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Surface integrals

Problem 12.459 · hard

Evaluate \( \displaystyle \iint_S 1 \, dS \), 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
\[ \int\limits_{0}^{1}\int\limits_{0}^{1 - u} \sqrt{3}\, dv\, du = \frac{\sqrt{3}}{2} \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved

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

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: pass
qwen3.6:27b-mlx: pass — The parametrization correctly models the plane in the first octant, and the bounds for u and v are correct. The calculation of the surface element magnitude and the final integral are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The parametrization correctly models the plane in the first octant, and the bounds for u and v are correct. The calculation of the surface element magnitude and the final integral are correct.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The parametrization is valid, the magnitude of the normal vector is correctly calculated as sqrt(3), and the integration bounds for the triangular region in the first octant are correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The parametrization r(u, v) = <u, v, -u - v + 1> is incorrect because it yields z = 1 - u - v, whereas the plane equation is x + y + z = 1, which implies z = 1 - x - y. With x=u and y=v, z should be 1 - u - v. Wait, the parametrization given is <u, v, -u - v + 1>, which is indeed 1 - u - v. So the parametrization is correct. Let's re-read carefully. r(u,v) = <u, v, 1-u-v>. r_u = <1, 0, -1>, r_v = <0, 1, -1>. Cross product r_u x r_v = <1, 1, 1>. Norm is sqrt(3). The integral is sqrt(3) * Area(D). D is the triangle in the uv-plane bounded by u=0, v=0, u+v=1. Area is 1/2. Result sqrt(3)/2. The solution seems correct. Why did I think it was wrong? Let me check the sentence again. 'Parametrize S by r(u, v) = <u, v, -u - v + 1>'. This is correct. 'dS = ||r_u x r_v|| du dv'. This is correct. The equations are marked unchecked but the prompt says 'do not re-check the algebra of a line marked proved'. Wait, lines 2 and 3 are marked [equation, unchecked]. The prompt says 'Each line of the solution is either an EQUATION, which a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)...'. It seems lines 2 and 3 are NOT marked proved, they are marked unchecked. However, the instruction says 'The sentences are what you are here for.' Line 1 is a sentence. Lines 2 and 3 are equations. I should focus on the sentences. Is there an error in the sentence? The sentence sets up the parametrization and the formula for dS. This is correct. The bounds are implicit in the integral limits in line 3. The sentence doesn't explicitly state the bounds, but line 3 provides them. Is the sentence misleading? No. Is it an error? No. Let me look closer at the prompt's constraints. 'Check that: the setup models the problem as stated... each theorem or test is applied where its hypotheses hold...'. The setup is correct. The answer is correct. Why would it fail? Maybe the prompt implies I should check the equations too if they are unchecked? 'Each line... is either an EQUATION... or a SENTENCE... The sentences are what you are here for.' This implies I ignore the algebra of equations. But if the equation is unchecked, maybe I should check it? No, the prompt says 'do not re-check the algebra of a line marked proved'. It doesn't say 'do not check unchecked lines'. But it also says 'The sentences are what you are here for.' This is a bit ambiguous. However, usually, in these tasks, if the algebra is wrong, it's an error. Let's check the algebra of line 2. (sqrt((0*(-1) + 1)**2 + (1 - 1*0)**2 + (0*0 + 1*1)**2))**2. This looks like a garbled attempt to calculate the norm squared. r_u = <1, 0, -1>, r_v = <0, 1, -1>. Cross product is <1, 1, 1>. Norm squared is 3. The expression in line 2 is weird. (0*-1 + 1) is 1. (1 - 1*0) is 1. (0*0 + 1*1) is 1. Sum of squares is 3. Sqrt(3) squared is 3. So the equation holds. Line 3: Integral(sqrt(3), (v, 0, 1-u), (u, 0, 1)). This is sqrt(3) * integral from 0 to 1 of (1-u) du = sqrt(3)/2. Correct. So the math is correct. The sentence is correct. Verdict should be pass.
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.