Home›Calculus 3›Double integrals in polar coordinates›Problem 11.39

Double integrals in polar coordinates

Problem 11.39 · easy

Evaluate \( \displaystyle \iint_D x^{2} + y^{2} + 1 \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 9 \) using polar coordinates.

In polar coordinates the disk is 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, and dA = r dr dθ.
\[ r^{2} \sin^{2}{\left(\theta \right)} + r^{2} \cos^{2}{\left(\theta \right)} + 1 = r^{2} + 1 \]
The integrand in r and θ.✓ Proved
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3} r \left(r^{2} + 1\right)\, dr\, d\theta = \frac{99 \pi}{2} \]
Integrate, remembering the extra r.✓ Proved

Answer \( \frac{99 \pi}{2} \)

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: fail (error) — Line 1 defines the radius bound as R but fails to specify that R=3 for the given disk x^2 + y^2 <= 9. This omission makes the setup incomplete and logically disconnected from the specific problem statement.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — Line 1 defines the radius bound as R but fails to specify that R=3 for the given disk x^2 + y^2 <= 9. This omission makes the setup incomplete and logically disconnected from the specific problem statement.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up the polar coordinate transformation, including the Jacobian factor r, and applies the correct bounds for the disk of radius 3. The final result is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly sets up the polar coordinate transformation, including the Jacobian factor r, and applies the correct bounds for the disk of radius 3. The final result is correct.
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.