» Required Reading » Table of Contents » Chapter 20 20.1 Area and Notation
	Chapter Index	Next Section

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). (Notice that this will be negative if either but not both of c and b-a are negative, as we have required) It is this approach which we will generalize to all sorts of contexts.