If we wanted to find the average position of a series of particles on an interval, we could find this position by adding up the values of the x-coordinate from each point (the only coordinate in one dimension) and dividing by the number of points. This is expressed as follows:
(Σnxn)/n
Where xn is the x coordinate of the position of the nth particle.
Suppose instead we wanted to find the center of mass of an interval as described in the first paragraph. Our expression for the x coordinate of the center of mass, given the mass mn of each particle is:
(Σnxnmn)/(Σnmn)
(Note that the denominator is equal to the total mass of the set of particles.)
Given a linear density function ρ(x), we can express this summation as
(Σnxn*ρ(xn)*hn)/(Σnρ(xn)*hn)
Where hn is the width of the nth particle.
For an interval with domain D composed of a very large number of particles, we can change this summation to a double integral, replacing hn with dx:
X = (∫Dx*ρ(x)dx)/(∫Dρ(x)dx).
This gives us the center of mass (X) of D.
