# Thread: Find the volume of this region!

1. ## Find the volume of this region!

Suppose we are working in $\displaystyle \mathbb{R}^n$ and that we write $\displaystyle x \in \mathbb{R}^n$ in the form $\displaystyle x=(x_1,...,x_n)$. Find the volume of the region

$\displaystyle 0 < x_1 < x_2 <...<x_j<1$
$\displaystyle x_{k}<1 \: \: \mbox{for } j<k\leq n$

for any $\displaystyle 1 \leq j \leq n$.

2. Hello,
Spoiler:
I think it's called the simplex.
The volume for $\displaystyle 0 < x_1 < x_2 <...<s ~,~ s>0$ is $\displaystyle \frac{s^n}{n!}$

So here it's $\displaystyle \frac{1}{n!}$

You can think of it as the volume of $\displaystyle [0,s]^n$, and that it's just a matter of permuting the coordinates...

3. Moo is correct. That volume can be written as the repeated integral
$\displaystyle \int_{x_n= 0}^1\int_{x_{n-1}= 0}^{x_n}\int_{x_{n-2}= 0}^{x_{n-1}+ x_n}\cdot\cdot\cdot\int_{x_1= 0}^{x_2+ x_3+\cdot\cdot\cdot x_n} dx_1dx_2\cdot\cdot\cdot dx_{n-1}dx_n$
and Moo's answer can be proved by induction on n.

4. I don't think it's the simplex, since, for example, in $\displaystyle \mathbb{R} ^2$ if $\displaystyle j=1$ then the region $\displaystyle R= \{(x_1,x_2) \in \mathbb{R} ^2 : 0<x_1 <1, x_2 <1 \}$ is unbounded (actually it is an open strip cut at $\displaystyle x_2=1$) and it's area is not finite (Acutally I believe it's only finite if $\displaystyle j=n$).
On another note, if I recall correctly a simplex is a set of the form $\displaystyle S(a_1,...,a_n)= \{ x=(x_1,...,x_n) \in \mathbb{R} ^n : x= \sum_{i=1}^n \ {b_i a_i}$ where $\displaystyle b_i \in [0,1]$ and $\displaystyle 1= \sum_{i=1}^n \ b_i \}$ where $\displaystyle a_i \in \mathbb{R} ^n$