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Surface integrals

Problem 12.479 · hard

Evaluate \( \displaystyle \iint_S x^{2} \, dS \), where \( \displaystyle S \) is the cylinder \( \displaystyle x^2 + y^2 = 4,\ 0 \le z \le 4 \).

Parametrize S by r(u, v) = <2*cos(u), 2*sin(u), v>; dS = ‖r_u × r_v‖ du dv.
Reviewed
\[ 4 \sin^{2}{\left(u \right)} + 4 \cos^{2}{\left(u \right)} = 4 \]
‖r_u × r_v‖² = (2)².✓ Proved
\[ \int\limits_{0}^{4}\int\limits_{0}^{2 \pi} 8 \cos^{2}{\left(u \right)}\, du\, dv = 32 \pi \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved

Answer \( 32 \pi \)

✓ 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. 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	f times area summed over a triangulated mesh of the surface

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The parametrization, surface element calculation, and integration limits are all correct. The solution is sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21 — The parametrization, surface element calculation, and integration limits are all correct. The solution is sound.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-20 — The parametrization correctly models the cylinder surface, and the surface element calculation is correct. The integration steps and final result are accurate.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The parametrization is correct for the given cylinder, the magnitude of the normal vector is correctly calculated as 2, and the resulting integral evaluates to the stated answer.
gpt-oss:20b: pass 2026-10-20

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 2026-10-21 with SymPy 1.14.0.