19.1 Introduction

If we want to study an unknown function defined by some physical or other situation, the method of analysis consists of studying its derivative, as a function of appropriate variables, and deducing from them an equation for it.

The equation will generally be an example of a differential equation, which is an equation that involves derivatives of an unknown function. The process of solving a differential equation is called integrating it.

One important special case of this problem occurs when we have an explicit formula for the derivative of a function f defined over an interval of a single variable x and this formula is depends only on the independent variable x.

We want to know as much as this information can provide us about the function f.

Thus we want to go from g(x), with for a < x < b, to f(x); we want to "undo" the differential operator , and this could be called undifferentiating to find f(x) itself or as much about it as we can; the standard name for this task is finding the anti-derivative of g(x).

This procedure is also called finding the indefinite integral of g(x) with respect to x.

We will not use this terminology below, because we want to define the definite integral in a completely different way, and want to avoid confusion between the two notions.

Of course you can anticipate from the similarity in names that there will be an intimate relation between the definite integral (here not defined at all) and the anti-derivative here. If you guess as much you are right. But this relation is a basic theorem, called the fundamental theorem of calculus.