Average value over a region
Problem 11.440 · easy
Find the average value of \( \displaystyle f(x, y) = e^{x + y} \) over \( \displaystyle R = [-1, 1] \times [0, 3] \).
- The average is (1/area) ∬_R f dA.
- \[ \int\limits_{0}^{3}\int\limits_{-1}^{1} e^{x + y}\, dx\, dy = \frac{1 - e^{2}}{e} + \left(-1 + e^{2}\right) e^{2} \]The double integral.✓ Proved
- \[ \frac{1 - e^{2}}{6 e} + \frac{\left(-1 + e^{2}\right) e^{2}}{6} = \frac{- e^{3} - e^{2} + 1 + e^{5}}{6 e} \]Divide by the area 6.✓ Proved
Answer \( \frac{- e^{3} - e^{2} + 1 + e^{5}}{6 e} \)
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 is computed incorrectly; the correct value is (e - e^{-1})(e^3 - 1), not the expression given.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: fail (error) 2026-10-17 — The double integral is computed incorrectly; the correct value is (e - e^{-1})(e^3 - 1), not the expression given.qwen3.6:27b-mlx: pass 2026-10-15gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the definition of the average value of a function over a region. The integration bounds and area calculation are correct, and the final algebraic simplification matches the stated answer.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.