Riemann sums for double integrals
Problem 11.464 · medium
Estimate \( \displaystyle \iint_R 8 x y + 4 x + 2 y \, dA \) over \( \displaystyle R = [0, 4] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 4 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the lower left corners.
- \[ 1 \cdot \frac{1}{2} \cdot 4 \cdot \frac{1}{4} = \frac{1}{2} \]ΔA = Δx Δy.✓ Proved
- The sample points: (0, 0), (0, 1/2), (1, 0), (1, 1/2), (2, 0), (2, 1/2), (3, 0), (3, 1/2).
- \[ 1 \cdot 1 \cdot \frac{1}{2} \left(0 + 1 + 4 + 8 + 9 + 12 + 17 + 25\right) = 38 \]ΔA times the sum of f at the sample points.✓ Proved
- \[ \int\limits_{0}^{1}\int\limits_{0}^{4} \left(8 x y + 4 x + 2 y\right)\, dx\, dy = 68 \]For comparison, the exact value is 68.✓ Proved
Answer \( 38 \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the grid dimensions, sample points, and function values. The arithmetic for the Riemann sum is correct, and the comparison with the exact integral is accurate.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the grid dimensions, sample points, and area element. The arithmetic for the Riemann sum is correct, and the comparison with the exact integral is accurate.gpt-oss:20b: fail (error) 2026-10-15 — The exact integral is 65, not 68; the comparison line is incorrect.
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.