Difficult change of variables: region of integration

Question

I'm trying to compute the value of the double integral $$ \int_{0} ^1 \int_{0} ^y \frac{\,dx \,dy}{y + \sqrt{xy}} $$ I'm instructed to use the change of variables $(x,y)=F(u,v) := (v(1+u)u, v(1+u)/u)$. Substituting this in and computing the Jacobian determinant seems routine, but I'm at a loss as to transforming the region of integration under this transformation. I'm also asked to describe as best as possible the region $F^{-1} ((0,1)^2)$ in the $(u,v)$-plane.
Any suggestions are greatly appreciated!

Answer 1

By letting $x=y z, dx = y\,dz$ the integral is transformed into $$ \int_{0}^{1}\int_{0}^{1}\frac{y}{y+y\sqrt{z}}\,dz\,dx =\int_{0}^{1}\frac{dz}{1+\sqrt{z}}\stackrel{z\mapsto u^2}{=}2\int_{0}^{1}\frac{u\,du}{1+u}=2(1-\log 2).$$