It is shown that the class of adequate mathematical models of dynamic objects of varying complexity is integrated model, built on the principle "input - output". It is important that this class of models, presented as a macro model of the object based on its input and output, suitable for linear and nonlinear modeling tasks productions. The most convenient form of descriptions of integrated macro models should be considered Volterra equations, numerical solution which is most often performed by a quadrature. It is noted that the main disadvantage of the use of quadrature formulas in the procedures of the approximate (numerical) integration is a significant error on large intervals of integration. Obvious known ways of minimizing errors of quadrature formulas on the selected class of functions should be considered a rational choice of the coefficients for quadrature formulas, as well as components and integration steps. Moreover, the accumulation of errors to increase the number of steps is determined not so much by the size of the step, as by replacing the original integral by a finite sum. The latter circumstance is important for modeling of objects in the real-time integration interval may be large or even unknown. To improve the accuracy of quadrature formulas procedure consisting in the expansion of the numerical integration of the original integral into two and apply for each of the last of its quadrature rules (for example, a combination of open and closed types of formulas of Newton-Cotes) can be offered. The increase in computing speed in the implementation of the integrated macro models based on Volterra models is achieved by using the method of degenerate cores, a feature of which is the same amount calculation step. Proposed fast algorithms for computing the design implementation of integrated macro models, providing calculations in real-time.