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Riemann sums for double integrals

Problem 11.458 · medium

Estimate \( \displaystyle \iint_R x^{2} + y \, dA \) over \( \displaystyle R = [0, 4] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 4 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the midpoint points.

\[ 1 \cdot \frac{1}{2} \cdot 4 \cdot \frac{1}{4} = \frac{1}{2} \]
ΔA = Δx Δy.✓ Proved
The sample points: (1/2, 1/4), (1/2, 3/4), (3/2, 1/4), (3/2, 3/4), (5/2, 1/4), (5/2, 3/4), (7/2, 1/4), (7/2, 3/4).
\[ 1 \cdot 1 \cdot \frac{1}{2} \left(1 \cdot \frac{1}{2} + 1 + \frac{5}{2} + 3 + \frac{13}{2} + 7 + \frac{25}{2} + 13\right) = 23 \]
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{4} \left(x^{2} + y\right)\, dx\, dy = \frac{70}{3} \]
For comparison, the exact value is 70/3.✓ Proved

Answer \( 23 \)

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. How we verify

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 sum recomputed in exact fractions

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution lists only 4 sample points in step 2, but the problem specifies m=4 and n=2, requiring 8 subrectangles and thus 8 sample points. The calculation in step 3 correctly sums 8 terms, but the explicit listing in step 2 is incomplete and misleading.
gpt-oss:20b: pass 2026-10-15

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/double_riemann, checked 2026-10-18 with SymPy 1.14.0.