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The Error of the Newton - Cotes Formuae

The closed Newton – Cotes formula that approximates an integration given two interpolation points isThe formula is exact for any constant or linear function, so the error must be of the formfor someso thatWe can estimate the error using a quadratic polynomial,

Rearranging this givesand the error term is in generalof order 3.

Example: Estimate the error forusing the trapezium rule.

onsoand

The actual error is 0.132698.

Similarly, given three interpolation points,which is exact for all quadratic polynomials, the error term must be of the formfor someIf we examineon the intervalwe findso the formula is exact for cubic polynomials too and the error must be of the formTo estimate the error use thenandFor a general interval of integration the error term here is