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Double integrals in polar coordinates

Problem 11.32 · easy

Evaluate \( \displaystyle \iint_D \sqrt{x^{2} + y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.

In polar coordinates the disk is 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, and dA = r dr dθ.
\[ \sqrt{r^{2} \sin^{2}{\left(\theta \right)} + r^{2} \cos^{2}{\left(\theta \right)}} = \left|{r}\right| \]
The integrand in r and θ.✓ Proved
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1} r \left|{r}\right|\, dr\, d\theta = \frac{2 \pi}{3} \]
Integrate, remembering the extra r.✓ Proved

Answer \( \frac{2 \pi}{3} \)

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	the integral done in x and y, numerically, without polar coordinates, agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the region and the Jacobian. It properly handles the absolute value of r (which is just r in the first quadrant/disk context) and computes the integral correctly.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-object
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_polar, checked 2026-09-28 with SymPy 1.14.0.