Exactness of quadrature formulas

The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases: Newton–Cotes, Clenshaw–Curtis, Gauss–Legendre, and Gauss–Hermite quadrature. Three further examples are mentioned more briefly.

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