Pictorial integration in the Calculus

Abstract

ABSTRACT Graphical analysis of the relationships between pairs of integrals and their corresponding derivatives resulted in the development of a procedure one may describe as ‘pictorial integration’. The arctangent of the derivative of various functions was analysed to confirm that this accurately reports the magnitude of angles made by integral functions from the horizontal, for polynomial, trigonometric, exponential, and transcendental forms. This was used to generate pictures of certain functions that have no known simple algebraic formula. The pictorial representations for the integral of and for the ellipse arc length integral are described. The pictures were confirmed through the use of Newton (Taylor) series expressions over appropriate domains, the second derivatives of the functions which are the first derivatives of their known derivatives, and electronic integration. Several powerful computer systems are available that quickly and easily graph such complicated integrals and confirm these results, but the analysis here explains for Calculus students the detailed graphical relationships between integrals and their derivatives.