A Comparative Study of Some Methods of Definite Numerical Integration Using Maple Software

Abstract

Numerical Integration is defined as how to find the approximate value of a given integral it is usually resorting to when there is difficulty or it is not possible to calculate the value of the integral using the usual methods, or when the formula of the function is not known but its values are known at a finite number of points. This study aims to compare between three numerical integration methods which are (the Trapezoid Method, the Simpson’s Method and Romberg's Method) to get to the best method in terms of accuracy of results and speed of convergence to exact value. And it also aims to effect of increasing the number of division of the period and the integration limit sign and type of the mathematical form of the function on the accuracy. The comparison was completed by using the Maple 18 Program in calculating the exact and approximate values (rounded to nine decimal digits after the comma) and calculate the absolute error of a number of single integrals of various functions in mathematical forms and continuing in its periods of integration. From the results reached is that the Trapezoid method have their results converging to the exact value of the integrals of trigonometric and multi-definition functions faster than other methods. As for the integrals of other functions (that are being studied) the Romberg's method is the best method and the Simpson's method comes in second place. Also increasing the number of division of the period does not always lead to an increase in the accuracy of the results for all integrals, And that when the function curve falls in the first quadrant, the methods of numerical integration gives very accurate results compared to whether the function curve falls in the first and second quadrants together or in the second quadrant which we got the lowest accuracy of the methods.