Standard-Slope Integration: A New Approach to Numerical Integration
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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 using the derivative, the results of which are on par with those of classical integration methods.
The printed edition includes formal derivations; the e-book does not.
Peter James Italia, MD
Peter James Italia, M.D. was born and raised in Florida and later received his Bachelor of Science from Emory University in Atlanta, Georgia, as well as earning degrees from Champlain College in Burlington, Vermont. He graduated from the Universidad Eugenio María de Hostos school of medicine in the Dominican Republic and later studied General Surgery in the United States. He has an extensive educational background that includes medicine, the natural sciences, computer science, business, and accounting. He is the author of the book "Quantum Medicine: True Stories Involving the Use of Time Travel in the Treatment and Prevention of Death and Disease," in which Dr. Italia describes his use of time travel and other methods far beyond current medical knowledge and abilities in the diagnosis and treatment of seriously ill and dying patients at major medical centers. He is also the author of the calculus monograph titled "Standard-Slope Integration: A New Approach to Numerical Integration," in which he describes a new method of determining the area under a curve that is both simple and (more) accurate and easily rivals classical integration methods, and the writer/composer of the inspirational song "Fly High" as well as the author of a love poem. On numerous occasions, he has also provided government, military and local authorities in the United States, as well as the news media and friends, with important information about "future" news and world events of major significance (only to be ignored and thus has given up), especially with regard to terrorism, crime, politics, war, and the weather, including the following: the 9/11 terrorist attacks, the anthrax letters, the shoe bomber, the Elizabeth Smart kidnapping, the D.C. snipers, the space shuttle "Columbia" disaster, the Madrid and London bombings, the small airplane crash in New York City involving Yankees pitcher Cory Lidle, the ambush and killing of U.S. troops in Iraq and Afghanistan, Hurricanes Katrina and Irene, the major earthquakes and tsunamis in Indonesia and Japan, the Secret Service scandal, and more. Rejected and ignored, a virtual outcast, a pariah among society, he is currently living in the USA.
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Standard-Slope Integration - Peter James Italia, MD
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
All rights reserved. No part of this book may be reproduced or transmitted in any
form or by any means, electronic or mechanical, including photocopying, recording,
or by any information storage and retrieval system, without prior written permission
from the publisher. For permission requests, contact [email protected].
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