Fredholm and Volterra integral equations appear for a number of branches of engineering sciences such as acoustic study, optics theory, laser, potential theory, and radioactive radiation theory, study of heart disease curves, fluid mechanics, and communication and so on. Since in most cases it is not possible to find an analytical solution of the problem, so it is necessary to find numerical solution of under consideration problem. In this research project, Simpson's quadrature method is first used to calculate the numerical solution of the first and second type of linear Volterra integral equations, and by using this method the solution is expressed as a recursive relation. That is, unlike conventional numerical methods, the linear Volterra integral equations of the first and second types are not converted to algebraic equations. Then, using the Romberg's quadrature method, two algorithms are expressed on the numerical solutions obtained equations from the trapezoidal Simpson's quadrature method and the Simpson's quadrature method. Numerical results express the accuracy of the method and show that without using additional points in numerical integration, the accuracy of the approximations can be increased to the required level.
