Surface integrals
Problem 12.453 · hard
Evaluate \( \displaystyle \iint_S x + y + z \, dS \), where \( \displaystyle S \) is the part of the plane \( \displaystyle x + y + z = 4 \) in the first octant.
- Parametrize S by r(u, v) = <u, v, -u - v + 4>; dS = ‖r_u × r_v‖ du dv.
- \[ \left(\sqrt{\left(0 \left(-1\right) + 1\right)^{2} + \left(1 - 1 \cdot 0\right)^{2} + \left(0 \cdot 0 + 1 \cdot 1\right)^{2}}\right)^{2} = 3 \]‖r_u × r_v‖² = (sqrt(3))².✓ Proved
- \[ \int\limits_{0}^{4}\int\limits_{0}^{4 - u} 4 \sqrt{3}\, dv\, du = 32 \sqrt{3} \]∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( 32 \sqrt{3} \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The parametrization r(u, v) = <u, v, -u - v + 4> is incorrect because it yields z = 4 - u - v, which does not satisfy the plane equation x + y + z =
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The parametrization r(u, v) = <u, v, -u - v + 4> is incorrect because it yields z = 4 - u - v, which does not satisfy the plane equation x + y + z =gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The parametrization r(u, v) = <u, v, -u - v + 4> is incorrect because it yields z = 4 - u - v, which does not satisfy the plane equation x + y + z =gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The parametrization r(u, v) = <u, v, -u - v + 4> is incorrect; it should be r(u, v) = <u, v, 4 - u - v> to satisfy the plane equation x + y + z = 4. Consequently, the integrand f(r(u, v)) is evaluated as 4 instead of the correct constant 4, but the logic for the bounds and the magnitude of the normal vector is sound. Wait, let me re-evaluate. If r(u,v) = <u, v, -u-v+4>, then x+y+z = u + v + (-u-v+4) = 4. The parametrization is actually correct algebraically. The integrand is x+y+z = 4. The magnitude of the cross product is sqrt(3). The integral is integral of 4*sqrt(3) over the triangle in the uv-plane with vertices (0,0), (4,0), (0,4). The area of this triangle is 1/2 * 4 * 4 = 8. So the integral is 4 * sqrt(3) * 8 = 32 * sqrt(3). The solution is actually correct. Let me double check the parametrization syntax. r(u, v) = <u, v, -u - v + 4>. This is standard. The cross product calculation in line 2 is a bit messy but yields sqrt(3). The integral setup in line 3 is correct. Why did I think it was wrong? Ah, I misread the sign in my head. -u-v+4 is correct. The solution is sound.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 2026-10-19 with SymPy 1.14.0.