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Riemann sums for double integrals

Problem 11.455 · medium

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

\[ 2 \cdot \frac{1}{2} \cdot 1 \cdot \frac{1}{4} = \frac{1}{4} \]
ΔA = Δx Δy.✓ Proved
The sample points: (1/8, 1/2), (1/8, 3/2), (3/8, 1/2), (3/8, 3/2), (5/8, 1/2), (5/8, 3/2), (7/8, 1/2), (7/8, 3/2).
Reviewed
\[ 1 \cdot 1 \cdot \frac{1}{4} \left(2 + 4 + 5 + 6 + 8 + 9 + 13 + 17\right) = 16 \]
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{1} \left(8 x y + 4 x + 2 y\right)\, dx\, dy = 16 \]
For comparison, the exact value is 16.✓ Proved

Answer \( 16 \)

✓ 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 subinterval widths, midpoints, and function values. The final sum is calculated correctly and matches the exact integral.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the subinterval widths, midpoints, and function values. The final sum is calculated correctly and matches the exact integral.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the grid dimensions, midpoints, and area element. The arithmetic in the Riemann sum is correct, and the comparison with the exact integral is accurate.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the grid dimensions, midpoints, and area element. The arithmetic in the Riemann sum is correct, and the comparison with the exact integral is accurate.
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.