To this point we have concentrated on just one central concept: instantaneous rate of change (the derivative) of a continuously varying function. We have seen some important problems — for example, optimization — for which we need derivatives of known functions, and others — for example, initial value problems — for which we need antiderivatives of known functions. What we needed to know about antiderivatives, we learned by studying derivatives.

In this chapter, we take up the second key concept of calculus: continuous accumulation. "Accumulation" is just a fancy word for adding to what you have already. Indeed, it is not an entirely new idea — it appeared, for example, in growth of populations and interest-bearing accounts. In fact, continuous accumulation can be viewed as a continuously varying function, and thus it has an instantaneous rate of change. That will be the connecting link between our two central concepts. And when the two rather different concepts are connected by that link, we will learn something very important: the Fundamental Theorem of Calculus.

Before we can link the second concept to the first, we need to know what the second concept is. We know what it means to add things together — but what does it mean to accumulate continuously? We answer that question in a familiar context that involves adding a lot of things together: averaging.