rotational dynamics - Area moment motivation

I have some intuition about the (second) moment of inertia, and there is some motivation to define this concept if we think about the $KE$ of a rotating body or the torque $\tau$ applied, for example, on a circular object with radius $r$:

$\tau = r \times F = rm\Delta p/\Delta t = mr^2\Delta w/\Delta t = I\alpha $

Then moment of inertia could be expressed as $I = \int y^2dm$, where $y$ is the distance between a point on the object and it's rotational axis.

I want to know the motivation for defining the (second) area moment $\int_{A}y^2dA$. What is the relationship between this moment and the moment of inertia?

Here it's clear that if we consider the density of the object $\rho_0$ to be uniform and equal to one ($\rho_0 = 1$), then we could say that both moments are the same. But why do we use area moment on problems regarding bending moments on beams instead of inertia moment?

Through this reasoning when I'm calculating the tensions that result from the bending moment $M$ in the formula $σ=(M×y)/I$, then I'm assuming the material is made out of water? Wouldn't $\rho$ be an important factor to take into consideration?

I still cannot relate the inertia moment (related with an object's rotation) with area moment that is used when we take into account a bending moment on an object. If possible use some graphics and intuitive examples and avoid using tensor calculus.