Surface integrals
Problem 12.367 · hard
Evaluate \( \displaystyle \iint_S x^{2} + z \, dS \), where \( \displaystyle S \) is the cylinder \( \displaystyle x^2 + y^2 = 9,\ 0 \le z \le 3 \).
- 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
- \[ \int\limits_{0}^{3}\int\limits_{0}^{2 \pi} \left(3 v + 27 \cos^{2}{\left(u \right)}\right)\, du\, dv = 108 \pi \]∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( 108 \pi \)
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: fail (error) — The double integral of 9cos²u+v over the cylinder gives 135π, not 108π. The computation of the integrand or the limits is incorrect.qwen3.6:27b-mlx: pass — The parametrization is correct for the given cylinder, including the bounds for u and v. The surface element magnitude is correctly calculated as 3, and the integrand substitution and final integration are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The parametrization is correct for the given cylinder, including the bounds for u and v. The surface element magnitude is correctly calculated as 3, and the integrand substitution and final integration are correct.gpt-oss:20b: fail (error) 2026-10-17 — The double integral of 9cos²u+v over the cylinder gives 135π, not 108π. The computation of the integrand or the limits is incorrect.qwen3.6:27b-mlx: pass 2026-10-15 — The parametrization correctly models the cylinder surface and bounds. The integrand substitution and magnitude of the normal vector are correct, leading to the right result.gpt-oss:20b: fail (error) 2026-10-15 — The integrand should be (9cos^2u+v)·3, giving 27cos^2u+3v. Integrating over u∈[0,2π], v∈[0,3] yields 27π+27, not 108π. The solution miscalculates the integral.qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "pass", "severity": "", "notes": "The parametrization correctly models the cylinder $x^2 + y^2 = 9$ with radius 3 and height 3. The surface element magnitude $\|r_u \times r_v\| = 3$ is cgpt-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.