Home›Calculus 3›Surface area by double integrals›Problem 11.423

Surface area by double integrals

Problem 11.423 · hard

Find the area of the part of the surface \( \displaystyle z = x^{2} + y^{2} \) that lies above the disk \( \displaystyle x^2 + y^2 \le \frac{1}{4} \).

Surface area = ∬ √(1 + f_x² + f_y²) dA.
Reviewed
\[ \left(\frac{\partial}{\partial x} \left(x^{2} + y^{2}\right)\right)^{2} + \left(\frac{\partial}{\partial y} \left(x^{2} + y^{2}\right)\right)^{2} + 1 = 4 x^{2} + 4 y^{2} + 1 \]
1 + f_x² + f_y².✓ Proved
Over a disk, use polar coordinates: 1 + 4x² + 4y² = 1 + 4r², and dA = r dr dθ.
Reviewed
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\frac{1}{2}} r \sqrt{4 r^{2} + 1}\, dr\, d\theta = \frac{\pi \left(-1 + 2 \sqrt{2}\right)}{6} \]
Integrate.✓ Proved

Answer \( \frac{\pi \left(-1 + 2 \sqrt{2}\right)}{6} \approx 0.95736 \)

✓ Nihil obstat Lines: 2 proved, 2 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
4	✓ 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 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14
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.