How to determine the new region after variable change

Question

Consider the double integral

$$\iint_\limits{E} e^\frac{y}{x+y}\, \, dx\, dy,$$ where $E$ is the region bounded by $x=0,y=0 $ and $x+y=1$.

Now if we change the variable by $x+y=u$ and $y=uv$,then how to determine $E'$, the new region over which the new integral is to be taken? It is clear that $J=u$, the Jacobian and hence the differential becomes $ue^v du dv$ but I cannot guess the general process to determine the ranges of $u$ and $v$. Can someone help me understand the general way?

Answer 1

Let consider the graphs for $u=x+y$ and $y=uv$ inside the given region, we can see that

• $0\le x+y\le 1 \implies 0\le u\le 1$

and for any fixed segment aligned with the line $u=x+y$

• $0\le y\le u \implies 0\le uv\le u \implies 0\le v\le 1$