Problem 1.240

Demonstrate that in the case of a thin plate of arbitrary shape there is the following relationship between the moments of inertia: $I_{1}+I_{2}=I_{3},$ where subindices $1,2,$ and 3 define three $\mathrm{mu}-$ tually perpendicular axes passing through one point, with axes 1 and 2 lying in the plane of the plate. Using this relationship, find the moment of inertia of a thin uniform round disc of radius $R$ and mass $m$ relative to the axis coinciding with one of its diameters.