The crucial idea that connects integration with the physical world is
the idea of density.
For example, suppose we are trying to find the total electric force on
a point charge, Q, due to a charged rod.
(Assume that Q is located away from the rod, but along its axis
so that we don't have to worry about vector integration.)
Each of the charges on the rod is a different distance away from Q, and
what's worse, the charge on the rod may not be distributed uniformly.
It is difficult to see, at first, how to use our simple force formula for
point charges,
to deal with this situation.
Well, if the formula only works for point charges, let's make sure we
only have point charges by breaking the rod up into little tiny pieces,
each approximately a point charge.
We find the force exerted on Q by each little piece, then add all of the
forces together to get the total.
The addition of infinitely many infinitesimally small pieces is, of course, integration.
This is conceptually quite simple, but when we try to carry it out we encounter a problem.
The formula for the little bit of force exerted on Q by a little bit of charge dq is
where r is the distance between Q and dq. But what is the formula for dq?
This is where the idea of a density enters.
If charges are distributed along a line or a curve, the important density is
the linear charge density, denoted by the symbol 
.
It tells us the charge per unit length at each point along the line,
and gives us a relation between dq and the length of each tiny
piece of the line, ds:
Similarly, if the charge is distributed over a surface, then the important density
is the area charge density, the charge per unit area 
.
In this case dq is given by
where dA is the area of a tiny patch of the surface containing charge dq.
Finally, if the charge is distributed throughout a volume, then
we must know the volume charge density, the charge per unit volume 
, which relates
the volume of each tiny piece, dV, with the charge of each
tiny piece, dq:
All that remains is to find expressions for ds, dA, or dV using geometry, substitute for dq in the formula for dF, supply the proper limits of integration, and carry out the integration to obtain the total force. Several examples of how to use these densities can be found in the integration article in the student packet.