Riemann sums for double integrals
Problem 11.454 · medium
Estimate \( \displaystyle \iint_R x^{2} + y \, dA \) over \( \displaystyle R = [0, 2] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the midpoint points.
- \[ 1 \cdot \frac{1}{2} \cdot 2 \cdot \frac{1}{2} = \frac{1}{2} \]ΔA = Δx Δy.✓ Proved
- The sample points: (1/2, 1/4), (1/2, 3/4), (3/2, 1/4), (3/2, 3/4).
- \[ 1 \cdot 1 \cdot \frac{1}{2} \left(1 \cdot \frac{1}{2} + 1 + \frac{5}{2} + 3\right) = \frac{7}{2} \]ΔA times the sum of f at the sample points.✓ Proved
- \[ \int\limits_{0}^{1}\int\limits_{0}^{2} \left(x^{2} + y\right)\, dx\, dy = \frac{11}{3} \]For comparison, the exact value is 11/3.✓ Proved
Answer \( \frac{7}{2} \)
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: fail (misleading) — The exact integral is 19/6, not 11/3; the comparison statement is incorrect.qwen3.6:27b-mlx: pass — The solution correctly identifies the subrectangle area, the midpoint sample points, and the resulting Riemann sum. The comparison with the exact integral is accurate.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the subrectangle area, the midpoint sample points, and the resulting Riemann sum. The comparison with the exact integral is accurate.gpt-oss:20b: fail (misleading) 2026-10-18 — The exact integral is 19/6, not 11/3; the comparison statement is incorrect.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the subrectangle area, the midpoint sample points, and computes the Riemann sum accurately. The comparison with the exact integral is correct and serves as a valid sanity check.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the subrectangle area, the midpoint sample points, and the resulting Riemann sum. The comparison with the exact integral is also correct.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.