The document discusses estimating thermodynamic properties using equations of state. It covers: - Evaluating integrals numerically using volume-explicit equations of state that can directly solve for the compressibility factor Z as a function of pressure P at constant temperature T. - Deriving residual properties from virial equations of state by substituting the two-term virial equation into the residual property equations. - Deriving residual properties from cubic equations of state by first recasting the generic cubic equation into a form that yields Z as a function of density at constant T, then substituting the resulting expressions into the residual property equations.