Trapezoidal, midpoint and Simpson's rules
Problem 4.981 · medium
Approximate \( \displaystyle \int_{1}^{2} \frac{1}{x^{2}}\, dx \) using the trapezoidal rule with \( \displaystyle n = 6 \).
- \[ 1 \cdot \frac{1}{6} = \frac{1}{6} \]Δx = (b − a)/n.✓ Proved
- The trapezoidal rule uses the points x = 1, 7/6, 4/3, 3/2, 5/3, 11/6, 2 with weights 1, 2, 2, 2, 2, 2, 1, all times 1/12.
- \[ \frac{1 \cdot 1 \cdot \frac{1}{4} + 2 \cdot 36 \cdot \frac{1}{121} + 2 \cdot 9 \cdot \frac{1}{25} + 2 \cdot 4 \cdot \frac{1}{9} + 1 \cdot 1 + 2 \cdot 9 \cdot \frac{1}{16} + 2 \cdot 36 \cdot \frac{1}{49}}{12} = \frac{64548859}{128066400} \]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 4.03e-03.✓ Proved
Answer \( T_{6} = \frac{64548859}{128066400} \approx 0.504026 \)
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 are assigned to the wrong points, so the computed trapezoidal approximation does not match the correct value.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The list of points in step 2 is missing x = 4/3 (or rather, the weights are misaligned with the points listed; there are 7 points listed but the wei
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The list of points in step 2 is missing x = 4/3 (or rather, the weights are misaligned with the points listed; there are 7 points listed but the weigpt-oss:20b: fail (error) 2026-10-12 — The weighted sum in step 3 is incorrect: the function values are assigned to the wrong points, so the computed trapezoidal approximation does not match the correct value.qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The list of points in step 2 is missing x = 4/3 (or rather, the function evaluation for x=4/3 is missing from the sum in step 3, and the weights aregpt-oss:20b: fail (error) 2026-10-12 — The function values are mis‑computed: e.g., f(7/6)=36/49 not 36/121, f(4/3)=9/16 not 9/25, etc. The weighted sum is therefore wrong, leading to an incorrect trapezoidal estimate.
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 2026-10-12 with SymPy 1.14.0.