Home›Calculus 3›Average value over a region›Problem 11.469

Average value over a region

Problem 11.469 · easy

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

The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{2}\int\limits_{-1}^{1} \left(3 x + y^{2}\right)\, dx\, dy = \frac{16}{3} \]
The double integral.✓ Proved
\[ \frac{16}{2 \cdot 2 \cdot 3} = \frac{4}{3} \]
Divide by the area 4.✓ Proved

Answer \( \frac{4}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the formula for the average value, computes the double integral over the specified rectangular region, and divides by the correct area of the region.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the formula for the average value, computes the double integral over the specified rectangular region, and divides by the correct area of the region.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the definition of the average value of a function over a region. The area calculation and integration are correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the formula for the average value, computes the double integral and area accurately, and performs the final division correctly.
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.