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 substitution
so that
and the lower and upper limits become
and
respectively and the integral becomes
This formula must hold for constant functions so, taking, e.g.
If we take
then
However, since for
the 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.