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Trapezoidal, midpoint and Simpson's rules

Problem 4.528 · medium

Approximate \( \displaystyle \int_{1}^{2} \frac{1}{x}\, dx \) using the midpoint rule with \( \displaystyle n = 5 \).

\[ 1 \cdot \frac{1}{5} = \frac{1}{5} \]
Δx = (b − a)/n.✓ Proved
The midpoint rule uses the points x = 11/10, 13/10, 3/2, 17/10, 19/10 with weights 1, 1, 1, 1, 1, all times 1/5.
\[ \frac{1 \cdot 10 \cdot \frac{1}{19} + 1 \cdot 10 \cdot \frac{1}{17} + 1 \cdot 2 \cdot \frac{1}{3} + 1 \cdot 10 \cdot \frac{1}{13} + 1 \cdot 10 \cdot \frac{1}{11}}{5} = \frac{479378}{692835} \]
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 1.24e-03.✓ Proved

Answer \( M_{5} = \frac{479378}{692835} \approx 0.691908 \)

Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 weighted sum in step 3 is computed incorrectly; the correct midpoint approximation is (10/11+10/13+2/3+10/17+10/19)/5 ≈ 0.6899, not 0.6919. The stated fraction 479378/692835 is not equal to this sum.
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

gpt-oss:20b: fail (error) 2026-10-06 — The weighted sum in step 3 is computed incorrectly; the correct midpoint approximation is (10/11+10/13+2/3+10/17+10/19)/5 ≈ 0.6899, not 0.6919. The stated fraction 479378/692835 is not equal to this sum.
qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-object
qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-object
gpt-oss:20b: fail (error) 2026-10-06 — The arithmetic in step 3 is incorrect: the average of the function values at the midpoints is about 0.6739, not 0.6919. The stated fraction 479378/692835 does not equal the correct midpoint approximation.

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 2026-10-06 with SymPy 1.14.0.