Trapezoidal, midpoint and Simpson's rules
Problem 4.982 · medium
Approximate \( \displaystyle \int_{0}^{1} \frac{4}{x^{2} + 1}\, dx \) using the midpoint rule with \( \displaystyle n = 6 \).
- \[ 1 \cdot \frac{1}{6} = \frac{1}{6} \]Δx = (b − a)/n.✓ Proved
- The midpoint rule uses the points x = 1/12, 1/4, 5/12, 7/12, 3/4, 11/12 with weights 1, 1, 1, 1, 1, 1, all times 1/6.
- \[ \frac{1 \cdot 576 \cdot \frac{1}{265} + 1 \cdot 64 \cdot \frac{1}{25} + 1 \cdot 576 \cdot \frac{1}{193} + 1 \cdot 576 \cdot \frac{1}{169} + 1 \cdot 64 \cdot \frac{1}{17} + 1 \cdot 576 \cdot \frac{1}{145}}{6} = \frac{66984845632}{21306239825} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{0}^{1} \frac{4}{x^{2} + 1}\, dx = \pi \]The exact value, ≈ 3.141593, for comparison: the error is about 2.31e-03.✓ Proved
Answer \( M_{6} = \frac{66984845632}{21306239825} \approx 3.143907 \)
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 first function value is miscomputed: f(1/12)=576/145, not 576/265, so the weighted sum and final approximation are incorrect.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-12 — The first function value is miscomputed: f(1/12)=576/145, not 576/265, so the weighted sum and final approximation are incorrect.qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the step size, the midpoints, and the function evaluations. The arithmetic is correct, and the comparison with the exact value is accurate.gpt-oss:20b: pass 2026-10-12
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.