Thermodynamic properties of a single chain in a confined space have been studied before with the polymer scaling theory and computer simulations. However, a comprehensive understanding of the entropic effects due to the molecular excluded volume and chain connectivity is emerging only recently, especially in the limit of large polymer packing densities as often encountered in biological systems. In this work, we propose a polymer density functional theory (DFT) to study the entropic forces for the confinement of single polymer chains in spherical cavities. At conditions accessible to Monte Carlo simulations, we show that the DFT predictions are in excellent agreement with the simulation results for the distributions of polymer segments as well as the free energy of confinement. The numerical efficiency of the DFT allows us to unify key conclusions from various theoretical analyses and experimental observations.