I am having problems with determining the integration limits for a pdf using the Jacobian.
Let $X$ and $Y$ be two independent random variables with uniform distribution on $[0,1].$
Let $Z = X -Y.\;$ I need to determine the pdf of $Z.$
Since the Jacobian need a square matrix, I added a second variable $U = Y.$
Then $|J| = 1.$
The combined pdf of $(Z,U)$ is $1.4\;$ (Right?).
To determining, the pdf of $Z,$ I need to integrate on $du.$ How do I determine the limits for $u?$
Comment. Sometimes plotting a few well-chosen points can help to establish the region of integration.
Below I mindlessly chose several thousand $(X,Y)$-pairs at random and plotted them. The plot at left below suggests the support of your bivariate distribution $(Z,U).$ Thinking more carefully, can you say what four $(x,y)$-pairs get transformed to the corners of the parallelogram?
The histogram suggests the distribution of $Z$ after you have 'integrated out' $U$. (Do you see that you need to have two integrals--one for $z < 0$ and one for $z > 0?)$
I hope this helps you with exact computation of the distribution of $Z.$ Can you see why the smallest and largest values of $Z$ would be $-1$ and $+1,$ respectively? Can you argue on intuitive grounds that $Z$ would have a mode at $0?$