Cavalieri Integration

  • ER Ackermann
  • TL Grobler
  • W Kleynhans
  • JC Olivier
  • BP Salmon
  • AJ van Zyl

Abstract

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).

Quaestiones Mathematicae 35(2012), 265–296

Author Biographies

ER Ackermann
Computational and Applied Mathematics Department, Rice University, Houston, Texas, TX 77005-1827, USA
TL Grobler
Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria, 0002, South Africa, and also Defence, Peace, Safety and Security, Council for Scienti c and Industrial Research, Pretoria, 0002, South-Africa
W Kleynhans
Remote Sensing Research Unit, Meraka Institute, Council for Scienti c and Industrial Research, Pretoria, 0002, South Africa, and also Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria, 0002, South-Africa
JC Olivier
School of Engineering, University of Tasmania, Hobart, 7001, Australia
BP Salmon
Remote Sensing Research Unit, Meraka Institute, Council for Scienti c and Industrial Research, Pretoria, 0002, South Africa, and also Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria, 0002, South-Africa
AJ van Zyl
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa
Section
Articles

Journal Identifiers


eISSN: 1727-933X
print ISSN: 1607-3606