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Average value over a region

Problem 11.467 · easy

Find the average value of \( \displaystyle f(x, y) = - x + 2 y \) over \( \displaystyle R = [0, 1] \times [0, 1] \).

The average is (1/area) ∬_R f dA.
\[ \int\limits_{0}^{1}\int\limits_{0}^{1} \left(- x + 2 y\right)\, dx\, dy = \frac{1}{2} \]
The double integral.✓ Proved
\[ 1 \frac{1}{1 \cdot 1 \cdot 2} = \frac{1}{2} \]
Divide by the area 1.✓ Proved

Answer \( \frac{1}{2} \)

Lines: 2 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ 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	two-dimensional numerical quadrature

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The double integral of f(x, y) = -x + 2y over the unit square is 1/2, but the average value is the integral divided by the area (which is 1). The solution correctly calculates the integral as 1/2, but then incorrectly divides by 2 in the final step (1/(2*((1*1)))), resulting in 1/2 by coincidence rather than correct logic. The average value should be (1/2) / 1 = 1/2, but the step shown implies dividing by 2, which is wrong.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The double integral of f(x, y) = -x + 2y over the unit square is 1/2, but the average value is the integral divided by the area (which is 1). The solution correctly calculates the integral as 1/2, but then incorrectly divides by 2 in the final step (1/(2*((1*1)))), resulting in 1/2 by coincidence rather than correct logic. The average value should be (1/2) / 1 = 1/2, but the step shown implies dividing by 2, which is wrong.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer returned a non-object
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/average_value_2var, checked 2026-10-18 with SymPy 1.14.0.