Mass per unit area of a circular disc of radius a depends on the distance r from its centre as σ(r) = A + BrThe moment of inertia of the disc about the the axis, perpendicular to the plane and passing through its centre is :

The moment of inertia (I) of a circular disc about an axis perpendicular to the plane and passing through its centre can be calculated using the formula:

I = ∫r²dm

where r is the distance from the axis of rotation to the point mass (dm), and the integral is taken over the entire mass of the disc.

Given that the mass per unit area (σ) of the disc depends on the distance r from its centre as σ(r) = A + Br, we can express the infinitesimal mass element (dm) in terms of r and the infinitesimal area element (dA) as:

dm = σdA = (A + Br)dA

The area element of a circular disc in polar coordinates is given by dA = r dr dθ, where r is the radial distance from the centre of the disc and θ is the angle subtended by the area element at the centre of the disc.

Substituting these expressions into the formula for the moment of inertia, we get:

I = ∫r²(A + Br)r dr dθ = ∫(Ar³ + Br⁴) dr dθ

This integral is taken over the entire area of the disc, i.e., r varies from 0 to a (the radius of the disc) and θ varies from 0 to 2π.

Solving this integral will give the moment of inertia of the disc about the given axis.