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Riemann sums for double integrals

Problem 11.465 · medium

Estimate \( \displaystyle \iint_R 8 x y + 4 x + 2 y \, dA \) over \( \displaystyle R = [0, 1] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the midpoint points.

\[ 1 \cdot \frac{1}{2} \cdot 1 \cdot \frac{1}{2} = \frac{1}{4} \]
ΔA = Δx Δy.✓ Proved
The sample points: (1/4, 1/4), (1/4, 3/4), (3/4, 1/4), (3/4, 3/4).
Reviewed
\[ 1 \cdot \frac{1}{2} \cdot 1 \cdot \frac{1}{2} \left(2 + 4 + 5 + 9\right) = 5 \]
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{1} \left(8 x y + 4 x + 2 y\right)\, dx\, dy = 5 \]
For comparison, the exact value is 5.✓ Proved

Answer \( 5 \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

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 sum recomputed in exact fractions

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the subregion areas and midpoint sample points. The function evaluations and final summation are correct, matching the exact integral value.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the subregion areas and midpoint sample points. The function evaluations and final summation are correct, matching the exact integral value.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the subregion areas, the midpoint sample points, and the function evaluations. The final summation yields the correct estimate, which coincidentally equals the exact integral value.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the area of each subrectangle and the midpoint sample points. The evaluation of the function at these points and the final summation are correct, matching the exact integral value.
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.