I need to calculate a double integral over a region as follows:
$$ 0 ≤ x − y ≤ 1 $$ $$2 ≤ x + y ≤ 3 $$
But I'm not sure what this region looks like, I'm having some trouble understanding in general how to sketch bounds. The inqequalities confuse me and I don't know how to correctly move around the $x$ and $y$. Is there any general approach to think of these bounds in order to sketch them?
If you want to see what the region looks like, proceed as follows: Adding $y$ to both sides of the first set of inequalities gives $$y\le x\le 1+y,$$ and subtracting $y$ from sides of the second set gives $$2-y\le x\le 3-y.$$
Then it becomes apparent that the concerned region is defined by the system $y\le x, x\le 1+y, 2-y\le x$ and $x\le 3-y.$
These are all linear. I'll pick the first one to illustrate; the others are like it. If you sketch the line defined by $y=x,$ then the region $y\le x$ belongs below this line and includes the points on the line because of the equality. The region you want is the intersection of the four subregions defined by the four inequalities in the system.
You should be able to proceed now.