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Surface integrals

Problem 12.438 · hard

Evaluate \( \displaystyle \iint_S x + y + z \, dS \), where \( \displaystyle S \) is the part of the plane \( \displaystyle x + y + z = 3 \) in the first octant.

Parametrize S by r(u, v) = <u, v, -u - v + 3>; 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
\[ \int\limits_{0}^{3}\int\limits_{0}^{3 - u} 3 \sqrt{3}\, dv\, du = \frac{27 \sqrt{3}}{2} \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved

Answer \( \frac{27 \sqrt{3}}{2} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The integrand is incorrect. The solution integrates the constant 3*sqrt(3), but the integrand x+y+z equals 3 on the surface, so the integral should be 3*sqrt(3) times the area of the projection, not just 3*sqrt(3) integrated over the projection without accounting for the variable bounds correctly in the sentence description (though the equation line 3 actually computes the correct value, the sentence in line 1 implies a generic setup, and line 3's equation shows the integrand as a constant 3*sqrt(3) which is correct for the value, but the description 'Integral(3*sqrt(3)...)' is ambiguous. Wait, let's look closer. Line 3 says `Integral(3*sqrt(3), ...)` which evaluates to `27*sqrt(3)/2`. This is mathematically correct because x+y+z=3 on the surface. However, the parametrization in line 1 is `r(u, v) = <u, v, -u - v + 3>`. The normal vector magnitude is `sqrt(3)`. The integrand `x+y+z` becomes `u + v + (-u-v+3) = 3`. So the integral is `∬ 3 * sqrt(3) dA`. The region is the triangle in the uv-plane with vertices (0,0), (3,0), (0,3). The area of this triangle is `1/2 * 3 * 3 = 9/2`. The integral is `3*sqrt(3) * 9/2 = 27*sqrt(3)/2`. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's constraints. 'Check that... each conclusion follows from the lines before it'. Line 1 sets up the parametrization. Line 2 calculates the magnitude squared. Line 3 calculates the integral. The sentence in line 1 is just a setup. The sentence in line 3 is missing? No, line 3 is marked [equation]. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 1 is a sentence. Line 2 is an equation. Line 3 is an equation. Is there a missing sentence explaining the bounds? The bounds `(v, 0, 3-u)` and `(u, 0, 3)` are in the equation line. This is acceptable. The solution is sound. Wait, look at line 1: `r(u, v) = <u, v, -u - v + 3>`. This is valid. `dS = ||r_u x r_v|| du dv`. Valid. Line 2 proves `||r_u x r_v|| = sqrt(3)`. Valid. Line 3 computes `Integral(3*sqrt(3), (v, 0, 3-u), (u, 0, 3))`. This represents `∫_0^3 ∫_0^{3-u} 3√3 dv du`. This is correct. The result is correct. The setup is correct. Verdict: pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The integrand is incorrect. The solution integrates the constant 3*sqrt(3), but the integrand x+y+z equals 3 on the surface, so the integral should be 3*sqrt(3) times the area of the projection, not just 3*sqrt(3) integrated over the projection without accounting for the variable bounds correctly in the sentence description (though the equation line 3 actually computes the correct value, the sentence in line 1 implies a generic setup, and line 3's equation shows the integrand as a constant 3*sqrt(3) which is correct for the value, but the description 'Integral(3*sqrt(3)...)' is ambiguous. Wait, let's look closer. Line 3 says `Integral(3*sqrt(3), ...)` which evaluates to `27*sqrt(3)/2`. This is mathematically correct because x+y+z=3 on the surface. However, the parametrization in line 1 is `r(u, v) = <u, v, -u - v + 3>`. The normal vector magnitude is `sqrt(3)`. The integrand `x+y+z` becomes `u + v + (-u-v+3) = 3`. So the integral is `∬ 3 * sqrt(3) dA`. The region is the triangle in the uv-plane with vertices (0,0), (3,0), (0,3). The area of this triangle is `1/2 * 3 * 3 = 9/2`. The integral is `3*sqrt(3) * 9/2 = 27*sqrt(3)/2`. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's constraints. 'Check that... each conclusion follows from the lines before it'. Line 1 sets up the parametrization. Line 2 calculates the magnitude squared. Line 3 calculates the integral. The sentence in line 1 is just a setup. The sentence in line 3 is missing? No, line 3 is marked [equation]. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 1 is a sentence. Line 2 is an equation. Line 3 is an equation. Is there a missing sentence explaining the bounds? The bounds `(v, 0, 3-u)` and `(u, 0, 3)` are in the equation line. This is acceptable. The solution is sound. Wait, look at line 1: `r(u, v) = <u, v, -u - v + 3>`. This is valid. `dS = ||r_u x r_v|| du dv`. Valid. Line 2 proves `||r_u x r_v|| = sqrt(3)`. Valid. Line 3 computes `Integral(3*sqrt(3), (v, 0, 3-u), (u, 0, 3))`. This represents `∫_0^3 ∫_0^{3-u} 3√3 dv du`. This is correct. The result is correct. The setup is correct. Verdict: pass.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The integrand in step 3 is incorrect. The problem asks for the integral of x + y + z, which equals 3 on the surface, but the solution integrates 3*sq
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15
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.