ON THE CONCEPT OF INTEGRAL, HISTORY OF ORIGIN AND SOME COUNTER-EXAMPLES

Abstract. In the article is given information about the integral concept and the history of its origin. In addition to the Riemann integral, which is covered in the school program, a number of other integrals (Riemann-Stiltes, Daniell, Alfred Haar, Henstock-Kurzweil, Ito and Stratonovich, Chokvet) are described. The difference between integrals is explained with the help of specific examples. The concept of definite integral and its use in practical problems were explained in detail, counter-examples of some theorems were presented. 

Keywords: infinitesimal, area, volume of a solid, integral, indefinite integral, surface, surface integral, sign of indefinite integral, counter derivative.

INTRODUCTION

 In mathematics, the integral defines the values of functions that describe displacement, area, volume, and other concepts resulting from the integration of infinitesimals. The process of finding integrals is called integration. Along with differentiation, integration is one of the basic, important concepts of mathematics and serves as a tool for solving problems involving the area of an arbitrary shape, the length of a curve, and the volume of a solid in mathematics and physics. 

RESEARCH MATERIALS AND METHODOLOGY

Integrals can be further generalized depending on the type of function, as well as the field in which the integration is performed. For example, a line integral for a function of two or more variables is defined and the interval of integration is replaced by the formula representation of the curve connecting the two endpoints of the interval. In surface integrals, the curve is calculated by replacing a part of the surface in three-dimensional space. The first documented technique capable of calculating integrals is the exhaustion method of the ancient Greek astronomer Eudoxus (ca. 370 BC), who is known to have tried to find areas and volumes by dividing them into infinitely many divisions. This method was further studied and developed by Archimedes in the 3rd century BC. It was used to calculate the area of a circle, the surface and volume of a sphere, the area of an ellipse, the area of the lower part of a parabola, and the volume of a segment. That is, it is not wrong to say that the revolution of integrals occurred during this period.

In addition, a similar method was developed in China in the 3rd century AD by Liu Hui, who used it to find the area of a circle. This method was later used in the 5th century by Chinese father and son mathematicians Zu Chongji and Zu Geng to find the volume of a sphere. And in the Middle East, Hasan Ibn al-Haytham, a man known in Latin countries as Alhazen (ca. 965 – 1040 AD), developed the formula for the sum of four powers. He used these results to calculate (create) what is now called the integral of a function, in which he was able to calculate the volume of a paraboloid using integral squares and sum-of-fours formulas. The next important advances in integral calculus did not appear until the 17th century.

At this time, Cavalieri’s work “By the method of indivisibles” and several works of Fermat began to form the basis of modern calculus. The next steps were taken at the beginning of the 17th century by Barrow and Torricelli, who put forward the first ideas about the connection between the operations of integration and differentiation. Barrow gives the first proof of the fundamental theorem of integral calculus. Wallis generalizes Cavalieri’s method to compute integrals of 𝑥 to general powers, including negative powers and fractional powers. The greatest success in the calculus of integrals occurred in the 17th century when Leibniz and Newton independently discovered the fundamental theorem of the integral calculus. They show the connection between integration and differentiation. This relation, together with the relative ease of differentiation, could also be used to calculate integrals. In particular, the fundamental theorem of calculus allows solving a much wider class of problems. Equally important was the comprehensive mathematical system developed by Leibniz and Newton. It allowed precise analysis of functions within continuous fields. This method eventually became the basis of modern calculus, and these calculations were taken directly from Leibniz’s writings.

Although Newton and Leibniz provided a systematic approach to the operation of integration, their work lacked a certain degree of rigor, and some mathematicians of their time considered these calculations to be non-general. A more robust result could be achieved by developing computational limits. The operation of integration was first officially recognized by Riemann in the creation of strict laws using definite limits. Although all finite piecewise continuous functions are Riemann integrable on a finite interval, later more general functions where Riemann’s definition does not apply were considered, especially in the concepts of Fourier analysis, and Lebesgue developed another gauge-based definition of the integral.