I need help with this problem:

"The joint distribution of amount of pollutant emmited from a smokestack without cleaning device(Y_1) and a similar smokestack with a cleaning device (Y_2) is

f(y_1,y_2)= 1, 0<=(less or equal to)y_1<=2, 0<=y_2<=1, 2y_2<=y_1

0, elsewhere

U = Y_1 - Y_2

Find the probability density function for U"

So I need to find F_U(u)=P(U<=u)=P(Y_1-Y_2<=u)=P(Y_1<=u+Y_2)

I drew the attachment, but I don't know how to proceed form there. I don't understand how to choose the region U<=u or where u splits.

I know for u<0, F_U(u)=0 since Y_1-Y_2<0 has no solution and for u>2, F_U(u)=1