Surface integrals
Problem 12.528 · hard
Evaluate \( \displaystyle \iint_S x \, dS \), where \( \displaystyle S \) is the part of the plane \( \displaystyle x + y + z = 2 \) in the first octant.
- Parametrize S by r(u, v) = <u, v, -u - v + 2>; 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
- \[ \int\limits_{0}^{2}\int\limits_{0}^{2 - u} \sqrt{3} u\, dv\, du = \frac{4 \sqrt{3}}{3} \]∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( \frac{4 \sqrt{3}}{3} \)
Lines: 2 proved, 1 reviewed. 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 | 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, surface element calculation, and integration bounds are all correct. The solution correctly evaluates the surface integral.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-09 — The parametrization, surface element calculation, and integration bounds are all correct. The solution correctly evaluates the surface integral.gpt-oss:20b: pass 2026-11-09qwen3.6:27b-mlx: pass 2026-11-08 — The parametrization and bounds correctly model the region in the first octant. The calculation of the surface element and the final integral are correct.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: inconclusive 2026-11-08 — reviewer returned a non-objectgpt-oss:20b: pass 2026-11-08
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 2026-11-09 with SymPy 1.14.0.