Surface integrals
Problem 12.446 · hard
Evaluate \( \displaystyle \iint_S x^{2} + z \, dS \), where \( \displaystyle S \) is the cylinder \( \displaystyle x^2 + y^2 = 9,\ 0 \le z \le 4 \).
- 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
- \[ \int\limits_{0}^{4}\int\limits_{0}^{2 \pi} \left(3 v + 27 \cos^{2}{\left(u \right)}\right)\, du\, dv = 156 \pi \]∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( 156 \pi \)
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 integration step is incorrect: the surface integral evaluates to 27π+24, not 156π. The solution misapplies the area element or the integrand.qwen3.6:27b-mlx: pass — The parametrization is correct for the cylinder, the magnitude of the normal vector is correctly calculated as 3, and the resulting double integral evaluates to the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The parametrization is correct for the cylinder, the magnitude of the normal vector is correctly calculated as 3, and the resulting double integral evaluates to the stated answer.gpt-oss:20b: fail (error) 2026-10-18 — The integration step is incorrect: the surface integral evaluates to 27π+24, not 156π. The solution misapplies the area element or the integrand.qwen3.6:27b-mlx: pass 2026-10-16 — The parametrization is correct for the given cylinder, and the surface element magnitude is correctly calculated as 3. The integrand substitution and subsequent integration are correct.gpt-oss:20b: fail (error) 2026-10-16 — The integrand after parametrization should be (9cos^2u+v)*3, whose integral over u∈[0,2π], v∈[0,4] equals 27π+24, not 156π. The solution miscomputes the integral.qwen3.6:27b-mlx: pass 2026-10-15 — The parametrization correctly models the cylinder, the surface element magnitude is calculated correctly, and the resulting integral evaluates to the stated answer.gpt-oss:20b: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The setup is correct, but the evaluation of the double integral is wrong. The integrand should be \(27\cos^2u+3v\), whose integral over \(u\in[0,2\pi],v\i
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 2026-10-18 with SymPy 1.14.0.