Trapezoidal, midpoint and Simpson's rules
Problem 4.382 · medium
Approximate \( \displaystyle \int_{0}^{1} \frac{1}{x^{2} + 1}\, dx \) using the midpoint rule with \( \displaystyle n = 5 \).
- \[ 1 \cdot \frac{1}{5} = \frac{1}{5} \]Δx = (b − a)/n.✓ Proved
- The midpoint rule uses the points x = 1/10, 3/10, 1/2, 7/10, 9/10 with weights 1, 1, 1, 1, 1, all times 1/5.Reviewed
- \[ \frac{1 \cdot 100 \cdot \frac{1}{181} + 1 \cdot 100 \cdot \frac{1}{149} + 1 \cdot 4 \cdot \frac{1}{5} + 1 \cdot 100 \cdot \frac{1}{109} + 1 \cdot 100 \cdot \frac{1}{101}}{5} = \frac{5835836884}{7422543025} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{0}^{1} \frac{1}{x^{2} + 1}\, dx = \frac{\pi}{4} \]The exact value, ≈ 0.785398, for comparison: the error is about 8.33e-04.✓ Proved
Answer \( M_{5} = \frac{5835836884}{7422543025} \approx 0.786231 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the subinterval width, the midpoints, and the function evaluations. The final approximation is accurate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the subinterval width, the midpoints, and the function evaluations. The final approximation is accurate.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the subinterval width, the midpoints, and the function evaluations. The final arithmetic and comparison with the exact value are correct.gpt-oss:20b: pass 2026-10-05
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.