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Surface area by double integrals

Problem 11.528 · hard

Find the area of the part of the surface \( \displaystyle z = - 2 x - 2 y - 2 \) that lies above the rectangle \( \displaystyle [0, 1] \times [0, 3] \).

Surface area = ∬ √(1 + f_x² + f_y²) dA.
\[ \left(\frac{\partial}{\partial x} \left(- 2 x - 2 y - 2\right)\right)^{2} + \left(\frac{\partial}{\partial y} \left(- 2 x - 2 y - 2\right)\right)^{2} + 1 = 9 \]
1 + f_x² + f_y².✓ Proved
\[ \int\limits_{0}^{3}\int\limits_{0}^{1} 3\, dx\, dy = 9 \]
Integrate.✓ Proved

Answer \( 9 \)

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: inconclusive — reviewer returned a non-object
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-21 — reviewer returned a non-object
gpt-oss:20b: inconclusive 2026-10-21 — reviewer returned a non-object
qwen3.6:27b-mlx: inconclusive 2026-10-20 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly identifies the surface area formula, computes the partial derivatives and the integrand constant (3), and integrates over the specified rectangular region to obtain the correct result.
gpt-oss:20b: pass 2026-10-20

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 2026-10-21 with SymPy 1.14.0.