Trapezoidal, midpoint and Simpson's rules
Problem 4.979 · medium
Approximate \( \displaystyle \int_{0}^{2} x^{2}\, dx \) using the Simpson's rule with \( \displaystyle n = 4 \).
- \[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]Δx = (b − a)/n.✓ Proved
- The Simpson's rule uses the points x = 0, 1/2, 1, 3/2, 2 with weights 1, 4, 2, 4, 1, all times 1/6.
- \[ \frac{1 \cdot 0 + 4 \cdot 1 \cdot \frac{1}{4} + 2 \cdot 1 + 1 \cdot 4 + 4 \cdot 9 \cdot \frac{1}{4}}{6} = \frac{8}{3} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{0}^{2} x^{2}\, dx = \frac{8}{3} \]The exact value, ≈ 2.666667, for comparison: the error is about 0.00e+00.✓ Proved
Answer \( S_{4} = \frac{8}{3} \approx 2.666667 \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly states that the weights are multiplied by 1/6. Simpson's Rule uses a factor of Δx/3 (which is 1/6 here), but the standard d
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly states that the weights are multiplied by 1/6. Simpson's Rule uses a factor of Δx/3 (which is 1/6 here), but the standard dqwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly states that the weights are multiplied by 1/6. The standard Simpson's rule formula is (Δx/3) * [f(x0) + 4f(x1) + ...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.