Find the moment of inertia of a lamina bounded by concentric circles of radii a and b about a diameter.

Let the mass of the lamina be m.

Therefore the density of the lamina § = m/¶(b²-a²).

Consider an element of thickness dr at adistance r from the centre.

                                                                             dm = §(2¶rdr)

Therefore moment of inertia about the centre O

                             I = dmr²

                              ⁼ §(2¶rdr)r²

                            ⁼ 2§¶r³dr

                                   = 2§¶(b⁴-a⁴)/4

                                   = 2m¶(b⁴-a⁴)/(4¶(b²-a²))

                            = m(a²+b²)/2

 Applying the perpendicular axis theoram