How to calculate volume in R5 Find the volume of $\Omega=\{(x,y,z,u,v):x^2+y^2+z^2+u^2+v^2)<=1\}$.
I have no idea what to do.

If you know the volume of a sphere, you can integrate like this using polar substitution:
$$\underset{x^2+y^2+z^2+u^2+v^2\leq1}{\int\!\!\!\int\!\!\!\int\!\!\!\int\!\!\!\int}\!\!\!\!\!\!1\ d(x,y,z,u,v)=\iint\limits_{u^2+v^2\leq1}\left(\ \iiint\limits_{x^2+y^2+z^2\leq1-u^2+v^2}\!\!\!\!\!\!\!\!\!\!\!1\ d(x,y,z)\right)d(u,v)=\\\iint\limits_{u^2+v^2\leq1}\!\!\frac43\pi\left(\sqrt{1-u^2-v^2}\right)^3\ d(u,v)=\iint\limits_{0\leq r\leq1\ \land\ 0\leq\varphi\leq2\pi}\!\!\!\!\!\!\!\!\!\!\frac43\pi\left(\sqrt{1-r^2}\right)^3\cdot r\ d(r,\varphi)=\\\frac23\pi\cdot2\pi\int\limits_{0\leq r\leq1}2r(1-r^2)^\frac32\ dr=\frac43\pi^2\left[-\frac25(1-r^2)^\frac52\right]_0^1=\frac43\pi^2\cdot\frac25=\frac8{15}\pi^2$$
