How to find the double integral limits of this problem after using Jacobian transformation?

Question

$$\int_{y=0}^{y=1} \int_{x=y}^{x=2-y} \dfrac{x+y}{x^2}e^{x+y}\text{d}x\text{d}y, \quad x+y = u, \frac yx = v$$

The $u, v$ limits in the guide answer is from $0$ to $2$ for $u$ and $0$ to $1$ for $v$. But I can't find how it is done.

Answer 1

Draw the domain in the $x-y$ plane. You can see that the limits for $x$ form some lines that happen to intersect where $x=y=1$. So the domain is the triangle between $(0,0), (1,1),(2,0)$. $u$ is along the direction perpendicular to the line from $(0,0)$ to $(1,1)$. You can split the domain into thin lines, parallel to the $(0,0)\to (1,1)$ line. Notice then that $u$ varies from $0+0=0$ (at the origin) to $1+1=2+0=2$ (the line in the triangle that is not going through the origin). In the perpendicular direction you are probably used to parametrize by $x-y$. But it's not necessary. A point on the perpendicular line can be parametrized by $u$ and any function $f$ of $x$ and $y$, as long as you can extract the original $x$ and $y$ from $u$ and $f$. In this particular case, the function chosen is $u=\frac yx$. Then we need to look at the limits for $v$. When $(x,y)$ is on the $(0,0)\to(2,0)$ line, you have $y=0$ so $u=0$. When $(x,y)$ is on the $(0,0)\to(1,1)$ line, then $y=x$ so $u=1$.