Sketch the domain $\mathcal{D}$ bounded by $y=x^2$, $y=\frac12 x^2$, and $y=3 x$. Use a change of variables with the map $x= uv$, $y=u^2$ to calculate $$\int\!\!\int _{\mathcal{D}} y^{-1}\,dx\,dy$$ This is an improper integral since $f(x,y)=y^{-1}$ is undefined at $(0,0)$, but it becomes proper after changing variables.

$\int\!\!\int _{\mathcal{D}} y^{-1}\,dx\,dy=$