Let $R= (x, y)\in R^2: x≥ 0, y ≥0, 1\le x+y\le 2$ Find a transformation that maps a rectangular region $D$ in the $uv$-plane onto the region $R$, and use it to evaluate $\iint_R \frac{y}{x + y} dA $.
This was asked in one of my exams (the exam is over) but I was unable to find a rectangular region $$D in the $uv$-plane which maps to region $R$.
I have tried to use the substitution $u=x+y$ and $v=y$ and got a trapezium. Other transformations which I have tried does not work.
Does there exist such a transformation? Any help will be appreciated.
Best Answer
Try $u=\frac{y}{x+y} \, , \, v=x+y$, so that $y=uv \, , \, x=v(1-u)$.
Then, $1\le x+y\le 2\,,\ x\ge 0,y\ge 0 \implies 1\le v\le 2 \,,\ 0 \le u \le 1$.
Absolute value of jacobian is $v$, which gives you
$$\iint_R \frac{y}{x+y}dx\,dy=\int_0^1 u\,du\int_1^2 v\,dv=\frac34$$