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Say I have an ideal gas that has a known $P_1$, $P_2$, $T_1$, and $T_2$ undergoing a reversible adiabatic process. I want to find the work done so I must use $PV = RT$ to get the change in $V$, so that's $$\frac {P_1V_1}{T_1} = \frac {P_2V_2}{T_2}$$, and also $$W = \int PdV = P(V_2 - V_1)$$.
My problem is, these are only 2 equations and 3 unknowns. How do I solve $V_1$, $V_2$, and $W$? Also, in the second equation, is the value of $P$ equal to $P_1$ or $P_2$?
You were onto it in the comments, so I might be late to offer anything new here. The pressure is irrelevant in this problem; it's a trick, I guess. A reversible adiabatic process is one in which there's no heat flow in or out of the gas, so all of the work done in the expansion/compression goes into the temperature change. Just calculate the change in energy like you were saying ($U_2-U_1$) and that's the work done (on it, not by it)! Good job figuring it out!
Also, as with all physics problems, make sure that the negative signs match your intuition. I can't always keep $-W$ and the like straight on paper, especially since I often mix up whether $W$ represents the work done by the gas or the work done on the gas. I'm pretty sure it's the former. So, if the gas expands, it does work, and $W>0$. If the gas contracts, work is done on it, so the work the gas does is negative.
