A calculus problem by Rocco Dalto

Calculus Level 4

The solution to the differential \( \dfrac{1}{2} \left(\dfrac{dy}{dx}\right)^2 - \dfrac{a}{y} = -\dfrac{1}{2} \) can be expressed as the arc \( x = f(\theta)\), \(y = g(\theta) \), and the curve passes thru the origin so that \( x = y = 0 \), when \( \theta = 0 \), and \( x(\pi) = a\pi , \: y(\pi) = 2a \), and \( f'(\theta) > 0 \) for all \( (0 < \theta < \pi) \).

The area of the surface formed by rotating the above arc \( x = f(\theta)\), \(y = g(\theta)\) about the \(x\)-axis from \( \theta = 0 \) to \( \theta = \pi \) can be expressed as \( \dfrac{m}{n}\pi a^2\), where \( m \) and \( n \) are coprime positive integers.

Find \( m + n \).

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