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.