Standard Article

Numerical Integration

  1. Gordon K. Smyth

Published Online: 15 JUL 2005

DOI: 10.1002/0470011815.b2a14026

Encyclopedia of Biostatistics

Encyclopedia of Biostatistics

How to Cite

Smyth, G. K. 2005. Numerical Integration. Encyclopedia of Biostatistics. 6.

Author Information

  1. Walter and Eliza Hall Institute of Medical Research, Melbourne, Victoria, Australia

Publication History

  1. Published Online: 15 JUL 2005


This article focuses on the process of approximating a definite integral from values of the integrand when exact mathematical integration is not available. This problem arises in statistics when marginal density functions or expected values of random variables are required. The article describes classical univariate quadrature methods, including the trapezoidal rule, Simpson's rule, Newton–Cotes formulas, Clenshaw–Curtis integration, and Gaussian quadrature. Refinements including adaptive methods, treatment of singularities, and progressive rules of the Gaussian type are also mentioned. A survey is given of the possibilities and limitations of multiple integration methods, including product rules, globally adaptive methods, rules of polynomial degree, lattice methods, and Monte Carlo integration. Detailed pointers are given to available software.


  • trapezoidal rule;
  • Simpson's rule;
  • Newton–Cotes formulas;
  • Clenshaw–Curtis integration;
  • Gaussian quadrature;
  • multiple integration;
  • lattice methods;
  • Monte Carlo methods