Integration

Test de modification Seb

The integral was naturally introduced into mathematics to calculate lengths, areas or volumes, in other words, to measure.

For example, to calculate the distance covered by a moving object on its trajectory, we integrate its (algebraic) speed. From the outset, the integral of a function can be interpreted as the increase in one of its primitives. For two centuries, although techniques for calculating integrals improved, the objects integrated remained the same: essentially analytic applications, then, at the beginning of the 19th century, continuous ones (Cauchy). At the same time, this same Cauchy tried to give meaning to the integral of a function that was not bounded, or defined on an interval that was not a segment: this notion corresponds to that of the improper integral. From this period (Fourier) comes the notation ${\displaystyle {\int }_a^{b}f\left(x\right)\mathrm{d}x}$

With the development of harmonic analysis, on the one hand, and the need to give a precise status to the operations of analysis, on the other, numerous attempts were made to define the integral of functions belonging to a fairly broad class and to determine its properties: let's mention Dirichlet, who sought to generalize the notion of improper integral, and above all Riemann, who defined an integral on a certain class of functions, the integrable functions in Riemann's sense, an integral that has remained very classical. The starting point is the same as Cauchy's, except that the function to be integrated is not a priori assumed to be continuous, or even fairly regular. In fact, what determines the integrability of the function is the convergence of the integration procedure alone. In reality, functions that are too irregular, too large or defined on sets that are too complicated or unbounded escape Riemann integration.

The end of the 19th century saw the development of the most general notions of function, and with it a taste for tools with the widest possible scope of application. The aim was to integrate functions that might be highly irregular. Darboux's superior integral, not unrelated to Lebesgue's integral, dates from this period.

Lebesgue starts from the following observation: Riemann cuts the starting interval into small intervals I _k , centered at x _k , and postulates that the integral of the function f is close to that of the...