Trapezoidal, midpoint and Simpson's rules
Problem 4.834 · medium
Approximate \( \displaystyle \int_{0}^{1} \frac{4}{x^{2} + 1}\, dx \) using the Simpson's rule with \( \displaystyle n = 4 \).
- \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]Δx = (b − a)/n.✓ Proved
- The Simpson's rule uses the points x = 0, 1/4, 1/2, 3/4, 1 with weights 1, 4, 2, 4, 1, all times 1/12.
- \[ \frac{1 \cdot 2 + 1 \cdot 4 + 2 \cdot 16 \cdot \frac{1}{5} + 4 \cdot 64 \cdot \frac{1}{25} + 4 \cdot 64 \cdot \frac{1}{17}}{12} = \frac{8011}{2550} \]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 2.40e-05.✓ Proved
Answer \( S_{4} = \frac{8011}{2550} \approx 3.141569 \)
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 weighted sum in step 3 is incorrect: the function values are 4, 64/17, 16/5, 64/25, 4, and the correct Simpson sum is (4+4+4·64/17+2·16/5+4·64/25)/12 ≈ 3.3082, not 8011/2550. The stated result is therefore wrong.qwen3.6:27b-mlx: pass — The solution correctly identifies the step size, nodes, and weights for Simpson's rule with n=4. The arithmetic is verified as correct, and the comparison with the exact value is accurate.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-10 — The weighted sum in step 3 is incorrect: the function values are 4, 64/17, 16/5, 64/25, 4, and the correct Simpson sum is (4+4+4·64/17+2·16/5+4·64/25)/12 ≈ 3.3082, not 8011/2550. The stated result is therefore wrong.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the step size, nodes, and weights for Simpson's rule with n=4. The arithmetic is verified as correct, and the comparison with the exact value is accurate.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the step size, nodes, and weights for Simpson's rule with n=4. The arithmetic in the weighted sum is correct, leading to the stated approximation.gpt-oss:20b: fail (error) 2026-10-10 — The first term in the weighted sum is incorrect: f(0)=4, not 2. This makes the Simpson estimate wrong.
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 2026-10-10 with SymPy 1.14.0.