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Surface area of revolution

Problem 5.250 · medium

Find the area of the surface obtained by rotating \( \displaystyle y = \sqrt{4 - x^{2}} \), \( \displaystyle -1 \le x \le 1 \), 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
\[ \int\limits_{-1}^{1} 4 \pi \sqrt{- \frac{1}{x^{2} - 4}} \sqrt{4 - x^{2}}\, dx = 8 \pi \]
Integrate.✓ Proved

Answer \( 8 \pi \approx 25.1327 \)

Lines: 3 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.

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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the surface area formula for rotation about the x-axis and the algebraic simplification of the integrand is valid on the given interval.

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 algebraic simplification of the integrand is valid on the given interval.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-object
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.