Volume of a particular $n$-dimensional region

Question

What is the $n$-dimensional volume of the region $\{ x \in \mathbb{R}^n|x_i \geq 0$ for all $i = 1,...,n$ and $x_1 + ... + x_n ≤ 1 \}$

I'm thinking of using a recurrence relation but I don't know how to start.

Answer 1

Look at the volume in 3D to see how to perform the integral:

$$V = \int\limits_{x_1=0}^1 d x_1 \int\limits_{x_2 =0}^{1-x_1} dx_2 \cdots \int\limits_{x_n = 0}^{1 - x_1 - x_2 - \cdots - x_{n-1}} dx_n$$