A homogeneous thin plate with uniform thickness weighs 20 lb. What is its mass moment of inertia about Y axis? Note, the shape is not a semi-circle. Assume all dimensions are in feet.

The mass moment of inertia about the y-axis for a uniform plate is defined as
     

where the dm is the mass of an infinitesimal volume. However, dm can be converted into x and y dimensions using t as the plate thickness and ρ as the mass density, or
   dm = ρ t dy dx

The limits of integration are -2 to 2 for x, and the plate bottom to the plate top for y. This gives,
     

The density of the plate is simply,
     ρ = m / (At) = (W/g) / (At) = W / (gAt)

The moment inertia becomes,
     I_y = 4.267 t [ W / (gAt) ]
        = 4.267 [20/ (32.2 A) ]
        = 2.650 / A

To find the plate area, just integrate as
     

The final moment of inertia is
     I_y = 2.650 slug / 5.333 ft³

I_y = 0.4970 slug-ft²