Intuition behind centre of mass equation for a rigid body

[Pochen Liu]

Homework Statement

I don't understand how we move from the discrete equation to the continuous one

Relevant Equations

I don't know how to type this in

So my basic understanding of an integral is that it finds the area underneath a graph.

I understand the idea behind an integral being the summation of f(x) * delta x, where delta x approaches zero.

If I look at the integral it's telling me that there's a change in mass that is being multiplied by its displacement?
Supposedly it's meant to represent a uniform distribution but I can't see it. If someone could clear up my intuition behind this that would be great :)

 

[haruspex]

change in mass that is being multiplied by its displacement?

No, it is an element of mass multiplied by its displacement from the origin.

it's meant to represent a uniform distribution

No, it is quite general.