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Surface area by double integrals

Problem 11.429 · hard

Find the area of the part of the surface \( \displaystyle z = 3 x + 2 y + 2 \) that lies above the rectangle \( \displaystyle [0, 2] \times [0, 2] \).

Surface area = ∬ √(1 + f_x² + f_y²) dA.
\[ \left(\frac{\partial}{\partial x} \left(3 x + 2 y + 2\right)\right)^{2} + \left(\frac{\partial}{\partial y} \left(3 x + 2 y + 2\right)\right)^{2} + 1 = 14 \]
1 + f_x² + f_y².✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{2} \sqrt{14}\, dx\, dy = 4 \sqrt{14} \]
Integrate.✓ Proved

Answer \( 4 \sqrt{14} \approx 14.96663 \)

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	the surface as a mesh of 28,800 triangles

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the surface area formula for a graph z = f(x,y). The partial derivatives are constant, leading to a constant integrand, and the integration over the rectangular region is performed correctly.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-object
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_area_double, checked 2026-10-17 with SymPy 1.14.0.