How the tackle this limited integral?

Question

I started with use of a new variable for the things under square root.

I would like to calculate the integral $$\iint_\Omega\sqrt{\sqrt{x}+\sqrt{y}}~\mathrm{d}x~\mathrm{d}y$$ over the respected area $$\Omega=\{(x,y)\in\mathbb R^2;x\geq 0,y\geq 0,\sqrt x+\sqrt y\leq 1\}$$

Answer 1

HINT:

Transform variables from $(x,y)$ to $(u,v)$ where $x=u^2$ and $y=v^2$. The Jacobian of transformation is $4uv$ so that

$$\int_0^1\int_0^{\left(1-\sqrt{y}\right)^2}\sqrt{\sqrt{x}+\sqrt{y}}\,dx\,dy=4\int_0^1\,v\,\left(\int_0^{1-v}\,u\,\sqrt{u+v}\,du\right)\,dv$$