Open Newton - Cotes Formulae

Open Newton-Cotes formulae are based on the interior points only as opposed to closed Newton - Cotes formulae based partly on the endpoints. For example, consider the open Newton - Cotes formula

We can make the substitutionso thatand the lower and upper limits becomeandrespectively and the integral becomes

This formula must hold for constant functions so, taking, e.g.If we takethenHowever, since forthe rule is exact for linear functions as well.

The general Newton – Cotes formula for 1 interior point evaluated at the midpoint is

Example: Estimate

We can estimate the error by evaluating the integral

The percentage error is

Better approximations can be found by taking more interior points into account, with the weights determined by the method of undetermined coefficients.