Home›Calculus 3›Surface area by double integrals›Problem 11.506

Surface area by double integrals

Problem 11.506 · hard

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

Surface area = ∬ √(1 + f_x² + f_y²) dA.
Reviewed
\[ \left(\frac{\partial}{\partial x} \left(- 2 x - y + 2\right)\right)^{2} + \left(\frac{\partial}{\partial y} \left(- 2 x - y + 2\right)\right)^{2} + 1 = 6 \]
1 + f_x² + f_y².✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{3} \sqrt{6}\, dx\, dy = 6 \sqrt{6} \]
Integrate.✓ Proved

Answer \( 6 \sqrt{6} \approx 14.69694 \)

✓ 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 — The solution correctly identifies the surface area formula, computes the partial derivatives and the integrand constant, and evaluates the double integral over the specified rectangular region.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the surface area formula, computes the partial derivatives and the integrand constant, and evaluates the double integral over the specified rectangular region.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — 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 rectangular region is performed accurately.
gpt-oss:20b: pass 2026-10-16
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 calculated correctly, the integrand is constant, and the integration over the rectangular region is performed accurately.
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_area_double, checked 2026-10-19 with SymPy 1.14.0.