computing integrals the major problem with this procedure is that we need to find the anti-derivative xcorresponding to given xfor some relatively simple integrands /finding xis doable taskbut it can very quickly become challengingeven impossiblethe method ( provides an exact or analytical value of the integral if we relax the requirement of the integral being exactand instead look for approximate valuesproduced by numerical methodsintegration becomes very straightforward task for any given (!the downside of numerical method is that it can only find an approximate answer leaving the exact for the approximate is mental barrier in the beginningbut remember that most real applications of integration will involve an xfunction that contains physical parameterswhich are measured with some error that isf xis very seldom exactand then it does not make sense to compute the integral with smaller error than the one already present in xanother advantage of numerical methods is that we can easily integrate function xthat is only known as samplesi discrete values at some pointsand not as continuous function of expressed through formula this is highly relevant when is measured in physical experiment basic ideas of numerical integration we consider the integral zb /dx ( most numerical methods for computing this integral split up the original integral into sum of several integralseach covering smaller part of the original integration interval oeabthis re-writing of the integral is based on selection of integration points xi that are distributed on the interval oeabintegration points mayor may notbe evenly distributed an even distribution simplifies expressions and is often sufficientso we will mostly restrict ourselves to that choice the integration points are then computed as xi ihi ( where ba ( given the integration pointsthe original integral is re-written as sum of integralseach integral being computed over the sub-interval between two consecutive integration points the integral in ( is thus expressed as hd zb zx /dx zx /dx note that and xn zxn /dx xn /dx (