A classical mechanics problem by Rocco Dalto

Let lamina $$S: x^2 + y^2 = 1, 0 \leq z \leq h$$, and axis $$A:$$ the line $$x = x_{0} + at, y = y_{0} +bt, z = z_{0} + ct$$ . 

Find a general formula for the moment of inertia $$I$$ of the lamina $$S$$ of unit density about $$A$$, then let $$x_{0} = y_{0} = z_{0} = 0$$ and $$a = b = c$$. 

If the moment of inertia $$I$$ can be represented as $$I = \dfrac{ \alpha \pi h}{\beta} ( \lambda + h^2)$$, where $$gcf( \alpha ,\beta ,\lambda) = 1$$. 

Find: $$\alpha + \beta + \lambda$$ 

Note: My intention here is to have the person find a general formula for $$I$$ given lamina $$S: x^2 + y^2 = 1, 0 \leq z \leq h$$ of unit density, and axis $$A:$$ the line $$x = x_{0} + at, y = y_{0} +bt, z = z_{0} + ct$$ , of course you need not do so, and just proceed using $$x_{0} = y_{0} = z_{0} = 0$$ and $$a = b = c$$. 

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