What is the volume of a hyper cylinder in d - dimension?

Question

Is it $L \times S^{d-1}$ where $S$ is the hyper sphere of $d-1$ dimension and $L$ is length in usual 1 dimension?

Answer 1

The volume of the Cartesian product of two sets is the product of volumes. The familiar solid cylinder is the Cartesian product of a two-dimensional ball $B^2$ and a line segment $L$. Consequently, its volume is $|B^2|\cdot |L| = \pi r^2 h$.

In $n$ dimensions, one can define cylinders as products $B^{n-1}\times L$ and get the volume formula $|B^{n-1}|\cdot |L|$. See Wikipedia for the volume of $B^{n-1}$.