Surface area of revolution
Problem 5.247 · medium
Find the area of the surface obtained by rotating \( \displaystyle y = \sqrt{4 - x^{2}} \), \( \displaystyle -1 \le x \le 0 \), about the \( \displaystyle x \)-axis.
- \[ \frac{d}{d x} \sqrt{4 - x^{2}} = - \frac{x}{\sqrt{4 - x^{2}}} \]y′.✓ Proved
- Rotating about the x-axis, the radius is y, so S = ∫ 2π·(radius)·√(1 + (y′)²) dx.Reviewed
- \[ 2 \pi \sqrt{4 - x^{2}} \sqrt{\frac{x^{2}}{4 - x^{2}} + 1} = 4 \pi \sqrt{- \frac{1}{x^{2} - 4}} \sqrt{4 - x^{2}} \]Simplify the integrand.✓ Proved
- \[ \int\limits_{-1}^{0} 4 \pi \sqrt{- \frac{1}{x^{2} - 4}} \sqrt{4 - x^{2}}\, dx = 4 \pi \]Integrate.✓ Proved
Answer \( 4 \pi \approx 12.5664 \)
✓ 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 surface area formula for rotation about the x-axis and the resulting integrand simplifies correctly to a constant, yielding the correct result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the surface area formula for rotation about the x-axis and the resulting integrand simplifies correctly to a constant, yielding the correct result.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the surface area formula for rotation about the x-axis, simplifies the integrand accurately, and computes the definite integral correctly.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.