Home›Calculus 2›Area and surface area with parametric curves›Problem 8.474

Area and surface area with parametric curves

Problem 8.474 · hard

Find the area of the surface made by rotating \( \displaystyle x = 2 t,\ y = t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.

S = ∫ 2πy √(x′(t)² + y′(t)²) dt.
Reviewed
\[ \left(\frac{d}{d t} t\right)^{2} + \left(\frac{d}{d t} 2 t\right)^{2} = 5 \]
x′² + y′², simplified.✓ Proved
\[ \int\limits_{0}^{1} 2 \sqrt{5} \pi t\, dt = \sqrt{5} \pi \]
Integrate.✓ Proved

Answer \( \sqrt{5} \pi \)

✓ Nihil obstat Lines: 2 proved, 1 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	✓ 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 curve as a 20,000-segment polyline: trapezoids under it, or the frusta it sweeps

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the surface area formula for rotation about the x-axis, computes the derivatives and integrand accurately, and evaluates the definite integral correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the surface area formula for rotation about the x-axis, computes the derivatives and integrand accurately, and evaluates the definite integral correctly.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the surface area formula for rotation about the x-axis, computes the derivatives and integrand accurately, and evaluates the definite integral correctly.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the formula for the surface area of revolution about the x-axis, computes the derivatives and their sum of squares accurately, and evaluates the definite integral correctly.
gpt-oss:20b: pass 2026-10-16

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/parametric_area_surface, checked 2026-10-19 with SymPy 1.14.0.