I have the following integral and with the following substitutions that I made:

$$\int\_{a}^{b}\int\_{c}^{d}xy\sqrt{x^2+y^2}\,dx\,dy$$

$u=x^2, v=y^2$

$du = 2xdx, dv = 2ydy$

Which led me to

$$\frac{1}{4}\int\_{a}^{b}\int\_{c}^{d}\sqrt{u+v}\,du\,dv$$

However this seems wrong to me. Don't I have to update $a,b,c,d$ now that I've made the substitution over $x$ and $y$? How do I know what to change them to? Do I square everything?