Riemann sums for double integrals
Problem 11.399 · medium
Estimate \( \displaystyle \iint_R x y + 1 \, dA \) over \( \displaystyle R = [0, 1] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 4 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the lower left corners.
- \[ 1 \cdot \frac{1}{2} \cdot 1 \cdot \frac{1}{4} = \frac{1}{8} \]ΔA = Δx Δy.✓ Proved
- The sample points: (0, 0), (0, 1/2), (1/4, 0), (1/4, 1/2), (1/2, 0), (1/2, 1/2), (3/4, 0), (3/4, 1/2).
- \[ 1 \cdot \frac{1}{4} \cdot 1 \cdot \frac{1}{2} \left(1 + 1 + 1 + 1 + 1 + \frac{9}{8} + \frac{5}{4} + \frac{11}{8}\right) = \frac{35}{32} \]ΔA times the sum of f at the sample points.✓ Proved
- \[ \int\limits_{0}^{1}\int\limits_{0}^{1} \left(x y + 1\right)\, dx\, dy = \frac{5}{4} \]For comparison, the exact value is 5/4.✓ Proved
Answer \( \frac{35}{32} \)
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 (error) — The sample points listed include points with y=1/2, which are not lower‑left corners for the subrectangles when n=2. The correct lower‑left corners are (0,0),(1/4,0),(1/2,0),(3/4,0).qwen3.6:27b-mlx: pass — The solution correctly identifies the grid dimensions, sample points, and function evaluations. The arithmetic is correct, and the final estimate matches the stated answer.
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 arithmetic is correct, and the final estimate matches the stated answer.gpt-oss:20b: fail (error) 2026-10-17 — The sample points listed include points with y=1/2, which are not lower‑left corners for the subrectangles when n=2. The correct lower‑left corners are (0,0),(1/4,0),(1/2,0),(3/4,0).qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the grid dimensions, sample points, and function evaluations. The arithmetic is correct, and the comparison with the exact integral is valid.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14gpt-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.