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.
