We study restricted Monte Carlo integration for anisotropic Hölder–Nikolskii classes. The results show that with clog2n random bits we have the same optimal order for the nth minimal Monte Carlo integration error as with arbitrary random numbers. We also study the computation of integration on anisotropic Sobolev classes in the quantum setting and present the optimal bound of nth minimal query error. The results show that the error bound of quantum algorithms is much smaller than that of deterministic and randomized algorithms.