This article describes a streamlined method for simultaneous integration of an entire family of interpolating functions that uses one and the same interpolation grid in one or more dimensions. A method for creating customized quadrature/cubature rules that takes advantage of certain special features of Mathematica's InterpolatingFunction objects is presented. The use of such rules leads to a new and more efficient implementation of the method for optimal stopping of stochastic systems that was developed in [1]. In particular, this new implementation allows one to extend the scope of the method to free boundary optimal stopping problems in higher dimensions. Concrete applications to finance--mainly to American-style financial derivatives--are presented. In particular, the price of an American put option that can be exercised with any one of two uncorrelated underlying assets is calculated as a function of the observed prices. This method is similar in nature to the well-known Longstaff-Schwartz algorithm, but does not involve Monte-Carlo simulation of any kind.