Advertisement

Integration

  • Robert Johansson
Chapter

Abstract

In this chapter we cover different aspects of integration, with the main focus on numerical integration. For historical reasons, numerical integration is also known as quadrature. Integration is significantly more difficult than its inverse operation – differentiation – and while there are many examples of integrals that can be calculated analytically, in general we have to resort to numerical methods. Depending on the properties of the integrand (the function being integrated) and the integration limits, it can be easy or difficult to numerically compute an integral. Integrals of continuous functions and with finite integration limits can in most cases be computed efficiently in one dimension, but integrable functions with singularities or integrals with infinite integration limits are examples of cases that can be difficult to handle numerically, even in a single dimension. Two-dimensional integrals (double integrals) and higher-order integrals can be numerically computed with repeated single-dimension integration or using methods that are multidimensional generalizations of the techniques used to solve single-dimensional integrals. However, the computational complexity grows quickly with the number of dimensions to integrate over, and in practice such methods are only feasible for low-dimensional integrals, such as double integrals or triple integrals. Integrals of higher dimension than that often require completely different techniques, such as Monte Carlo sampling algorithms.

Copyright information

© Robert Johansson 2019

Authors and Affiliations

  • Robert Johansson
    • 1
  1. 1.Urayasu-shi, ChibaJapan

Personalised recommendations