∫(xydy-y2dx)=0∫1xydy-0∫1y2dx, because c defines the limits for x and y as the boundaries of the unit square.
0∫1xydy is to be integrated between y=0 and y=1. In this interval x is always 1, so xdy=dy, and the integral becomes 0∫1ydy=[y2/2]01=½.
0∫1y2dx is to be integrated between x=0 and x=1. In this interval y is always 1, so y2dx=dx, and the integral becomes 0∫1dx=[x]01=1.
∫(xydy-y2dx)=½-1=-½.