Trapezoidal, midpoint and Simpson's rules
Problem 4.977 · medium
Approximate \( \displaystyle \int_{1}^{2} \frac{1}{x}\, dx \) using the midpoint rule with \( \displaystyle n = 4 \).
- \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]Δx = (b − a)/n.✓ Proved
- The midpoint rule uses the points x = 9/8, 11/8, 13/8, 15/8 with weights 1, 1, 1, 1, all times 1/4.
- \[ \frac{1 \cdot 8 \cdot \frac{1}{15} + 1 \cdot 8 \cdot \frac{1}{13} + 1 \cdot 8 \cdot \frac{1}{11} + 1 \cdot 8 \cdot \frac{1}{9}}{4} = \frac{4448}{6435} \]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 1.93e-03.✓ Proved
Answer \( M_{4} = \frac{4448}{6435} \approx 0.691220 \)
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 returned a non-object
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer returned a non-object
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.