Finding solution to a double integral Well, here it is
$$\int\int_{[0,1]^{2}}\sqrt{1+x^2+y^2}\mathrm dx\mathrm dy\tag*{}$$
Maybe there's a really nice way of doing it, hopefully someone knows. Good luck!

If you use sostitution $\sqrt{x^2+y^2+1}=x+t$ squaring $x^2+y^2+1=x^2+2xt+t^2\iff x=\frac{y^2-t^2+1}{2t}$ and $dx=-\frac{y^2+t^2+1}{2t^2}dt $
So you have $$\int_0^1 \sqrt{x^2+y^2+1}dx=-\int_{\sqrt{y^2+1}}^{\sqrt{y^2+2}-1} \left(t+\frac{y^2-t^2+1}{2t}\right)\frac{y^2+t^2+1}{2t^2}dt$$ that is an easy fraction integral! Then solving the y-integral maybe you have to use another sostitution. 
