Why is the formula for calculating the moment of inertia this integral
$$ \int r^2 dm~? $$ I understand the way we derived this formula from looking at the distribution of kinetic energy of a rotating object. I believe classical mechanics should have intuitive sense, and that everything in every formula has a reason it is the way it is and not something else(I repeat, in classical mechanics). Inertia (rotational) as I understand it, tells us how "hard" is it to rotate an object around a perpendicular axis. I i just don't see why $\int r^2 dm$ tell's us that.
For example the center of mass, similar formula but it makes much more sense (to me) $$ \sum_{i=1}^n m_i r_i, $$ because the more mass the $i$-th component of mass has, the closer the center of mass will be to that particular point, and also the further away a given mass is from our origin the further the center of mass, this is all perfectly logical and every part of the formula makes complete sense but in the case of the moment of inertia I simply cannot see why $ \int r^2 dm$ tells me how hard is it to rotate.