Let $D$ be the region that's bound by $y=x^2, y=2x^2, x=y^2, x=3y^2$. $D$ corresponds to the region $E$ where $u=\frac{x^2}{y}$ and $v=\frac{y^2}{x}$.
- Sketch $D$ and $E$
- Find the solution to the integral $$\iint_D xy\ dxdy$$
Okay, I'm pretty lost here. Obviously I know how to sketch D but I'm pretty clueless on how to sketch E.
I found the determinant of the Jacobian I got from $\frac{\partial (u,v)}{\partial (x,y}$ to be 3 but I don't really know how to use it here (I just know that you're suppose to put in the integral when changing variables). Some help would be appreciated since I can't figure out myself how to tackle this problem by my own.
HINT
Translate the curves. If $u = x^2/y$ and $v = y^2/x$, then $y=x^2$ translates to $u=1$, and $x=y^2$ to $v=1$. Can you translate the others?
UPDATE
You are almost correct in your comments, so the boundaries translate to a box $1 \le u \le 2$ and $1/3 \le v \le 1$, and the integrated fund is $xy=uv$, so the integral becomes $$ \iint_D xy dx dy = \int_{u=1}^{u=2} \int_{v=1/3}^{v=1} uv J(u,v) dudv, $$ where $J(u,v)$ denotes the Jacobian of the transformation. Can you find it and complete the problem?