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Relativistic confining potential models, endowed with bag constants associated to volume energy terms, are investigated. In contrast to the usual bag model, these potential bags are distinguished by having smeared bag surfaces. Based on the dynamical assumptions underlying the fuzzy bag model, these bag constants are derived from the corresponding energy-momentum tensor. Explicit expressions for the single-quark energies and for the nucleon bag constant are obtained by means of an improved analytical version of the saddle-point variational method for the Dirac equation with confining power-law potentials of the scalar plus vector (S + V) or pure scalar (S) type.