In this section we introduce the notions of left sum, right sum, trapezoidal sum, midpoint sum and Simpson sum of a given function f over a partition of an interval We begin with a brief review of the definition of a Riemann integral.
In a first course in integral calculus, the Riemann integral of a bounded function f on an interval is described as the limit of a sequence of sums of the type where and where, for each we have Sums of this type are called Riemann sums of the function f over the interval The sense in which the limit is taken is that if we define the mesh of the partition to be the largest of the lengths of the intervals then the above Riemann Sum can be made as close as we like to by making this mesh small enough.
The simplest type of partition of a given interval is a partition for which all of the intervals have the same length. In this case the partition is said to be regular and for each we have and Since the mesh of this partition is we make it approach 0 by letting
Given a<b and a positive integer n, we have described the regular partition as being the finite sequence of numbers defined by for each Before we give this definition to Scientific W orkPlace, we shall make the notation a bit more precise. We shall replace the notation xj by in order to account for the fact that this number depends also upon the value of n and upon the interval that is being partitioned. Accordingly, the first step in the procedure is to point at the equation and to click on Define and New Definition.
Since the various approximating sums all depend upon the function f that we wish to integrate, we need to let Scientific WorkPlace know that the symbol f stands for a function before we write down the definitions of the approximating sums. In order to achieve this, we make the nominal definition by pointing at the equation and clicking on Define and New Definition.
Note that this definition of f is purely temporary. We can change it at any time and all the sums will change accordingly.