Sometimes one can calculate a definite integral exactly by antidifferentiating and using the fundamental theorem of calculus. However, sometimes this method does not work, or is not appropriate. (The second might be the case if, for example, one was integrating not an explicit function but an unspecified function about which one had certain information.) Also, in many applications one does not actually need to compute an integral exactly; merely obtaining a sufficiently good approximation to that integral, or even just an upper bound on its magnitude, may be sufficient in many applications (particularly those in analysis).
This article gives links to further articles about techniques for approximating or bounding integrals.