Surface area of revolution
Problem 5.251 · medium
Find the area of the surface obtained by rotating \( \displaystyle y = x^{2} \), \( \displaystyle 0 \le x \le 1 \), about the \( \displaystyle y \)-axis.
- \[ \frac{d}{d x} x^{2} = 2 x \]y′.✓ Proved
- Rotating about the y-axis, the radius is x, so S = ∫ 2π·(radius)·√(1 + (y′)²) dx.Reviewed
- \[ 2 \pi x \sqrt{4 x^{2} + 1} \]Simplify the integrand.✓ Proved
- \[ \int\limits_{0}^{1} 2 \pi x \sqrt{4 x^{2} + 1}\, dx = \frac{\pi \left(-1 + 5 \sqrt{5}\right)}{6} \]Integrate.✓ Proved
Answer \( \frac{\pi \left(-1 + 5 \sqrt{5}\right)}{6} \approx 5.3304 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | numerical quadrature of the unsimplified integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the formula for surface area of revolution about the y-axis using x as the variable of integration. The setup, simplification, and final integration are all correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the formula for surface area of revolution about the y-axis using x as the variable of integration. The setup, simplification, and final integration are all correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the formula for surface area of revolution about the y-axis using x as the variable of integration. The setup, simplification, and final integration are all correct.gpt-oss:20b: pass 2026-10-05
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_revolution, checked 2026-10-05 with SymPy 1.14.0.