Double integrals over general regions
Problem 11.113 · easy
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
- D is Type I: for each x in [0, 1], y runs from 0 to sqrt(x).Reviewed
- \[ \int\limits_{0}^{\sqrt{x}} \left(x + y\right)\, dy = x^{\frac{3}{2}} + \frac{x}{2} \]The inner integral.✓ Proved
- \[ \int\limits_{0}^{1} \left(x^{\frac{3}{2}} + \frac{x}{2}\right)\, dx = \frac{13}{20} \]The outer integral.✓ Proved
Answer \( \frac{13}{20} \)
✓ 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 | nested numerical quadrature over the region agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The setup correctly identifies the region D as Type I with appropriate bounds. The integration steps are algebraically correct and lead to the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The setup correctly identifies the region D as Type I with appropriate bounds. The integration steps are algebraically correct and lead to the stated answer.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The setup correctly identifies the region D as Type I with appropriate bounds. The integration steps are algebraically correct and yield the stated answer.gpt-oss:20b: pass 2026-09-26
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_region, checked 2026-09-26 with SymPy 1.14.0.