Teaching demonstration of the integral calculus

Abstract

A simple in-class demonstration of integral Calculus for first-time students is described for straightforward whole number area magnitudes, for ease of understanding. Following the Second Fundamental Theorem of the Calculus, macroscopic differences in ordinal values of several integrals, ΔF(x), are compared to the regions of area traced out from the horizontal axis by the derivative functions f(x) over various domains. In addition, microscopic incremental differentials of an integral at a particular position, dF(x), are compared to corresponding values of the derivative function f(x) multiplied by various horizontal shifts, dx. For any area to exist for a derivative function, dx > 0, but the difference between these compared magnitudes collapses to zero as long as dx widths are small. The demonstration readily confirms, both arithmetically and graphically for trigonometric, polynomial, and transcendental functions, the Newton discoveries that (1) the rate that area accumulates under a function is proportional to the ordinal value of the function itself, and (2) changes in elevation along an integral function automatically equal the exact net area traced out by its derivative from the X-axis.