Integral-Differential Calculus

TL;DR

This paper provides a detailed exposition of Newton-Leibniz calculus, covering definitions, proofs, and fundamental theorems of integral and differential calculus using Riemann sums.

Contribution

It offers a comprehensive, step-by-step presentation of classical calculus foundations with rigorous proofs and explanations.

Findings

Defined the integral as a limit of Riemann sums

Verified integrals of standard functions through direct manipulation

Proved the Fundamental Theorem of Calculus

Abstract

We give an exposition of the Newton-Leibniz calculus. We begin by defining the integral as a limit of Riemann sums, verify the integrals of the standard catalog of functions by direct manipulation, prove the substitution lemmas as theorems about Riemann sums, cross the Fundamental Theorem of Calculus, and harvest the differential calculus on the other side.