Trapezoidal, midpoint and Simpson's rules
Problem 4.526 · medium
Approximate \( \displaystyle \int_{1}^{2} \frac{1}{x^{2}}\, dx \) 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 = 13/12, 5/4, 17/12, 19/12, 7/4, 23/12 with weights 1, 1, 1, 1, 1, 1, all times 1/6.
- \[ \frac{1 \cdot 144 \cdot \frac{1}{529} + 1 \cdot 16 \cdot \frac{1}{49} + 1 \cdot 144 \cdot \frac{1}{361} + 1 \cdot 144 \cdot \frac{1}{289} + 1 \cdot 16 \cdot \frac{1}{25} + 1 \cdot 144 \cdot \frac{1}{169}}{6} = \frac{17069881376368}{34277154714075} \]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 2.00e-03.✓ Proved
Answer \( M_{6} = \frac{17069881376368}{34277154714075} \approx 0.497996 \)
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 function values at the midpoints are incorrect: e.g., f(13/12)=144/169, not 144/529. This miscomputes the midpoint rule sum.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The function values at the midpoints are incorrect: e.g., f(13/12)=144/169, not 144/529. This miscomputes the midpoint rule sum.qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-objectqwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-06 — The function value at the first midpoint 13/12 was computed incorrectly as 144/529 instead of 144/169, swapping the values for the first and last midpoints. This makes the weighted sum—and thus the midpoint approximation—incorrect.
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.