Two classes of methods for numerical integration are presented : The class of Newton-Cotes methods and the class of Gauss methods. In the class of Newton-Cotes methods, the trapezoidal rule, Simpson's rule and other higher order rules like Romberg's method have been derived . The methods are clarified by examples. Mathematica modules are designed to derive Newton Cotes methods of any accuracy and to apply them for evaluation of integrals. Gauss methods are derived in general form with n Gauss-Legendres knots and in particular with n=1,2,3,4 knots. Example illustrating the methods are presented. The exact analysis of both methods of numerical integration has been carrying out. A set of questions is enclosed in the chapter.
Keywords: Newtons-Cotes formulas, Gauss formulas.