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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