f(x,y) = e^(-y) , 0<=x<=y

I want to find the expected value of XY, so

E(XY) = DoubleIntegral[xye^(-y) dxdy]

What are the boundaries for this integration? My thinking is that

integrate x from 0 to y

integrate y from x to infinity

But my boundaries for y are incorrect. It should be

integrate y from 0 to infinity

Why?