Trapezoidal, midpoint and Simpson's rules
Problem 4.530 · medium
Approximate \( \displaystyle \int_{0}^{1} \frac{1}{x^{2} + 1}\, dx \) using the Simpson's rule with \( \displaystyle n = 6 \).
- \[ 1 \cdot \frac{1}{6} = \frac{1}{6} \]Δx = (b − a)/n.✓ Proved
- The Simpson's rule uses the points x = 0, 1/6, 1/3, 1/2, 2/3, 5/6, 1 with weights 1, 4, 2, 4, 2, 4, 1, all times 1/18.
- \[ \frac{1 \cdot 1 \cdot \frac{1}{2} + 1 \cdot 1 + 2 \cdot 9 \cdot \frac{1}{13} + 2 \cdot 9 \cdot \frac{1}{10} + 4 \cdot 36 \cdot \frac{1}{61} + 4 \cdot 4 \cdot \frac{1}{5} + 4 \cdot 36 \cdot \frac{1}{37}}{18} = \frac{829597}{1056276} \]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 2.18e-07.✓ Proved
Answer \( S_{6} = \frac{829597}{1056276} \approx 0.785398 \)
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: fail (error) — The weighted sum in step 3 is incorrect: the function values and their corresponding weights are mismatched. The correct Simpson sum should use f(0)=1, f(1/6)=36/37, f(1/3)=9/10, f(1/2)=4/5, f(2/3)=9/13, f(5/6)=36/61, f(1)=1 with weights 1,4,2,4,2,4,1. The given expression misorders the terms and thus gives an incorrect numerical result.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The weights listed in step 2 are incorrect for Simpson's Rule with n=6; the sequence should be 1, 4, 2, 4, 2, 4, 1, but the solution lists 1, 4, 2,
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The weighted sum in step 3 is incorrect: the function values and their corresponding weights are mismatched. The correct Simpson sum should use f(0)=1, f(1/6)=36/37, f(1/3)=9/10, f(1/2)=4/5, f(2/3)=9/13, f(5/6)=36/61, f(1)=1 with weights 1,4,2,4,2,4,1. The given expression misorders the terms and thus gives an incorrect numerical result.qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The weights listed in step 2 are incorrect for Simpson's Rule with n=6; the sequence should be 1, 4, 2, 4, 2, 4, 1, but the solution lists 1, 4, 2,qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly assigns the weights for Simpson's Rule. The weights for n=6 should be 1, 4, 2, 4, 2, 4, 1, but the solution lists them as 1, 4, 2, 4, 2, 4, 1 in the text description while the equation in step 3 applies weights 1, 1, 2, 2, 4, 4, 4 to the function values (or rather, it seems to have garbled the weights and function evaluations entirely, e.g., using f(0)=1 with weight 1/2? No, looking closely at step 3: it sums terms like 1*(1/2) which is wrong for f(0)=1. Actually, step 3 is a mess of incorrect arithmetic and weight application. Specifically, f(0)=1, but the term is 1*(1/2). f(1)=1/2, but the term is 1*1. The weights and function values are swapped or miscalculated. Furthermore, the multiplier is 1/18, but Simpson's rule with h=1/6 uses h/3 = 1/18, so the multiplier is correct, but the weighted sum inside is wrong.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.