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
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.