newtonian mechanics - Moment of Inertia, why $r^2$and not $r$?

So my engineering mechanics book includes a brief discussion on area moments of inertia.

Unfortunately, the ensuing chapter is predominately computational in nature. I don't have a thorough grasp of where the equations for this come from. The equation in question is expressed as:

$$I_x = \int_A y^2dA \space \mathrm{ and } \space I_y = \int_A x^2dA$$

Because of the simple equation for torque $\tau=Fd$, it is easy to imagine that the angular momentum of an object being rotated about an axis is greater the further it's mass is from the axis.

Simply divide any continous object up into a set of differential elements and violá, you have an integral that can be vaguely matched to the angular momentum.

The problem, however, is I can't possibly derive this. In fact, it makes more sense to me to integrate over simply $x$ or $y$ rather than $x^2$ or $y^2$.

Due to a logistical nightmare, I have no access to a physics book to look this up. I would be grateful to anyone who could explain (ie, explain the motivation of) this to me!