Problem $5$: $(10$ pts$)$ Consider a process during which the pressure and volume are linearly related, i$.$e$.$

$P=k_{1}v+k_2$

with $k_1$ and $k_2$ constant

$($a$)$ Show that if this process carries a fluid from an initial state defined by $(P_1,v_1)$ to a final state defined by $(P_2,v_2)$ then the work performed is equal to the following.

$w=\frac{1}{2}(P_1+P_2)(v_1-v_2)$

$($b$)$ If $1\mathrm{.}5$ kg of water undergoes this process beginning at $200$ kPa and $150C$ and ending at $600$ kPa and $350C,$ how much work was performed?