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Riemann sums for double integrals

Problem 11.443 · medium

Estimate \( \displaystyle \iint_R x y + 1 \, dA \) 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 midpoint points.

\[ 2 \cdot \frac{1}{2} \cdot 4 \cdot \frac{1}{4} = 1 \]
ΔA = Δx Δy.✓ Proved
The sample points: (1/2, 1/2), (1/2, 3/2), (3/2, 1/2), (3/2, 3/2), (5/2, 1/2), (5/2, 3/2), (7/2, 1/2), (7/2, 3/2).
Reviewed
\[ 1 \cdot 1 \left(\frac{5}{4} + \frac{7}{4} + \frac{7}{4} + \frac{9}{4} + \frac{11}{4} + \frac{13}{4} + \frac{19}{4} + \frac{25}{4}\right) = 24 \]
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{4} \left(x y + 1\right)\, dx\, dy = 24 \]
For comparison, the exact value is 24.✓ Proved

Answer \( 24 \)

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 subrectangle area, lists the correct midpoint sample points, and computes the Riemann sum accurately. The comparison with the exact integral is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the subrectangle area, lists the correct midpoint sample points, and computes the Riemann sum accurately. The comparison with the exact integral is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the subinterval widths, midpoints, and function evaluations. The calculation is accurate and matches the exact integral.
gpt-oss:20b: fail (error) 2026-10-15 — The Riemann sum calculation is incorrect: the sum of f(x,y) at the midpoints is 95/4=23.75, not 24. The final value should be 23.75, not 24.
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the subrectangle area, lists the correct midpoint sample points, and computes the Riemann sum accurately. The comparison with the exact integral is also correct.
gpt-oss:20b: fail (error) 2026-10-14 — The Riemann sum calculation is incorrect: the function values at the midpoints sum to 95/4, giving 23.75, not 24. The conclusion that the approximate integral equals 24 is therefore false.

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.