The　authors　study　the　Rayleigh-Taylor　instability　for　two　incompressible　immiscible　fluids　with　or　without　surface　tension,　evolving　with　a　free　interface　in　the　presence　of　a　uniform　gravitational　field　in　Eulerian　coordinates.　To　deal　with　the　free　surface,　instead　of　using　the　transformation　to　Lagrangian　coordinates,　the　perturbed　equations　in　Eulerian　coordinates　are　transformed　to　an　integral　form　and　the　two-fluid　flow　is　formulated　as　a　single-fluid　flow　in　a　fixed　domain,　thus　offering　an　alternative　approach　to　deal　with　the　jump　conditions　at　the　free　interface.　First,　the　linearized　problem　around　the　steady　state　which　describes　a　denser　immiscible　fluid　lying　above　a　light　one　with　a　free　interface　separating　the　two　fluids,　both　fluids　being　in　(unstable)　equilibrium　is　analyzed.　By　a　general　method　of　studying　a　family　of　modes,　the　smooth　(when　restricted　to　each　fluid　domain)　solutions　to　the　linearized　problem　that　grow　exponentially　fast　in　time　in　Sobolev　spaces　are　constructed,　thus　leading　to　a　global　instability　result　for　the　linearized　problem.　Then,　by　using　these　pathological　solutions,　the　global　instability　for　the　corresponding　nonlinear　problem　in　an　appropriate　sense　is　demonstrated.