# 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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