Derivation of Romberg's Rule Using N-Subdivisions and Applications of Romberg's Rule

By using a large number of subdivisions n to obtain good accuracy in approximating the integral  we note that an increase in the value of n does not necessarily mean an increase in accuracy in the result or even an improvement in the result. In addition, increasing the number n means an increase in the number of times the function values are calculated. The importance of Romberg's rule lies in improving the results obtained using the previous rules and obtaining high accuracy in arriving at approximate results faster and with a small amount of error. The reason for this is the high rank of the Romberg rule compared to the trapezoid rule, for example, which is not accurate in its results because the cutting error for this rule is of the order only.