Riemann sums for double integrals
Problem 11.447 · medium
Estimate \( \displaystyle \iint_R x + y \, dA \) over \( \displaystyle R = [0, 2] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 4 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the lower left corners.
- \[ 1 \cdot \frac{1}{2} \cdot 2 \cdot \frac{1}{4} = \frac{1}{4} \]ΔA = Δx Δy.✓ Proved
- The sample points: (0, 0), (0, 1/2), (1/2, 0), (1/2, 1/2), (1, 0), (1, 1/2), (3/2, 0), (3/2, 1/2).
- \[ 1 \cdot \frac{1}{2} \cdot 1 \cdot \frac{1}{2} \left(0 + 1 \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} + 1 + 1 + \frac{3}{2} + \frac{3}{2} + 2\right) = 2 \]ΔA times the sum of f at the sample points.✓ Proved
- \[ \int\limits_{0}^{1}\int\limits_{0}^{2} \left(x + y\right)\, dx\, dy = 3 \]For comparison, the exact value is 3.✓ Proved
Answer \( 2 \)
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 (error) — The sample points listed in sentence 2 are incorrect: the lower‑left corners of the 4×2 grid are (0,0),(0.5,0),(1,0),(1.5,0) and (0,0.5),(0.5,0.5),(1,0.5),(1.5,0.5). The solution omitted (0,0) and (0,0.5) and included an extra (0,1/2).qwen3.6:27b-mlx: pass — The solution correctly identifies the grid dimensions, sample points, and area element. The calculation of the Riemann sum is accurate, and the comparison with the exact integral is valid.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the grid dimensions, sample points, and area element. The calculation of the Riemann sum is accurate, and the comparison with the exact integral is valid.gpt-oss:20b: fail (error) 2026-10-18 — The sample points listed in sentence 2 are incorrect: the lower‑left corners of the 4×2 grid are (0,0),(0.5,0),(1,0),(1.5,0) and (0,0.5),(0.5,0.5),(1,0.5),(1.5,0.5). The solution omitted (0,0) and (0,0.5) and included an extra (0,1/2).qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The sample points listed in step 2 are incorrect for 'lower left corners'. For the grid defined by m=4 and n=2, the lower left corners should have y-coordinates of 0 for all points, but the list includes points with y=1/2 (which are upper-left corners for the bottom row of subrectangles). Consequently, the sum in step 3 uses incorrect function values.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-objectgpt-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 2026-10-18 with SymPy 1.14.0.