Trapezoidal, midpoint and Simpson's rules
Problem 4.383 · medium
Approximate \( \displaystyle \int_{1}^{3} \frac{1}{x}\, dx \) using the trapezoidal rule with \( \displaystyle n = 4 \).
- \[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]Δx = (b − a)/n.✓ Proved
- The trapezoidal rule uses the points x = 1, 3/2, 2, 5/2, 3 with weights 1, 2, 2, 2, 1, all times 1/4.
- \[ \frac{1 \cdot 1 \cdot \frac{1}{3} + 2 \cdot 2 \cdot \frac{1}{5} + 1 \cdot 1 + 2 \cdot 1 \cdot \frac{1}{2} + 2 \cdot 2 \cdot \frac{1}{3}}{4} = \frac{67}{60} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{1}^{3} \frac{1}{x}\, dx = \ln{\left(3 \right)} \]The exact value, ≈ 1.098612, for comparison: the error is about 1.81e-02.✓ Proved
Answer \( T_{4} = \frac{67}{60} \approx 1.116667 \)
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 function values are mis‑assigned: f(1.5)=2/3, not 2/5, and the weighted sum is incorrectly ordered. The trapezoidal formula is applied with the wrong values, leading to an incorrect approximation.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-05 — The function values are mis‑assigned: f(1.5)=2/3, not 2/5, and the weighted sum is incorrectly ordered. The trapezoidal formula is applied with the wrong values, leading to an incorrect approximation.qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution lists the correct grid points but applies incorrect function values in the calculation (e.g., using 1/3 for x=1 instead of 1, and 1 for x=2 instead of 1/2). Although the final numerical result is coincidentally correct, the intermediate arithmetic is wrong.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.