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Average value over a region

Problem 11.480 · easy

Find the average value of \( \displaystyle f(x, y) = 3 x + y^{2} \) over \( \displaystyle R = [1, 3] \times [0, 3] \).

The average is (1/area) ∬_R f dA.
\[ \int\limits_{0}^{3}\int\limits_{1}^{3} \left(3 x + y^{2}\right)\, dx\, dy = 54 \]
The double integral.✓ Proved
\[ \frac{54}{2 \cdot 3} = 9 \]
Divide by the area 6.✓ Proved

Answer \( 9 \)

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: fail (error) — The double integral was computed incorrectly; the correct value is 30, not 54, so the average value is 30/6=5, not 9.
qwen3.6:27b-mlx: pass — The solution correctly applies the definition of the average value of a function over a region. The area calculation and the final division are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the definition of the average value of a function over a region. The area calculation and the final division are correct.
gpt-oss:20b: fail (error) 2026-10-18 — The double integral was computed incorrectly; the correct value is 30, not 54, so the average value is 30/6=5, not 9.
qwen3.6:27b-mlx: pass 2026-10-16
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the definition of the average value of a function over a region. The area calculation and the final division are correct.
gpt-oss:20b: fail (error) 2026-10-15 — The double integral was miscomputed: ∫₁³∫₀³(3x+y²)dy dx = 30, not 54. Consequently the average value is 30/6 = 5, not 9.

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.