We use Cavalieri’s principle to develop a novel integration technique
which we call Cavalieri integration. Cavalieri integrals differ from Riemann integrals
in that non-rectangular integration strips are used. In this way we can use single
Cavalieri integrals to find the areas of some interesting regions for which it is difficult
to construct single Riemann integrals.
We also present two methods of evaluating a Cavalieri integral by first transforming
it to either an equivalent Riemann or Riemann-Stieltjes integral by using special transformation functions h(x) and its inverse g(x), respectively. Interestingly enough
it is often very difficult to find the transformation function h(x), whereas it is very
simple to obtain its inverse g(x).