Consider a step function and a dissection of [a,b] such that the function has the constant value ck on each subinterval of the dissection. Then:
The integral of s from a to b is the number .
This means that the integral is simply the sum of the products of the constants ck times the length of the corresponding subintervals. In particular if f is constant and positive the integral is the area of a rectangle.
The two symbols and are used without distinction: we prefer the first, or sometimes , because it is more compact, but the second one has some advantages, in particular when we deal with multi-variable functions or in the use of the substitution rule. It is important, at any rate, to remember that the integral depends only on the function s and the interval [a,b]: the symbol "dx" has no particular meaning and is not a differential.
Observe in particular that the values of the step functions at the points of the dissection are unimportant: these numbers are never used in the previous definition. This means that if you modify the values of a step function in a finite number of points (obviously changing also the dissection of the interval [a,b]) the value of the integral is not affected. This agrees with the fact that the area of a segment is zero. From now on we'll no more graph these points.
It's also important to observe that continuity of the functions is not important for the concept of integral: step functions are never continuous (except when constant!).
In the above definition of integral the number a is less than b. This definition is usually extended, in order to allow also a>b, as follows: if a>b then . We also define .