## Introduction to Closed Newton - Cotes Formulae

Newton – Cotes formulae use interpolating polynomials toto evaluateusing leading to the formulawhere theare equally spaced, so

The simplest example of this method is the trapezium rule. It calculates the area of the trapezium formed by approximatingusing linear interpolation. This is shown below forthat interpolates onlyand

The trapezium rule can be obtained by integrating the linear interpolation function

over the interval

We can generate other integration rules by using higher order Lagrange polynomials.

Theare labelledand called weights, thus we obtain

Closed Newton – Cotes formulae include the endpointsandin the interpolating polynomial.