Home›Calculus 3›Double integrals in polar coordinates›Problem 11.35

Double integrals in polar coordinates

Problem 11.35 · easy

Evaluate \( \displaystyle \iint_D x^{2} + y^{2} + 1 \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 4 \) using polar coordinates.

In polar coordinates the disk is 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, and dA = r dr dθ.
Reviewed
\[ r^{2} \sin^{2}{\left(\theta \right)} + r^{2} \cos^{2}{\left(\theta \right)} + 1 = r^{2} + 1 \]
The integrand in r and θ.✓ Proved
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2} r \left(r^{2} + 1\right)\, dr\, d\theta = 12 \pi \]
Integrate, remembering the extra r.✓ Proved

Answer \( 12 \pi \)

Lines: 2 proved, 1 reviewed. 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	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	the integral done in x and y, numerically, without polar coordinates, agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly sets up the polar coordinate transformation, including the Jacobian factor r, and applies the correct bounds for the disk of radius 2. The final result is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly sets up the polar coordinate transformation, including the Jacobian factor r, and applies the correct bounds for the disk of radius 2. The final result is correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution sets up the radial bound as 0 ≤ r ≤ R in the text but uses 2 in the integral without explicitly defining R=2. While the numerical result is correct, the sentence fails to specify the value of R derived from the problem statement (x^2 + y^2 ≤ 4 implies R=2), leaving a gap in the logical connection between the problem setup and the integration limits.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Line 1 defines the radial bound as R but never specifies R=2, leaving the setup incomplete. Line 3 implicitly uses r=2 without stating it, which is a logical gap in the derivation.
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.