Let $(X,Y)$ be a random vector with the density $f_{XY}(x,y)$ on the trapezoid $T$ with vertices $(1,0),(2,0),(2,2),(1,1)$.
Based on some note that I got if $Z=X+Y$ then the CDF of $Z$ should be:
For $1 \leq z_0 \leq 2$:
$$F_z(z_0)=\int_{1}^{z_o}\left(\int_{0}^{z_o-x}f_{xy}dy\right)dx$$
For $2 \leq z_0 \leq 4$:
$$F_z(z_0)=\int_{1}^{z_o/2}\left(\int_{0}^{x}f_{xy}dy\right)dx+\int_{z_0/2}^{2}\left(\int_{0}^{z_o-x}f_{xy}dy\right)dx$$
I can't understand the integration bounds of $F_z(z_0)$. Do You have any suggestions?