If SC313.

That approach is spot on, but you need to consider X/2 < Y in which the order of integration is 0 to Y/2 for X and 0 to 1 for Y.

Changing the order of integration gives 0 to 1/2 for X and 0 to 1 - 2X for Y.

Integrating this with respect to dy with limits (0 to 1-2x) first gives (1/2)*(1-2X)^2 and integrating this over dx with limits (0 to 1/2) gives 1/6.

Double check my calculations for the integration limits, but the approach you had to setup the PDF is right (and if I'm wrong about the integration please inform).