You could make a graph of the size of the smallest interval versus x and y. We know that the height of the graph is 0 along all the edges of the unit square where it is defined and on the line x=y. It achieves its maximum of 1/3 at (1/3,2/3) and (2/3,1/3). In between these positions the surface forms a series of planes. Thus the volume under it consists of 2 pyramids each with an altitude of 1/3 and an (isosceles triangular) base of area 1/2, yielding a total volume of 1/9.