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 \( y = x, z = 0 \) .

If the moment of inertia \( I \) of the lamina \( S \) of unit density about \( A \) can be expressed as \( I = \pi h \left (\dfrac{a}{b} h^2 + c\right) \), where \( a \), \(b \), and \( c \) are coprime positive integers, find: \( a + b + c \).


Refer to previous problem. ..
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