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Thread: Find the volume of this region!

  1. #1
    MHF Contributor Bruno J.'s Avatar
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    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 $\displaystyle 1 \leq j \leq n$.
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  2. #2
    Moo
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    A Cute Angle Moo's Avatar
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    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...
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  3. #3
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    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.
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  4. #4
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    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 $\displaystyle a_i \in \mathbb{R} ^n$
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