How much information does the sequence of integer moments carry about the correspond- ing unknown absolutely continuous distribution? We prove that a reliable evaluation of the corresponding Shannon entropy can be done by exploiting some known theoretical results on the entropy convergence, uniquely involving exact moments without solving the underlying moment problem. All the procedure essentially rests on the solution of lin- ear systems, with nearly singular matrices, and hence it requires both calculations in high precision and a pre-conditioning technique. Numerical examples are provided to support the theoretical results.