Surface integrals
Problem 12.448 · hard
Evaluate \( \displaystyle \iint_S x^{2} + y^{2} \, dS \), where \( \displaystyle S \) is the upper hemisphere \( \displaystyle x^2 + y^2 + z^2 = 4,\ z \ge 0 \).
- Parametrize S by r(u, v) = <2*sin(u)*cos(v), 2*sin(u)*sin(v), 2*cos(u)>; dS = ‖r_u × r_v‖ du dv.Reviewed
- \[ \left(4 \sin{\left(u \right)} \sin^{2}{\left(v \right)} \cos{\left(u \right)} + 4 \sin{\left(u \right)} \cos{\left(u \right)} \cos^{2}{\left(v \right)}\right)^{2} + 16 \sin^{4}{\left(u \right)} \sin^{2}{\left(v \right)} + 16 \sin^{4}{\left(u \right)} \cos^{2}{\left(v \right)} = 16 \sin^{2}{\left(u \right)} \]‖r_u × r_v‖² = (4*sin(u))².✓ Proved
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\frac{\pi}{2}} 16 \sin^{3}{\left(u \right)}\, du\, dv = \frac{64 \pi}{3} \]∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( \frac{64 \pi}{3} \)
✓ 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: passqwen3.6:27b-mlx: pass — The parametrization correctly models the upper hemisphere of radius 2, and the scalar surface element calculation is correct. The integration limits and result are accurate.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The parametrization correctly models the upper hemisphere of radius 2, and the scalar surface element calculation is correct. The integration limits and result are accurate.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The parametrization correctly models the upper hemisphere with appropriate bounds. The surface element calculation and subsequent integration are correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The parametrization, bounds, and magnitude of the normal vector are correct. The integration steps and final result are accurate.gpt-oss:20b: pass 2026-10-15
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.