}\, dx = \pi \]
The exact value, ≈ 3.141593, for comparison: the error is about 1.04e-02.✓ Proved

Answer \( T_{4} = \frac{5323}{1700} \approx 3.131176 \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the step size, grid points, and weights for the trapezoidal rule. The arithmetic in the weighted sum is correct, leading to the stated approximation.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the step size, grid points, and weights for the trapezoidal rule. The arithmetic in the weighted sum is correct, leading to the stated approximation.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly applies a factor of 1/8 instead of the correct factor of Δx/2 = 1/8 * 2 = 1/4? No, the formula is (Δx/2) * sum. Δx =
gpt-oss:20b: pass 2026-10-08

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 2026-10-08 with SymPy 1.14.0.