We study diffusion-controlled reactions in which a reactive solute diffuses through a medium containing static, reactive, spherical traps of many different sizes. We focus on the cases of impenetrable, i.e., nonoverlapping traps, and of randomly overlapping traps. Bounds for the trapping rate are derived using trial functions of the kind developed by Doi, and by Weissberg and Prager. It is shown that the bounds for trapping rate are relatively insensitive to dispersivity of trap size when they are plotted against the proper scaling variable. The trap volume fraction is the proper scaling variable for randomly overlapping traps. The mean density of traps ρ〈a〉 is shown to be the proper scaling variable for nonoverlapping traps. It is shown that the low-density limits of both the Doi and the Weissberg-Prager bounds fail to reproduce the radius-averaged single-trap solution of Smoluchowski. We give a generalization of these classes of bounds that has the proper behavior. In evaluating trapping bounds for impenetrable traps, the material correlation functions are evaluated for the first time for a nontrivial polydisperse system.