Integration

Abstract

The theory of integration has developed from simple computations of length, area, and volume to some very abstract constructions. We start with some intuitive calculations of area and volume giving, for instance, Archimedes’ formulas for the volume and area of a ball, but after a reprimand from A. himself, we become more serious and turn to the Riemann integral. With both differentiation and integration at our disposal, we can then go through some of the fundamental results of analysis at a brisk pace. By means of a series of well-chosen examples we shall at the same time prove the basic properties of the Fourier transform, one of the most versatile tools of mathematics. After this there are sections on the Stieltjes integral and on integration on manifolds. Only the simplest properties of manifolds and differential forms are used. I chose this way in order to be able to give the right proof of Green’s formula, and at the same time demonstrate one of the magic tricks of analysis, Stokes’s formula in all its generality.