Let $B(n,r)$ be the ball of radius $r$ in $\mathbb{R}^n$. Note that $volB(2,r)= \pi r^2$ and $volB(3,r)= \dfrac{4\pi r^3}{3}$. Show that $volB(4,r)= \dfrac{\pi^2 r^4}{2}$.
Hint: We know that $volB(4,r)= \int_{B(4,r)} dx~\wedge~dy~\wedge~dz~\wedge~dt$.Parametrize the ball $B(4,r)$ by
$$(x,y,z,t)=F(\sigma,\psi,\theta,\phi)=(\sigma\sin\psi\sin\phi\cos\theta,\sigma\sin\psi\sin\phi\sin\theta,\sigma\sin\psi\cos\phi,\sigma\cos\psi)$$
whit $(\sigma,\psi,\theta,\phi) \in ~[0,r] \times[0,\pi] \times [0,2\pi] \times [0, \pi]$. Use the stokes theorem.
Ps. Fixed $r>0$, we have $volB(n,r) \rightarrow 0$ when $n \rightarrow \infty$.