Trapezoidal, midpoint and Simpson's rules
Problem 4.976 · medium
Approximate \( \displaystyle \int_{1}^{2} \frac{1}{x}\, dx \) using the trapezoidal rule with \( \displaystyle n = 6 \).
- \[ 1 \cdot \frac{1}{6} = \frac{1}{6} \]Δx = (b − a)/n.✓ Proved
- The trapezoidal rule uses the points x = 1, 7/6, 4/3, 3/2, 5/3, 11/6, 2 with weights 1, 2, 2, 2, 2, 2, 1, all times 1/12.
- \[ \frac{1 \cdot 1 \cdot \frac{1}{2} + 1 \cdot 1 + 2 \cdot 6 \cdot \frac{1}{11} + 2 \cdot 3 \cdot \frac{1}{5} + 2 \cdot 2 \cdot \frac{1}{3} + 2 \cdot 3 \cdot \frac{1}{4} + 2 \cdot 6 \cdot \frac{1}{7}}{12} = \frac{9631}{13860} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{1}^{2} \frac{1}{x}\, dx = \ln{\left(2 \right)} \]The exact value, ≈ 0.693147, for comparison: the error is about 1.73e-03.✓ Proved
Answer \( T_{6} = \frac{9631}{13860} \approx 0.694877 \)
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 solution misstates the function values at the endpoints: it uses 1/2 for f(1) and 1 for f(7/6), which are incorrect. The correct values are f(1)=1 and f(7/6)=6/7. This error invalidates the computed trapezoidal sum.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The solution misstates the function values at the endpoints: it uses 1/2 for f(1) and 1 for f(7/6), which are incorrect. The correct values are f(1)=1 and f(7/6)=6/7. This error invalidates the computed trapezoidal sum.qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The function values in line 3 do not correspond to the x-values listed in line 2. For example, f(7/6) should be 6/7, but the term used is 1/2 (which is f(2)). The calculation mixes up the function evaluations, leading to an incorrect numerical result.
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.