The Meaning of Integration - I

Integration is the 'inverse of differentiation' is what we are told in our beginning calculus course. It is indeed the case when the given function is continuousl. Augustin Cauchy2 (1789 1857), a famous mathematician, was the first person to establish that any continuous function is the derivative of a differentiable function. The proof of this result, known as the Fundamental theorem of Calculus (see below) is based on the concept of the area of planar regions bounded by 'suitable' curves. But in practice, it takes a few years of study to understand the concept of integration. In fact, one grasps the idea of integration only on studying the theory of Riemann3 integration (R-integration) which is the generalization of Cauchy integration to the class of functions which are not necessarily continuous (Figure 1b).