In a Steam Engine the Pressure and Volume of Steam $P V ^ { 1 / 4 } = k$

In a steam engine the pressure and volume of steam satisfy the equation $P V ^ { 1 / 4 } = k$ , 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.) Calculate the work done by the engine (in ft-lb) during a cycle when the steam starts at a pressure of $160 \mathrm { lb } / \mathrm { in } ^ { 2 }$ and a volume of $100 \mathrm { in } ^ { 3 }$ and expands to a volume of $1000 \mathrm { in } ^ { 3 }$ . Use the fact that the work done by the gas when the volume expands from $V _ { 1 }$ to volume $V _ { 2 }$ is
$$W = \int _ { V _ { 1 } } ^ { V _ { 2 } } P d V$$