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Surface integrals

Problem 12.366 · hard

Evaluate $ \displaystyle \iint_S x^{2} \, dS $, where $ \displaystyle S $ is the cylinder $ \displaystyle x^2 + y^2 = 9,\ 0 \le z \le 2 $.

Parametrize S by r(u, v) = <3*cos(u), 3*sin(u), v>; dS = ‖r_u × r_v‖ du dv.
\[ 9 \sin^{2}{\left(u \right)} + 9 \cos^{2}{\left(u \right)} = 9 \]
‖r_u × r_v‖² = (3)².✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{2 \pi} 27 \cos^{2}{\left(u \right)}\, du\, dv = 54 \pi \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved

Answer $ 54 \pi $

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	f times area summed over a triangulated mesh of the surface

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "pass", "severity": "", "notes": "The parametrization correctly models the cylinder $x^2+y^2=9$ with radius 3. The bounds for $u$ ($0$ to $2\pi$) and $v$ ($0$ to $2$) cover the surface ex

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "pass", "severity": "", "notes": "The parametrization correctly models the cylinder $x^2+y^2=9$ with radius 3. The bounds for $u$ ($0$ to $2\pi$) and $v$ ($0$ to $2$) cover the surface ex
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The parametrization correctly models the cylinder of radius 3 and height 2. The magnitude of the normal vector is correctly calculated as 3, and the resulting integral evaluates to 54π.
gpt-oss:20b: pass 2026-10-14

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/surface_integral, checked 2026-10-17 with SymPy 1.14.0.