THE DEFINITION OF THE DEFINITE INTEGRAL IN TERMS OF DARBOUX SUMS AND THE CONDITIONS FOR INTEGRABILITY

This paper presents a rigorous development of the definite integral through the framework of Darboux sums, emphasizing both theoretical foundations and practical implications. The concepts of upper and lower Darboux sums are introduced as fundamental tools for approximating the area under a bounded function over a closed interval. By examining partitions of the interval and the behavior of supremum and infimum values on subintervals, the study establishes the formal definition of the definite integral. A key focus is placed on the necessary and sufficient conditions for integrability, particularly the convergence of upper and lower sums to a common limit.