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Average value over a region

Problem 11.538 · easy

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

The average is (1/area) ∬_R f dA.
\[ \int\limits_{0}^{1}\int\limits_{-1}^{1} x^{2} y\, dx\, dy = \frac{1}{3} \]
The double integral.✓ Proved
\[ 1 \frac{1}{1 \cdot 2 \cdot 3} = \frac{1}{6} \]
Divide by the area 2.✓ Proved

Answer \( \frac{1}{6} \)

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 area of the region R is 2, but the solution divides by 3 * (2*1) = 6 instead of just 2. This results in an incorrect denominator for the average value formula.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-23 — The area of the region R is 2, but the solution divides by 3 * (2*1) = 6 instead of just 2. This results in an incorrect denominator for the average value formula.
gpt-oss:20b: pass 2026-10-23
qwen3.6:27b-mlx: fail (error) 2026-10-22 — The area of the region R is 2, but the solution divides by 3 * (2*1) = 6 instead of just 2. This results in an incorrect denominator for the average value formula.
gpt-oss:20b: pass 2026-10-22
qwen3.6:27b-mlx: fail (error) 2026-10-22 — The area of the region R is calculated incorrectly as 3*2=6 in the denominator, whereas the actual area is (1 - (-1)) * (1 - 0) = 2. This leads to an incorrect final answer.
gpt-oss:20b: pass 2026-10-22

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 2026-10-23 with SymPy 1.14.0.