All I'm trying to do is find the CDF of $f_{X+Y}(z)=\begin{cases}\frac z2 & 0\le z\le1\\ \frac12 & 1\le z\le2\\ \frac{(3-z)}{2} & 2\le z\le 3\\ 0 & \text{elsewhere} \end{cases}$
But I cant seem to figure it out. I know what the answers are but for whatever reason I cant get them. I thought to get them we simply do: $\int_0^z\frac x2dx$ Which I know is right, but then I go to the next one and tried a variety of different bounds and I couldn't get the right answer. So if someone could explain in general the process of finding CDF's and point out where I was going wrong that would be great.