7.8 Antidifferentiation

We have seen already that given a function f( x) we can create a new function f ?( x) by taking its derivative. If we run this process backward, it is called antidifferentiation. The goal now is to find a function with a given derivative. The Fundamental Theorem of Calculus, which as the name, implies must be pretty significant, relates antidifferentiation to integration in an extremely important way. If f( x) is a function with antiderivative F( x) (so that F ?( x) = f( x)), then the Fundamental Theorem of Calculus states that

This is really a pretty amazing theorem. On the left-hand side, we have the integral of f( x) over the interval [ a, b] which remember is defined as the limit of all possible Riemann sums as the subdivision of the interval [ a, b] grows finer and finer. On the right-hand side, we have the difference of the values of the antiderivative of f( x) at the endpoints of the interval. At first glance, it seems totally unlikely that these things would be related at all!

The Fundamental Theorem is quite useful because it allows us to trade integration (i.e., computing Riemann sums, and perhaps wondering how accurate our estimate for the integral is) with antidifferentiation. On the other hand, before you think this is the death knell for integration, be warned that lots...