An Approximation of Gaussian Integral Over (-1, 1)

Abstract

The Gaussian integral is useful in many branches of science, statistics and probability theory. There are some standard methods to evaluate the famous Gaussian integral \(\int{_-}_\infty^\infty\) \(\mathit{e^{-x^2}}\) \(\mathit{dx}\). However, the evaluation of \(\int\) \(\mathit{e^{-x^2}}\) \(\mathit{dx}\) over a finite interval is a complex task. This chapter is devoted to approximating the integral \(\int{_-}_1^1\) \(\mathit{e^{-x^2}}\) \(\mathit{dx}\) by establishing its bounds. Our method may be used for approximation of \(\int\) \(\mathit{e^{-x^2}}\) \(\mathit{dx}\) over other finite intervals.