Home›Calculus 3›Average value over a region›Problem 11.513

Average value over a region

Problem 11.513 · easy

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

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

Answer \( \frac{e \left(- e^{2} - e + 1 + e^{3}\right)}{2} \)

✓ 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19
gpt-oss:20b: pass 2026-10-19
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 setup, integration bounds, and algebraic simplification are all correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly sets up the average value formula, computes the double integral, and divides by the area of the rectangle. The algebraic simplification matches the stated answer.
gpt-oss:20b: pass 2026-10-16

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 2026-10-19 with SymPy 1.14.0.