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Double integrals over general regions

Problem 11.135 · easy

Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).

D is Type I: for each x in [0, 1], y runs from 0 to x**2.
Reviewed
\[ \int\limits_{0}^{x^{2}} \left(x + y\right)\, dy = \frac{x^{4}}{2} + x^{3} \]
The inner integral.✓ Proved
\[ \int\limits_{0}^{1} \left(\frac{x^{4}}{2} + x^{3}\right)\, dx = \frac{7}{20} \]
The outer integral.✓ Proved

Answer \( \frac{7}{20} \)

✓ 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: pass
qwen3.6:27b-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 2026-09-27 — 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 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the region D as Type I with appropriate bounds and computes the iterated integral 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_region, checked 2026-09-27 with SymPy 1.14.0.