Numerical computation of integrals in higher dimensions

Quadrature refers to any method for numerically approximating the value of defi-bnite integral a f (x)dx. The goal is to attain a given level of precision with the fewestfunction evaluations.The factors that control the difficulty of a numerical integration problem are thedimension of the integral and the smoothness of the integrand f .Any quadrature method relies on evaluating the integrand f on a finite set of points(called the abscissas or quadrature points), and after processing these evaluations toproduce an approximation to the integral. Usually this involves taking a weightedaverage.The goal is to determine which points to evaluate and what weight to use so as tomaximize performance over a broad class of integrands.This study reviews Monte Carlo and Newton-Cotes methods of numerical approx-imation of integrals on both rectangular and nonrectangular regions and containsnew routines that can evaluate integrals up to 7 dimensions over arbitrary regions inMATLAB.The work aims to compare the methods and give some approximation results usingour self-written code.