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Riemann sums: left, right and midpoint

Problem 4.405 · medium

Compute the midpoint Riemann sum \( \displaystyle M_{4} \) for \( \displaystyle f(x) = - x^{2} - x + 13 \) on \( \displaystyle [0, 1] \).

\[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]
Δx = (b − a)/n.✓ Proved
The midpoint points are x = 1/8, 3/8, 5/8, 7/8.
\[ \frac{\frac{727}{64} + \frac{767}{64} + \frac{799}{64} + \frac{823}{64}}{4} = \frac{779}{64} \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 73/6 ≈ 12.1667.

Answer \( M_{4} = \frac{779}{64} \approx 12.1719 \)

Lines: 2 proved, 2 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	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the sum recomputed in 30-digit floating point

Reviewers

gpt-oss:20b: fail (error) — The midpoint values are mis‑computed: the correct function values are 823/64, 799/64, 767/64, 727/64, giving a sum of 3216/64. Multiplying by Δx=1/4 yields M₄=3216/256=12.5625, not 779/64. The solution’s arithmetic is incorrect.
qwen3.6:27b-mlx: pass

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05
gpt-oss:20b: fail (error) 2026-10-05 — The midpoint values are mis‑computed: the correct function values are 823/64, 799/64, 767/64, 727/64, giving a sum of 3216/64. Multiplying by Δx=1/4 yields M₄=3216/256=12.5625, not 779/64. The solution’s arithmetic is incorrect.
qwen3.6:27b-mlx: pass 2026-10-05
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/riemann_sums, checked 2026-10-05 with SymPy 1.14.0.