need a deravation for  parallel axis theorem

In physics, the parallel axis theorem or Huygens–Steiner theorem can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular distance (r) between the axes.

The moment of inertia about the new axis z is given by:

where:

is the moment of inertia of the object about an axis passing through its centre of mass;

is the object's mass;

is the perpendicular distance between the two axes.

This rule can be applied with the stretch rule and perpendicular axis theorem to find moments of inertia for a variety of shapes.

Parallel axes rule for area moment of inertia

The parallel axes rule also applies to the second moment of area (area moment of inertia) for a plane region D:

where:

is the area moment of inertia of D relative to the parallel axis;

is the area moment of inertia of D relative to its centroid;

is the area of the plane region D;

is the distance from the new axis z to the centroid of the plane region D.