7.7 Integration

The second pillar of calculus is integration, which you may recall is defined via Riemann sums. Remember that the Riemann sum of a function f( x) on an interval ( a, b) is defined as follows. First, the interval is subdivided into a finite number of subintervals. These subintervals do not all need to be the same length, although in practice it is simplest to adopt this procedure and that is what we will do from now on. If there are n subintervals then each has length ( b ? a)/ n and we will call this quantity ? x (read "delta x"). Next, we "sample" the function at a point in each subinterval. This means that we pick a point in each of the subintervals and compute the value of the function at that point. The sampling set does not have to be chosen in any particular way. We could even choose randomly in each subinterval. But again, there are a few standard practices: we could always sample at the left endpoint of each subinterval, or at the midpoint, or at the right endpoint, for example. Finally, all the sampled values are added together and the sum is multiplied times ? x. This is called a Reimann sum. There are infinitely many different ways to form the sum, so what you get definitely depends on how the subintervals and sampling points are chosen. The big idea...