What is the center of mass of a uniform sphere? That isn't
too bad. By symmetry it's at the sphere's center. But what
if the sphere is not of uniform density? The top hemisphere
is made out of balsa wood and the bottom hemisphere is made
out of roquefort cheese? That's a bit more tricky, and to
solve this sort of problem, we'll have to formulate it in
terms of integration. Let's start off with the one dimensional
case. Suppose that you have a rod that has a mass per unit
length 
that can vary with position x.
We choose coordinates so that
the left hand side of the rod is at 0 and the right hand
side is at L. To obtain the center of mass, we break
up the rod into millions of tiny section each of length
. Then the center of mass is the sum of all
the individual masses 
times their corresponding
position 
. So
Now one of these little masses 
so
You might remember from calculus expressions of this kind. If we take the
limit as 
then these become integrals
Since 
, an infinitessimal of mass, this integral
can be more elegantly, but less usefully written as
