Home›Calculus 3›Surface area by double integrals›Problem 11.418

Surface area by double integrals

Problem 11.418 · hard

Find the area of the part of the surface \( \displaystyle z = \frac{2 x^{\frac{3}{2}}}{3} + \frac{2 y^{\frac{3}{2}}}{3} \) that lies above the square \( \displaystyle [0, 1] \times [0, 1] \).

Surface area = ∬ √(1 + f_x² + f_y²) dA.
Reviewed
\[ \left(\frac{\partial}{\partial x} \left(\frac{2 x^{\frac{3}{2}}}{3} + \frac{2 y^{\frac{3}{2}}}{3}\right)\right)^{2} + \left(\frac{\partial}{\partial y} \left(\frac{2 x^{\frac{3}{2}}}{3} + \frac{2 y^{\frac{3}{2}}}{3}\right)\right)^{2} + 1 = x + y + 1 \]
1 + f_x² + f_y².✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{1} \sqrt{x + y + 1}\, dx\, dy = - \frac{32 \sqrt{2}}{15} + \frac{4}{15} + \frac{12 \sqrt{3}}{5} \]
Integrate.✓ Proved

Answer \( - \frac{32 \sqrt{2}}{15} + \frac{4}{15} + \frac{12 \sqrt{3}}{5} \approx 1.40660 \)

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	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 accurately, and sets up the double integral over the specified square region. The final integration result matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the surface area formula, computes the partial derivatives and the integrand accurately, and sets up the double integral over the specified square region. The final integration result matches the stated answer.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the surface area formula, computes the partial derivatives and the integrand accurately, and evaluates the double integral over the specified square region to obtain the correct result.
gpt-oss:20b: inconclusive 2026-10-15 — reviewer returned a non-object
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the surface area formula, computes the partial derivatives and their squares accurately, and sets up the double integral over the specified square region. The final integration result matches the stated answer.
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.