Volume of a certain set in $\mathbb{R}^n$

Question

For any dimension $n \ge 1$ and any real number $a \ge 0$

$$S(a)=\{x\in \mathbb{R}^n \mid 0 \le x_i \text{ for all } i \text{ and}\, x_1+...+x_n\le a\}$$

Find the volume of the set S(a).

I tried writing it as a multiple integral but it got pretty complicated.

Answer 1

First space: $$ H: a \ge x_1 + \dotsb + x_n = (x_1, \dotsc, x_n) \cdot (1, \dotsc, 1) $$ is the equation of a half space $H$ including the plane $E$ with normal vector $n = (1, \dotsc, 1)$ and distance $a / \sqrt{n}$ from the origin.

Second space: $$ A: x \ge 0 $$

E.g. for $n=3$ the intersection $H \cap A$ is the tetraeder with vertices $O$, $a \, e_1$, $a \, e_2$, $a \, e_3$.