Trapezoidal, midpoint and Simpson's rules
Problem 4.675 · medium
Approximate \( \displaystyle \int_{1}^{2} \frac{1}{x}\, dx \) using the Simpson's rule with \( \displaystyle n = 4 \).
- \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]Δx = (b − a)/n.✓ Proved
- The Simpson's rule uses the points x = 1, 5/4, 3/2, 7/4, 2 with weights 1, 4, 2, 4, 1, all times 1/12.
- \[ \frac{1 \cdot 1 \cdot \frac{1}{2} + 1 \cdot 1 + 2 \cdot 2 \cdot \frac{1}{3} + 4 \cdot 4 \cdot \frac{1}{7} + 4 \cdot 4 \cdot \frac{1}{5}}{12} = \frac{1747}{2520} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{1}^{2} \frac{1}{x}\, dx = \ln{\left(2 \right)} \]The exact value, ≈ 0.693147, for comparison: the error is about 1.07e-04.✓ Proved
Answer \( S_{4} = \frac{1747}{2520} \approx 0.693254 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 uses incorrect function values: it substitutes 1/2 for f(1) and misorders the terms, so the computed Simpson estimate is not derived correctly from the given points.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The weights in line 2 are incorrect for Simpson's Rule with n=4; they should be 1, 4, 2, 4, 1, but the solution lists 1, 4, 2, 4, 1 (which is actual
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The weights in line 2 are incorrect for Simpson's Rule with n=4; they should be 1, 4, 2, 4, 1, but the solution lists 1, 4, 2, 4, 1 (which is actualgpt-oss:20b: fail (error) 2026-10-08 — The weighted sum in step 3 uses incorrect function values: it substitutes 1/2 for f(1) and misorders the terms, so the computed Simpson estimate is not derived correctly from the given points.gpt-oss:20b: fail (error) 2026-10-08 — The weighted sum in step 3 mixes up the function values and the Simpson weights; the correct sum is 1·f(1)+4·f(1.25)+2·f(1.5)+4·f(1.75)+1·f(2).qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly assigns Simpson's weights to the function values. It applies weight 1 to f(1) and f(5/4), whereas the standard pattern for n=4 is 1, 4, 2, 4, 1. Additionally, the equation in step 3 does not match the weights described in step 2 or the correct calculation.
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.