Moment of inertia of a pointed cubic-profile solid of revolution
= Moment of inertia of a pointed cubic-profile solid of revolution
Consider the uniform <solid of revolution> about the $x_3$-axis whose boundary is
$$
x_1^2+x_2^2=x_3^2-\frac{x_3^3}{h},
\qquad 0\leq x_3\leq h.
$$
For total mass $M$, the center of mass lies a distance $3h/5$ from the pointed origin, and the principal moments about that origin are
$$
I_1=I_2=\frac37Mh^2,
\qquad
I_3=\frac2{35}Mh^2.
$$