The N-Point Definite Integral Approximation Formula (N-POINT DIAF)

Francis Oketch Ochieng’

doi:10.11648/j.acm.20170601.11

Abstract

Various authors have discovered formulae for numerical integration approximation. However these formulae always result to some amount of error which may differ in size depending on the formula. It’s therefore important that a formula with highest precision has been discovered and should be implemented for use in numerical integration approximations problems, especially for the definite integrals which cannot be evaluated by applying the analytical techniques. The present paper therefore explores the derivation of the N-point Definite Integral Approximation Formula (N-point DIAF) which amounts to the discovery of the 2-Point DIAF. This formula will assist in almost accurate evaluation of all definite integrals numerically. The proof of the formula is given, a specific test problem is then solved using the discovered 2-Point DIAF to obtain the solution numerically, which has the highest precision compared to other numerical methods of integration. Further the error terms are obtained and compared with the existing methods. Finally, the effectiveness of the proposed formula is illustrated by means of a numerical example.

Highlights

IntroductionA where F (x) is differentiable function whose derivative is f (x) i.e. Often need arises for evaluating the definite integral of functions that does not have explicit antiderivative, in other circumstances the function is not known explicitly but is given empirically by a set of measured or tabulated values
Integrals of most analytical functions can be evaluated as I = b ∫ f ( x)dx F (b) − (a) (1)a where F (x) is differentiable function whose derivative is f (x) i.e. F / (x)= f (x)
We present a formula to approximate definite integrals

Summary

Introduction

A where F (x) is differentiable function whose derivative is f (x) i.e. Often need arises for evaluating the definite integral of functions that does not have explicit antiderivative, in other circumstances the function is not known explicitly but is given empirically by a set of measured or tabulated values. Of central interest is the process of approximating a definite integral from values of the integrand when exact mathematical integration is not available. Integral Calculus is a fundamental field of study in Mathematics and is widely used to model physical processes by scientists and engineers [3]. It has widespread uses in science, engineering and economics and can solve many problems that Algebra alone cannot [1]. Some integrals cannot be evaluated analytically and need to be approximated This means that we have to apply numerical methods in order to get an approximate solution. It is worthwhile to note further that the choice of the variables used in the N-point DIAF, stated below, does not in any way suggest any correlation with the neighborhood method used in Regression Analysis (i.e. the k-Nearest Neighbours Regression algorithm)

Statement of the Formula

Formula Proof

Numerical Illustration

Precision and Stability

Conclusion