Trapezoidal, midpoint and Simpson's rules
Problem 4.385 · medium
Approximate \( \displaystyle \int_{1}^{2} \frac{1}{x^{2}}\, dx \) 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 = 1, 7/6, 4/3, 3/2, 5/3, 11/6, 2 with weights 1, 4, 2, 4, 2, 4, 1, all times 1/18.
- \[ \frac{1 \cdot 1 \cdot \frac{1}{4} + 2 \cdot 9 \cdot \frac{1}{25} + 1 \cdot 1 + 2 \cdot 9 \cdot \frac{1}{16} + 4 \cdot 36 \cdot \frac{1}{121} + 4 \cdot 4 \cdot \frac{1}{9} + 4 \cdot 36 \cdot \frac{1}{49}}{18} = \frac{96067259}{192099600} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{1}^{2} \frac{1}{x^{2}}\, dx = \frac{1}{2} \]The exact value, ≈ 0.500000, for comparison: the error is about 9.09e-05.✓ Proved
Answer \( S_{6} = \frac{96067259}{192099600} \approx 0.500091 \)
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 is incorrect: the function values and weights are mis‑calculated (e.g., f(7/6)=36/49, not 1/4, and the Simpson weights are 1,4,2,4,2,4,1).qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly applies Simpson's Rule weights. It lists 7 points but assigns weights 1, 4, 2, 4, 2, 4, 1 (which sum to 18) to a sum that is
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 applies Simpson's Rule weights. It lists 7 points but assigns weights 1, 4, 2, 4, 2, 4, 1 (which sum to 18) to a sum that isgpt-oss:20b: fail (error) 2026-10-05 — The weighted sum in step 3 is incorrect: the function values and weights are mis‑calculated (e.g., f(7/6)=36/49, not 1/4, and the Simpson weights are 1,4,2,4,2,4,1).qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution incorrectly applies Simpson's Rule weights. The weights should be 1, 4, 2, 4, 2, 4, 1, but the calculation in line 3 uses weights 1, 2, 1, 2, 4, 4, 4 (or similar incorrect mapping) and divides by 18 instead of 3*Δx = 1/2. Specifically, the term for x=1 (f(1)=1) is multiplied by 1/4 in the numerator before dividing by 18, implying a weight of 1/72, whereas it should be 1/6 * 1 = 1/6. The entire weighted sum is computed incorrectly.gpt-oss:20b: fail (error) 2026-10-05 — The weighted sum in step 3 misapplies Simpson’s weights: it uses 1,2,1,2,4,4,4 instead of 1,4,2,4,2,4,1, leading to an incorrect numerical result.
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.