Article Details

A Research on the Fundamental Theorem of Calculus and Its Proof | Original Article

Sandeep .*, in Journal of Advances and Scholarly Researches in Allied Education | Multidisciplinary Academic Research

ABSTRACT:

The Fundamental Theorem of Calculus (FTC) and its proof provide an illuminating but also curious example. The propositional content of the statements, which are connected to this name, varies. Consequently, also the proofs differ. The formulations of different versions of “the” FTC cannot be understood in isolation from its historical context. In this paper we will formulate one of the most important results of calculus, the Fundamental Theorem. This result will link together the notions of an integral and a derivative. This paper contains a new elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral. The hardest part of our proof simply concerns the convergence in L1 of a certain sequence of step functions, and we prove it using only basic elements from Lebesgue integration theory.