Interval arithmetic applied to simulation of dynamical systems has attracted a great deal of interest in recent years. Much of this research has been carried out in the calculation of fixed points or low-period windows for nonlinear discrete maps. This study proposes a novel interval computation based on a piecewise method to calculate periodic orbits for the logistic map. Using the cobweb plot, three rounding situations have been applied to a correct outward rounding, as required by interval arithmetic. The proposed method is compared with results in the literature and with the results obtained by means of the Matlab toolbox Intlab. The comparison is accomplished for nine case studies using the logistic map. Numerical results explicitly indicate that the proposed method produces intervals that are substantially narrower than those obtained with the traditional techniques.