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Riemann sums for double integrals

Problem 11.400 · medium

Estimate \( \displaystyle \iint_R 3 x - y^{2} + 5 \, dA \) over \( \displaystyle R = [0, 4] \times [0, 2] \) with a Riemann sum, \( \displaystyle m = 4 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the lower left corners.

\[ 2 \cdot \frac{1}{2} \cdot 4 \cdot \frac{1}{4} = 1 \]
ΔA = Δx Δy.✓ Proved
The sample points: (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1).
Reviewed
\[ 1 \cdot 1 \left(4 + 5 + 7 + 8 + 10 + 11 + 13 + 14\right) = 72 \]
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{4} \left(3 x - y^{2} + 5\right)\, dx\, dy = \frac{232}{3} \]
For comparison, the exact value is 232/3.✓ Proved

Answer \( 72 \)

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. 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 grid dimensions, sample points, and function evaluations. The final Riemann sum calculation is accurate.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the grid dimensions, sample points, and function evaluations. The final Riemann sum calculation is accurate.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the grid dimensions, sample points, and function evaluations. The final Riemann sum calculation is accurate.
gpt-oss:20b: fail (error) 2026-10-15 — The sample point (0,1) evaluates to 4, not 4? The sum used in step 3 is incorrect; the correct Riemann sum is 1*(5+4+8+7+11+10+14+13)=72, which matches the answer, but the explanation of the values is misleading. The solution should explicitly list the correct function values at each sample point.
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the subrectangle area, the sample points for lower-left corners, and the function evaluations. The final sum is correct.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.