Numerical integration

Numerical integration approximates analytic integration, and is particularly useful because:

• Analytic integrals may be hard to find if they exist at all!

• Integration tends to damp experimental noise (in contrast with differentiation which tends to amplify it)

In 1D, integration is simply finding the area under the curve \(s = \int_a^b f(x) \ dx\) in the range [a,b]:

![Integral_as_region_under_curve]

For definite integrals (i.e.: with finite limits), numerical integration is called numerical quadrature.

Aside: The integral sign \(\int\) looks like an elongated ‘S’ because that ‘summa’ (latin for summation) is exactly what we are doing!

The methods discussed in this section consider 2 cases:

• The function \(f(x)\) is available

• The data \(f(x), x\) is known at a set of points.