Trapezoidal, midpoint and Simpson's rules
Problem 4.384 · medium
Approximate \( \displaystyle \int_{0}^{2} x^{3} + 1\, dx \) using the trapezoidal rule with \( \displaystyle n = 5 \).
- \[ \frac{2}{5} \]Δx = (b − a)/n.✓ Proved
- The trapezoidal rule uses the points x = 0, 2/5, 4/5, 6/5, 8/5, 2 with weights 1, 2, 2, 2, 2, 1, all times 1/5.
- \[ \frac{1 \cdot 1 + 2 \cdot 133 \cdot \frac{1}{125} + 2 \cdot 189 \cdot \frac{1}{125} + 2 \cdot 341 \cdot \frac{1}{125} + 1 \cdot 9 + 2 \cdot 637 \cdot \frac{1}{125}}{5} = \frac{154}{25} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{0}^{2} \left(x^{3} + 1\right)\, dx = 6 \]The exact value, ≈ 6.000000, for comparison: the error is about 1.60e-01.✓ Proved
Answer \( T_{5} = \frac{154}{25} \approx 6.160000 \)
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 uses a multiplier of 1/5 in step 2 and step 3. The trapezoidal rule formula requires multiplying the weighted sum by Δx/2,
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly uses a multiplier of 1/5 in step 2 and step 3. The trapezoidal rule formula requires multiplying the weighted sum by Δx/2,gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution incorrectly uses a multiplier of 1/5 in line 2 and line 3, whereas the trapezoidal rule formula requires a multiplier of Δx/2 = 1/5. This results in an answer that is exactly double the correct value.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.