Newton-Cotes formulas (so named after Isaac Newton and Roger Cotes) are a group of formulas used for numerical integration (quadrature), a subject of numerical analysis. They deal with classifying a number of possible solutions to this problem under the assumption that you have gained samples of a function on equidistant points such that and . Under this assumption Newton-Cotes formula holds that:
The :s are called weights and are derived from computing the corresponding Lagrange polynomials thusly:
Here the :s are fix constants depending only on the chosen :s. Computing this integral for different numbers of points will yield some famous quadrature formulas:
These are the actual Newton-Cotes formulas. The :s on the left signify on which monomial bases the solution is exact, e.g. a solution that is exact for will be exact on the monomials in the set . Note that the formulas' exactness increase in odd steps, so using the trapezoid rule for integrating is actually no better than using the midpoint rule, and just a waste of time. In the same manner, Simpson's rule will be exact for monomials in which will suffice for most functions found in practice.
If you also let the distance between each sample point vary, you obtain another group of quadrature formulas, the best of which is know as Gaussian quadrature.
on 
equidistant points 
such that 
and 
. Under this assumption Newton-Cotes formula holds that:
:s are called weights and are derived from computing the corresponding 
:s. Computing this integral for different numbers of points 
:s on the left signify on which 
will be exact on the monomials in the set 
. Note that the formulas' exactness increase in odd steps, so using the trapezoid rule for integrating is actually no better than using the midpoint rule, and just a waste of time. In the same manner, 
which will suffice for most functions found in practice.
between each sample point vary, you obtain another group of quadrature formulas, the best of which is know as