High volumetric-efficiency thermally integrated systems for capturing a target gas from a process gas stream include a monolithic body and a distribution system. The monolithic body includes a first plurality of channels and a second plurality of channels each having sorbent surfaces that reversibly adsorb the target gas. The channels are in thermal communication such that heat from an exothermic adsorption of target gas in one plurality of channels is used by an endothermic desorption of target gas from the other plurality of channels. Methods for separating a target gas from a process gas stream include switching the high volumetric-efficiency thermally integrated systems between a first state and a second state. In the first state, the first plurality of channels undergoes desorption while the second undergoes adsorption. In the second state, the second plurality of channels undergoes desorption while the first plurality undergoes adsorption.