Numerical quadrature approximates a definite integral by a weighted sum of function values:

$$\int_a^b f(x),dx\approx\sum_{i=0}^n w_i f(x_i).$$

The nodes $x_i$ specify where the function is sampled and the weights $w_i$ determine how those samples contribute.

Many quadrature rules can be derived by replacing $f$ with a simpler interpolating function that can be integrated exactly.

A quadrature rule has degree of exactness $m$ when it integrates every polynomial of degree at most $m$ exactly.

For general smooth functions the remaining error depends on the step size, the rule and higher derivatives of $f$. Repeating a rule over many smaller subintervals gives a composite quadrature method.

Quadrature is useful when an antiderivative is unavailable, inconvenient or when the integrand is known only through numerical evaluations.