Surface integrals
Problem 12.441 · hard
Evaluate \( \displaystyle \iint_S z \, dS \), where \( \displaystyle S \) is the part of the plane \( \displaystyle x + y + z = 1 \) in the first octant.
- Parametrize S by r(u, v) = <u, v, -u - v + 1>; dS = ‖r_u × r_v‖ du dv.Reviewed
- \[ \left(\sqrt{\left(0 \left(-1\right) + 1\right)^{2} + \left(1 - 1 \cdot 0\right)^{2} + \left(0 \cdot 0 + 1 \cdot 1\right)^{2}}\right)^{2} = 3 \]‖r_u × r_v‖² = (sqrt(3))².✓ Proved
- \[ \int\limits_{0}^{1}\int\limits_{0}^{1 - u} \sqrt{3} \left(- u - v + 1\right)\, dv\, du = \frac{\sqrt{3}}{6} \]∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( \frac{\sqrt{3}}{6} \)
✓ 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 plane in the first octant, and the bounds for u and v are correct. The magnitude of the normal vector is calculated correctly, and the final integration yields the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The parametrization correctly models the plane in the first octant, and the bounds for u and v are correct. The magnitude of the normal vector is calculated correctly, and the final integration yields the stated answer.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The parametrization correctly models the plane in the first octant, and the surface area element calculation is correct. The integration bounds and result are accurate.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The parametrization is correct for the plane in the first octant, the magnitude of the normal vector is correctly calculated as sqrt(3), and the integration limits and result are correct.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.