Double integrals over rectangles
Problem 11.134 · easy
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 2] \).
- By Fubini's theorem, integrate in y first, then in x.Reviewed
- \[ \int\limits_{0}^{2} \left(x + 2 y\right)\, dy = 2 x + 4 \]The inner integral.✓ Proved
- \[ \int\limits_{0}^{3} \left(2 x + 4\right)\, dx = 21 \]The outer integral.✓ Proved
Answer \( 21 \)
✓ 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.
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 | 2-D numerical quadrature (mpmath) agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies Fubini's theorem to the rectangular region and performs the integration steps accurately.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies Fubini's theorem to the rectangular region and performs the integration steps accurately.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies Fubini's theorem to the rectangular region and performs the integration steps accurately.gpt-oss:20b: pass 2026-09-27
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/double_integral_rectangle, checked 2026-09-27 with SymPy 1.14.0.