Newton-Cotes formulae

The two methods derived above can be written in the general form

, which is known as a quadrature formula. Typically, and . One can derive formulae involving more points in the integration interval. In general, with equidistant points and , and we call the formula

,

the point closed Newton-Cotes formula when the formula is derived from integrating the Lagrange Polynomial through the data points.

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There also exists open Newton-Cotes formulae where but these are less accurate and less often used.

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The following Theorem holds:

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Theorem 13

Definition

Some formulae