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Trapezoidal, midpoint and Simpson's rules

Problem 4.684 · medium

Approximate \( \displaystyle \int_{0}^{1} \frac{1}{x^{2} + 1}\, dx \) 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
\[ \frac{1 \cdot 1 \cdot \frac{1}{2} + 1 \cdot 1 + 2 \cdot 16 \cdot \frac{1}{25} + 2 \cdot 4 \cdot \frac{1}{5} + 2 \cdot 16 \cdot \frac{1}{17}}{8} = \frac{5323}{6800} \]
Weighted sum of the function values.✓ Proved
\[ \int\limits_{0}^{1} \frac{1}{x^{2} + 1}\, dx = \frac{\pi}{4} \]
The exact value, ≈ 0.785398, for comparison: the error is about 2.60e-03.✓ Proved

Answer \( T_{4} = \frac{5323}{6800} \approx 0.782794 \)

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 with n=4. The arithmetic is verified as correct, and the comparison with the exact value is accurate.

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 with n=4. The arithmetic is verified as correct, and the comparison with the exact value is accurate.
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 states that the weights are multiplied by 1/8. The trapezoidal rule formula is (Δx/2) * [f(x0) + 2f(x1) + ... + f(xn)
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.