To put it simply, this course is about addition. That makes it sound pretty simple but there are, of course, some subtleties; namely, we will be adding infinitely many numbers together. At first, it seems like that job can never be completed. However, as we did last term, we can form approximate answers and then make our approximations better and better. This process will be called integration.
To speak in the same way, we might say that last term was really about subtraction. We were then primarily concerned with understanding change and we devised a useful tool, the derivative, to help us. To measure the change in a quantity, we subtract the value of the quantity at one time from the value at another. In fact, this was built into the definition of the derivative when we wrote
![\[ f^\prime(x_0) = \lim_{h\to 0} \frac{f(x_0 + h) - f(x_0)}{h} \]](preview_1.gif)
The important relationship between these two processes, which will be expressed in the Fundamental Theorem of Calculus, says that the operations of differentiation and integration are inverses of one another. That is, just as subtraction undoes the process of addition (and vice-versa), so will the process of differentiation undo integration.
Furthermore, there were a variety of interpretations of differentiation, the principle ones being (1) as the slope of a curve and (2) as a rate of change. In the same way, integration will possess both a geometric interpretation (as the area under a graph) and a more analytical one (as an average value).
This discussion is meant merely to point us in the appropriate direction; the ideas will become more clear as we move along. Now let's jump in and discuss integration from a geometric perspective.