The Newton-Cotes formulas are an ex

The Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.

To integrate a function f(x) over some interval [a,b], divide it into n equal parts such that f_n=f(x_n) and h=(b-a)/n. Then find polynomials which approximate the tabulated function, and integrate them to approximate the area under the curve. To find the fitting polynomials, use Lagrange interpolating polynomials. The resulting formulas are called Newton-Cotes formulas, or quadrature formulas.

Newton-Cotes formulas may be "closed" if the interval [x_1,x_n] is included in the fit, "open" if the points [x_2,x_(n-1)] are used, or a variation of these two. If the formula uses n points (closed or open), the coefficients of terms sum to n-1.

If the function f(x) is given explicitly instead of simply being tabulated at the values x_i, the best numerical method of integration is called Gaussian quadrature. By picking the intervals at which to sample the function, this procedure produces more accurate approximations (but is significantly more complicated to implement).