\[-w = \frac{P_1 V_1 - P_2V_2}{\gamma - 1}\]

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\[-w = \frac{P_1 V_1 - P_2V_2}{\gamma - 1}\]

Mathematics
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Alright, so the P-V curve is given by \(PV^{\gamma} = \rm constant \). Fair enough.
this looks bit like thermodynamics, reverse adiabatic expansion?
Yes.

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\[w = - \int P dV = - {\rm constant }\int \limits _{V_1}^{V_2} \frac{dV}{V^{\gamma }} = -{\rm constant } \left[ \frac{V^{1 - \gamma }}{1- \gamma } \right]_{V_1 }^{V_2}\]\[= {\rm constant }\frac{V_2^{1-\gamma}-V_1^{1-\gamma }}{\gamma - 1}\]\[= \frac{PV^{\gamma} (V_2^{1-\gamma} - V_1^{1-\gamma})}{\gamma-1 }\]\[= \frac{P_2 V_2 - P_1 V_1}{\gamma - 1}\]Incredible, I did it!

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