A treatment of resonance absorption intermediate between the usual narrow and wide resonance approximations is developed for homogeneous systems. An arbitrary parameter, λ, is introduced into the flux and two distinct approximations are employed to determine λ as a function of the resonance parameters. One is based upon a method of equating successive orders of approximation and the other is based upon a variational principle. Formulas are given, from which the resonance integral may be calculated. The parameter λ characterizes, in essence, the location between the narrow and wide resonance extremes, of the actual resonance. When λ is set equal to 0 or 1, the usual first order wide or narrow resonance integrals are obtained. Sample calculations are carried out for a good intermediate case (the 192 ev resonance of U238 in a 1:1 atomic mixture with hydrogen) using linear and nonlinear trial functions for both types of approximations. All results agree to within less than one percent of 0.172 barns. In comparison, the usual extreme energy-loss assumptions yield results which differ by more than a factor of 2 (0.121 barns for the narrow resonance approximation and 0.253 barns for the wide resonance approximation).