We propose a deterministic method to eval- uate the integral of a positive function based on soft-binning functions that smoothly cut the integral into smaller integrals that are easier to approximate. In combination with mean-field approximations for each individ- ual sub-part this leads to a tractable algo- rithm that alternates between the optimiza- tion of the bins and the approximation of the local integrals. We introduce suitable choices for the binning functions such that a stan- dard mean field approximation can be ex- tended to a split mean field approximation without the need for extra derivations. The method can be seen as a revival of the ideas underlying the mixture mean field approach. The latter can be obtained as a special case by taking soft-max functions for the binning.