Suppose we have a non-negative function f of the variable x, defined in some domain that includes the interval [a, b] with a < b.
If f is sufficiently well behaved, there is a well defined area enclosed between the lines x = a, x = b, y = 0 and the curve y = f(x).
That area is called the definite integral of f dx between x = a and x = b (of course only for those functions for which it makes sense).
It is usually written as
If c lies between a and b we obviously have
In order to make this equation hold for arbitrary c, we require that when b is less than a the symbols above represent the negative of the area indicated.
Where the function f is sometimes negative, we define the definite integral and the same symbols to represent the area between the x-axis and y = f(x) where f is positive minus the area between the two when f is negative (when a is less than b).
To make this definition mathematical we must give a procedure for computing the area, at least in theory, and some indication of what functions f we can and cannot define it for.
Here f is called the integrand, and it is said to be integrated "ds".
Our approach to defining the area is based on the fact that we know what the area of a rectangle is, namely it is the product of the lengths of its sides. If the function f(x) is a constant c, then the area in question will be a rectangle and the area will be c(b - a).
That's all we need to define area for a constant function.
Our task is to generalize this definition to functions that are not constant.