Gas in a Cylinder with Piston

$N$ molecules of an ideal monoatomic gas at initial volume $V_1$ and pressure $P_1$ expands in a cylinder quasi-statically and adiabatically (so that $dU + PdV = 0$) to volume $V_2$ and pressure $P_2$. All answers should be expressed in terms of these quantities, $k$ and $\gamma = C_p/C_v$. Calculate the work done two ways:

a) Integrate $dW = -C_v dT$ (and put the answer in a form with only the variables noted above). I did this in lecture for you.

b) Integrate $dW = P dV$ directly, using the equation of state satisfied by an adiabatically expanding gas. Show that the two answers are equal (I noted the equality in class and assigned this problem for homework).