We study the power dissipated by the tip of an oscillating micron-scale cantilever as it interacts with a sample using a nonlinear model of the tip-surface force interactions that includes attractive, adhesive, repulsive, and capillary contributions. The force interactions of the model are entirely conservative and the dissipated power is due to the hysteretic nature of the interaction with the capillary fluid layer. Using numerical techniques tailored for nonlinear and discontinuous dynamical systems we compute the exact dissipated power over a range of experimentally relevant conditions. This is accomplished by computing precisely the fraction of oscillations that break the fluid meniscus. We find that the dissipated power as a function of the equilibrium cantilever-surface separation has a characteristic shape that we directly relate to the cantilever dynamics. Even for regions where the cantilever dynamics are highly irregular the fraction of oscillations breaking the fluid meniscus exhibits a simple trend. Using our results we also explore the accuracy of the often used harmonic approximation in determining dissipated power.