Find a function that describes $D$

Here, is an image of a region$D$ bounded by curves $y=x^3$, $9y=x^3$, $x=y^3$, $16x=y^3$. And I need to find a mapping that maps this equation to $(u, v)=F(x, y)$, where $F(D)$ of $D$ is a rectangle in terms of $uv$. So I need to figure out $uv$. But there is a mysterious connection between $xy$...Because according to this, $16x=y^3=x$, $9y=x^3=y$. So the only case is when $x=y=0$. But this is not an equation and I need to make a rectangle by finding $u, v$. So maybe $0 \leq u \leq 3$ and $0 \leq v \leq 4$ is the rectangle shape to map to $F(D)$?

Solution 1:

Region $D$ is described as $$ D=\{(x,y):x,y>0,y\le x^3\le 9y,x\le y^3\le 16x\} $$ or $$ D=\{(x,y):x,y>0,1\le \frac{x^3}{y}\le 9,1\le \frac{y^3}{x}\le 16\}. $$ Now defining $u=\frac{x^3}{y}$ and $v=\frac{y^3}{x}$, maps $D$ to rectangle $[1,9]\times [1,16]$. Any other transformations are performed easily.