I have no understanding of what you are trying to say here, but I can tell you as one of Physics Forums experts on thermodynamics that I agree totally with what Philip Kosck and and Terry Bing have said. If what you are saying disagrees with them, then I too disagree with you.No I am not calculating integral pdx, I am just using mechanics consideration. And even this mechanical work is to be assumed to be done quasi statically. We use work energy theorem of Mechanics to conclude that the energy of the system on which work has been done must increase its energy when the final equilibrium state is reached. To conclude the change in internal energy we will have to consider another reversible process of heating quasi statically to achieve the final state with the help of a reservoir of infinite thermal capacity passing through series of equlibrium states of increasing temperature.