Your region of integration is bound by the triangle whose vertices are at (0,0), (1,1), and (1,0).

If x=0 & y=0,the (u, v) = (0, 0). If x=1 & y=1, then (u, v) = (1, 0). If x=1 & y=0, then u = 1/2 & v = 1/2 .

Integrate 0 ≤ u ≤ 1, 0 ≤ v ≤ (1/2) - |x - (1/2)|