a very useful one---polynomials are very simple to integrate and this argument does not by itself guarantee better accuracy on integrating other functions. Clenshaw–Curtis quadrature is based on approximating f by a polynomial interpolant at Chebyshev nodes and integrates polynomials of degree up to n exactly when given n samples. Even though it does not integrate polynomials or other functions that are analytic in a large neighborhood of as well as Gauss–Legendre quadrature, Clenshaw–Curtis converges at approximately the same rate as Gauss–Legendre quadrature for most (non-analytic) integrands. Also, Clenshaw–Curtis shares the properties that Gauss–Legendre quadrature enjoys of convergence for all continuous integrands f and robustness to numerical rounding errors. Clenshaw–Curtis is straightforward to implement in time by FFT-based methods. Newton–Cotes quadrature is based on approximating f by a polynomial interpolant at equally-spaced points in , and like Clenshaw–Curtis also integrates polynomials of degree up to n exactly when given n samples. However, it suffers from Runge's phenomenon as n increases; Newton–Cotes does not converge for some continuous integrands f, and when it does converge it may suffer from numerical rounding errors. Thus, both Clenshaw–Curtis and Gauss–Legendre enjoy substantial benefits over Newton–Cotes for moderate to large n. References Numerical integration (quadrature)