Moment of Disk or Solid Cylinder about Axis

We presume the axis of rotation is perpendicular to the disk and through its center and coincident with the axis of the cylinder. Consent the radius of the disk be R and a mass element δm = σδS. A surface element is able to be defined as δS = r dθ dr. Alternative the surface element

Into the equation for a mass element we have δm = σr dr dθ. Since an element of inertia is able to be expressed δI = δmr²  we are able to write      

δI = σr²rδr δθ                                      and,

δI = σr³δr δθ

Summing the elements and taking limits confer

Performing the integrations

                                                            I = σ2π(R⁴/4)

I = σπ(R⁴/2).

The mass of the disk is σπR²

                                                            I =1/2(mR²).