Let us assume that one is given a finite collection of data values 
located at a discrete set of points 
, where i=0,1,...,N and we desire to
find approximations to the one-dimensional integral between fixed end points a
andb:
The basic idea is to approximate the integral as a weighted sum of the given
values of :
where 
is the number of values used to produce the approximation to
the integral over the given interval and 
is the weight attached to the ith
value. The goal is to determine the consistent with a desired accuracy for
the final result. The numerical analysis literature is voluminous on the rigors of
how best to find the weights and so we shall not attempt to delve into such issues.
Our goal here is to present a few practical methods and to leave the reader with a
reasonable impression as to how these methods are obtained.
In what follows, a general strategy is to think of the function f(x) whose sampled values
obey 
as an unknown but existing object which possesses enough derivatives for
our use. Namely, we shall be using the assumption that the function is expandable in a Taylor
series. We shall develop a few methods which utilize this to obtain a systematic set of
methods of varying degrees of accuracy.