Home›Calculus 2›Trapezoidal, midpoint and Simpson's rules›Problem 4.679

Trapezoidal, midpoint and Simpson's rules

Problem 4.679 · medium

Approximate \( \displaystyle \int_{1}^{3} \frac{1}{x}\, dx \) using the midpoint rule with \( \displaystyle n = 4 \).

\[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]
Δx = (b − a)/n.✓ Proved
The midpoint rule uses the points x = 5/4, 7/4, 9/4, 11/4 with weights 1, 1, 1, 1, all times 1/2.
Reviewed
\[ \frac{1 \cdot 4 \cdot \frac{1}{11} + 1 \cdot 4 \cdot \frac{1}{9} + 1 \cdot 4 \cdot \frac{1}{7} + 1 \cdot 4 \cdot \frac{1}{5}}{2} = \frac{3776}{3465} \]
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 8.86e-03.✓ Proved

Answer \( M_{4} = \frac{3776}{3465} \approx 1.089755 \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the subinterval width and midpoints, applies the midpoint rule formula accurately, and provides the correct final approximation.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the subinterval width and midpoints, applies the midpoint rule formula accurately, and provides the correct final approximation.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-08

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 2026-10-08 with SymPy 1.14.0.