Assume the origin to be the center of the disk. We will weigh the distance by the mass: dm = ρdA with dA = 2πrdr, a thin ring around the origin. Let ρ = M/πa2 where M is the mass of the disk.
ravg = 1/M ∫rρ(2πr)dr integrated from 0 to a.
ravg = (2πρ/M) ∫r2 dr = (2πρ/M) [(1/3) r3] from 0 to a.
ravg = (2π/M)(M/πa2)(a3/3) = 2a/3