When gas expands in a cylinder with radius $r$, the pressure at any given time is a function of the volume: $P = P(V)$. The force exerted by the gas on the piston is the product of the pressure and the area: $F=\pi r^2 P.$ One can then show that the work done by the gas when the volume expands from volume $V_1$ to volume $V_2$ is $$W=\int_{V_1}^{\,V_2} {P}\,dV$$

In a steam engine the pressure $P$ and volume $V$ of steam satisfy the equation $PV^{1.4}=k,$ where $k$ is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Use the information given above to calculate the work done (in ft-lb) by the engine during a cycle when the steam starts at a pressure of $160\, lb/in^2$ and a volume of $100\, in^3$ and expands to a volume of $800\, in^3.$

Work done = ft-lb