In the following p-V diagram the equation of state along the curved...

From the equation of state $(V-2)^2 = 4ap$, we get $p = \frac{(V-2)^2}{4a}$.
At V = 1: $p = \frac{1}{4a}$ and at V = 3: $p = \frac{1}{4a}$.
The curved path A→B→C follows the given equation, while C→A is a straight line at constant pressure.
Work along the curved path:
$W_{ABC} = \int_1^3 \frac{(V-2)^2}{4a} dV = \frac{1}{4a}\left[\frac{(V-2)^3}{3}\right]_1^3 = \frac{1}{4a} \cdot \frac{2}{3} = \frac{1}{6a}$.
Work along C→A at constant pressure:
$W_{CA} = \frac{1}{4a}(1-3) = -\frac{1}{2a}$.
Total work: $W_{total} = \frac{1}{6a} - \frac{1}{2a} = -\frac{1}{3a}$.