Let $X$ be the region of $\mathbf{R}^3$ bounded by the hyperboloid $x^2+y^2-z^2=1$ and the planes $z=0$ and $z=1$. Calculate $\int_X z\,dx\,dy\,dz$.
I managed to solve this using cylindrical coordinates, but I would like to know if there is a way to do it using Fubini's theorem.
I tried to decompose $X$ as a $x$ or $y$-simple region (something like $-\sqrt{1+z^2-x^2}\leq y\leq \sqrt{1+z^2-x^2}$) and $-\sqrt{1-y^2}\leq x\leq \sqrt{1-y^2}$ and $z\in[0,1]$, and get something like $\int_0^1 \int_{-\sqrt{1+z^2-x^2}}^{\sqrt{1+z^2-x^2}} \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}z\,dx\,dy\,dz$, but this doesn't work because of the order of variables.
Could someone provide some help?