Standard-Slope Integration: A New Approach to Numerical Integration

S t a n d a r d - S l o p e I n t e g r a t i o n

A New Approach to Numerical Integration

Peter James Italia, M.D.

Layout and design: Peter J. Italia

Cover design: Peter J. Italia

Copyright © 2023, 2020, 2013, 2007 by Peter J. Italia

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ISBN: 9781301818419

Published by Peter J. Italia at Draft2Digital

Abridged and Revised Edition

In loving memory of Susan Elizabeth Palmer,

my childhood friend.

Standard-Slope Integration

Abstract: The integral (or antiderivative) has long been the backbone of calculus, serving as a powerful means by which the area under a curve can be determined. Despite its great usefulness, however, finding anything other than the simplest antiderivatives can prove to be extremely challenging as well as difficult to evaluate, or they may not even exist. In such cases, it is often necessary to turn to other methods of numerical integration, which in themselves can be rather cumbersome to apply and may offer only a crude estimate of the actual area under the curve (AUC). Standard-slope integration (SSI) is a fast, easy, and more accurate method of numerical integration, the results of which are on par with those of classical integration methods.

Key Words: Calculus, numerical integration, integral calculus, differential calculus, standard-slope integration, standard slope, integral, derivative, antiderivative, area under the curve.

Since its first introduction by Isaac Newton and G. W. Leibniz during the 17th century, the integral (or antiderivative, as it is often called) has been the backbone of calculus, serving as a powerful means by which the area under (or above) a curve can be determined [1,2,3]. Despite its great usefulness, however, the integral has its own share of problems, too.

Among the most noteworthy of these problems is the fact that finding anything more than the simplest antiderivatives proves more often than not to be an extremely challenging task in itself. While finding the antiderivative can be a difficult chore indeed, once found, it may yet again be quite cumbersome to evaluate. Finally, there always exists the possibility that we may ultimately come to the realization after much time and effort has been expended that the integral for which we have been searching simply does not exist.

In such cases, it is often necessary to turn to other methods of numerical integration that have since been developed. Some of these methods, however, are themselves often rather cumbersome to apply and may offer only a crude estimate of the actual area