Applied Mathematics, 2014, 5, 1412-1426 Published Online June 2014 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2014.510133 How to cite this paper: Marinov, T., et al. (2014) Behavior of the Numerical Integration Error. Applied Mathematics, 5, 1412-1426. http://dx.doi.org/10.4236/am.2014.510133 Behavior of the Numerical Integration Error Tchavdar Marinov * , Joe Omojola, Quintel Washington, LaQunia Banks Department of Natural Sciences, Southern University at New Orleans, New Orleans, USA Email: * tmarinov@suno.edu Received 11 February 2014; revised 11 March 2014; accepted 18 March 2014 Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract In this work, we consider different numerical methods for the approximation of definite integrals. The three basic methods used here are the Midpoint, the Trapezoidal, and Simpson’s rules. We trace the behavior of the error when we refine the mesh and show that Richardson’s extrapolation improves the rate of convergence of the basic methods when the integrands are sufficiently diffe- rentiable many times. However, Richardson’s extrapolation does not work when we approximate improper integrals or even proper integrals from functions without smooth derivatives. In order to save computational resources, we construct an adaptive recursive procedure. We also show that there is a lower limit to the error during computations with floating point arithmetic. Keywords Numerical Integration, Algorithms with Automatic Result Verification, Roundoff Error 1. Introduction Suppose ( ) f x is a real function of the real variable x , defined for all [ ] , x ab ∈ . The definite integral from the function ( ) f x from a to b is defined as the limit ( ) ( ) max 0 1 d lim , k k b n k k k a I f x x f x ∆ → = = = ∆ ∑ ∫ (1) where 0 1 n a x x x b = < < < =  and 1 , 1, 2, ,. k k k x x k n − ∆ = − =  In the case when the anti-derivative ( ) F x of ( ) f x is known, we can calculate the definite integral using the fundamental theorem of Calculus: ( ) () ( ) d . b a f x x Fb Fa = − ∫ (2) * Corresponding author.