A thin disc of mass M and radius R has per unit area σ(r)=kr², where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is :

IDisc = ∫0R (dm)r² ⇒ IDisc = ∫0R (σ 2πrdr)r²
IDisc = ∫0R (kr² πrdr)r²
Mass of disc
IDisc = 2πk ∫0R r⁵dr
M = ∫0R 2πrdrkr²
IDisc = 2πk(r⁶/6)|0R
M = 2πk ∫0R r³dr
IDisc = 2πkR⁶/6
M = 2πk(r⁴/4)|0R
IDisc = πkR⁶/3
M = 2πkR⁴/4
IDisc = (πkR⁴/2)R²/3
M = πkR⁴/2
IDisc = M(2R²/3)
IDisc = 2/3 × MR²