Find the volume of this region!

**Bruno J.** · June 20th 2009, 10:45 AM

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

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

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

**Moo** · June 20th 2009, 11:02 AM

Hello,

Spoiler:

**HallsofIvy** · June 21st 2009, 07:31 AM

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

**Jose27** · June 23rd 2009, 01:24 PM

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

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