Integration is fundamentally about chopping and adding. Yet this point of view is not adequately emphasized in calculus courses.

As already noted, there is a disconnect between the meaning of integration and the process used to compute integrals. Furthermore, the “meaning” of integration is emphasized largely during the more-or-less rigorous construction of integrals as Riemann sums, a process which most students will never use again, rather than the more geometric notion of simply adding up small pieces.

We explore these ideas further in the next several sections. But there is a little-known fundamental difference between the way mathematicians and physicists regard integration, which we discuss first.

What is it that gets chopped?

For a straight wire or a flat metallic plate, there is little confusion. But for line and surface integrals — which are needed for other shapes — the physicist will chop up the physical wire or plate, whereas the mathematician will chop up the coordinate domain.

For example, when finding the volume of a paraboloidal reflector, which is most easily done in cylindrical coordinates, the physicist will imagine chopping up the actual reflector, whereas the mathematician will chop up the circular disk which corresponds to the limits of integration. The mathematician regards both the mass density $\sigma$ and the element of integration $dA$ as being parametric functions on the disk, whereas the physicist regards these quantities, as well as cylindrical coordinates themselves, as functions directly on on the paraboloid.