Let us consider small area of half disc as shown in figure whose radius is r and width dr.
mass of this shaded portion, m = ρ × Π × r × dr
where ρ is mass per unit area.
moment of inertia of this shaded area of disc about the axis passing through the centre of disc is mr2 = ρ × Π × r × dr × r2
Hence moment of inertia of whole disc is given by
mass of the given disc M is