Surface integrals
Problem 12.364 · hard
Evaluate \( \displaystyle \iint_S x + y + z \, dS \), 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.
- \[ \left(\sqrt{\left(0 \left(-1\right) + 1\right)^{2} + \left(1 - 1 \cdot 0\right)^{2} + \left(0 \cdot 0 + 1 \cdot 1\right)^{2}}\right)^{2} = 3 \]‖r_u × r_v‖² = (sqrt(3))².✓ Proved
- \[ \int\limits_{0}^{2}\int\limits_{0}^{2 - u} 2 \sqrt{3}\, dv\, du = 4 \sqrt{3} \]∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( 4 \sqrt{3} \)
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 + 2> is incorrect because it yields z = 2 - u - v, which does not satisfy the plane equation x + y + z =
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The parametrization r(u, v) = <u, v, -u - v + 2> is incorrect because it yields z = 2 - u - v, which does not satisfy the plane equation x + y + z =gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The parametrization r(u, v) = <u, v, -u - v + 2> is incorrect because it yields z = 2 - u - v, which does not satisfy the plane equation x + y + z = 2 (it sums to 2 - u - v + u + v = 2, wait, actually x+y+z = u + v + (2-u-v) = 2. The parametrization is correct. Let me re-read carefully. r(u,v) = <u, v, 2-u-v>. Then x+y+z = u+v+2-u-v = 2. This is correct. However, the integrand in step 3 is just 2*sqrt(3). The original integrand is x+y+z. On the surface S, x+y+z = 2. So the integrand becomes 2. Then multiplied by dS factor sqrt(3). So the integral is ∬ 2 * sqrt(3) dA. This part is correct. Let's check the bounds. The region is in the first octant, so u>=0, v>=0, 2-u-v>=0 => u+v<=2. The bounds in step 3 are v from 0 to 2-u, u from 0 to 2. This is correct. The calculation: Integral(2*sqrt(3), (v, 0, 2-u), (u, 0, 2)). Inner integral: 2*sqrt(3) * (2-u). Outer integral: 2*sqrt(3) * [2u - u^2/2] from 0 to 2 = 2*sqrt(3) * (4 - 2) = 4*sqrt(3). The math seems correct. Why did I think it was wrong? Let me re-read the prompt's solution text. Step 1: r(u, v) = <u, v, -u - v + 2>. This is <u, v, 2-u-v>. Correct. Step 2: Norm squared is 3. Correct. Step 3: Integral of 2*sqrt(3) over the triangle. Correct. Wait, is there a subtle error? "Parametrize S by r(u, v) = <u, v, -u - v + 2>". Usually, we project onto the xy-plane. The bounds are correct. Is the sentence misleading? Let's look at the notation in Step 3: `Integral(2*sqrt(3), (v, 0, 2 - u), (u, 0, 2))`. This looks like a CAS output format. Is the answer 4*sqrt(3) correct? Area of triangle in xy plane is 1/2 * 2 * 2 = 2. Surface area is Area * sqrt(1 + (dz/dx)^2 + (dz/dy)^2) = 2 * sqrt(1+1+1) = 2*sqrt(3). Integral of (x+y+z) dS. Since x+y+z=2 on the surface, Integral = 2 * Surface Area = 2 * 2*sqrt(3) = 4*sqrt(3). The solution appears completely correct. Why would I fail it? Let me check the parametrization vector again. r_u = <1, 0, -1> r_v = <0, 1, -1> r_u x r_v = <1, 1, 1> ||r_u x r_v|| = sqrt(3). This is correct. Maybe the issue is with the sentence "Parametrize S by...". It doesn't specify the domain D for u and v. Step 3 implies the domain, but Step 1 is a sentence that sets up the model. A complete setup should define the domain. However, Step 3 provides the bounds. Let's look closer at Step 1. "Parametrize S by r(u, v) = <u, v, -u - v + 2>; dS = ‖r_u × r_v‖ du dv." This is a standard setup. Is there any ambiguity? "first octant". u>=0, v>=0, z>=0 => 2-u-v>=0. I will mark it as pass. It is sound.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The parametrization r(u, v) = <u, v, -u - v + 2> is incorrect; it should be r(u, v) = <u, v, 2 - u - v> to satisfy the plane equation x + y + z = 2. The current parametrization yields x + y + z = 2 - 2u - 2v, which does not lie on the plane.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.