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Surface area by double integrals

Problem 11.535 · hard

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

Surface area = ∬ √(1 + f_x² + f_y²) dA.
Reviewed
\[ \left(\frac{\partial}{\partial x} \left(- x - 3 y - 1\right)\right)^{2} + \left(\frac{\partial}{\partial y} \left(- x - 3 y - 1\right)\right)^{2} + 1 = 11 \]
1 + f_x² + f_y².✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{3} \sqrt{11}\, dx\, dy = 3 \sqrt{11} \]
Integrate.✓ Proved

Answer \( 3 \sqrt{11} \approx 9.94987 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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	the surface as a mesh of 28,800 triangles

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the surface area formula for a graph z = f(x,y). The partial derivatives are computed correctly, the integrand is constant, and the integration over the specified rectangular region is correct.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the surface area formula for a graph z = f(x,y). The partial derivatives are computed correctly, the integrand is constant, and the integration over the specified rectangular region yields 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.