Double integrals over rectangles
Problem 11.150 · easy
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 3] \).
- By Fubini's theorem, integrate in y first, then in x.Reviewed
- \[ \int\limits_{0}^{3} 6 x^{2} y\, dy = 27 x^{2} \]The inner integral.✓ Proved
- \[ \int\limits_{0}^{3} 27 x^{2}\, dx = 243 \]The outer integral.✓ Proved
Answer \( 243 \)
✓ 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 evaluate the iterated integral over the rectangular region. The algebraic steps are verified and the final result is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies Fubini's theorem to evaluate the iterated integral over the rectangular region. The algebraic steps are verified and the final result is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies Fubini's theorem to the rectangular region and performs the integration steps accurately.gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.