Numerical Integration — Five Techniques

Martin McBride
Technological Singularity
11 min readMay 7, 2024


In mathematics, integration is the opposite process to differentiation. Where differentiation can be used to find the rate of change of a curve, integration is commonly used to find the area under a curve, or the volume under a surface, or the equivalent in higher dimensions. Integration is also used to solve differential equations, amongst many other applications.

We know how to differentiate many different types of mathematical functions because we can often find derivatives from first principles. We can’t generally integrate a function from first principles, so we can usually only find the integral of a function by applying what we know about derivatives. For example, if we know that differentiating f(x) results in g(x), we can determine (subject to certain conditions) that the integral of g(x) is f(x).

A consequence of this is that there are many functions that we cannot integrate analytically.

But sometimes we only need to know the approximate value of a definite integral (that is, the area under a curve between two points a and b). Numerical methods provide ways to calculate approximate integrals, often accurate to many significant figures, that are fine for most practical purposes.

This article will look at some of those methods:

  • Rectangular approximations
  • Trapezium rule
  • Simpson’s rule
  • Monte Carlo methods
  • Integration by series…