proceeding from ( )the different integration methods will differ in the way they approximate each integral on the right hand side the fundamental idea is that each term is an integral over small interval oexi xi and over this small intervalit makes sense to approximate by simple shapesay constanta straight lineor parabolawhich we can easily integrate by hand the details will become clear in the coming examples computational example to understand and compare the numerical integration methodsit is advantageous to use specific integral for computations and graphical illustrations in particularwe want to use an integral that we can calculate by hand such that the accuracy of the approximation methods can easily be assessed our specific integral is taken from basic physics assume that you speed up your car from rest and wonder how far you go in seconds the distance is given by the rt integral /dtwhere tis the velocity as function of time rapidly increasing velocity function might be td ( the distance after one second is /dt( which is the integral we aim to compute by numerical methods fortunatelythe chosen expression of the velocity has form that makes it easy to calculate the anti-derivative as td ( we can therefore compute the exact value of the integral as : (rounded to decimals for convenience the composite trapezoidal rule rb the integral /dx may be interpreted as the area between the axis and the graph xof the integrand figure illustrates this area for the choice ( computing the integral /dt amounts to computing the area of the hatched region if we replace the true graph in fig by set of straight line segmentswe may view the area rather as composed of trapezoidsthe areas of which are easy to compute this is illustrated in fig where straight line segments give rise to trapezoidscovering the time intervals oe : /oe : : /oe : : and oe : : note that we have taken the opportunity here to demonstrate the computations with time intervals that differ in size