Is there any method of finding volume in a high dimensional cube between two planes?I want to know the volume of this region:In R n , ( x 1 , x 2 , . . . , x n ) 0 &lt; x i &lt; 1 for i = 1 , 2 , … , n (just a cube), and 0 &lt; ϵ 1 &lt; x 1 + x 2 + ⋯ + x n &lt; ϵ 2 &lt; n (two planes).Is there any method or formula to get this volume??

Question

Is there any method of finding volume in a high dimensional cube between two planes?I want to know the volume of this region:In             R        n  , (      x    1    ,      x    2    ,  .  .  .  ,      x    n  )  0  &amp;lt;      x    i    &amp;lt;  1 for   i  =  1  ,  2  ,  …  ,  n (just a cube), and  0  &amp;lt;      ϵ    1    &amp;lt;      x    1    +      x    2    +  ⋯  +      x    n    &amp;lt;      ϵ    2    &amp;lt;  n (two planes).Is there any method or formula to get this volume??

Kyle George · Accepted Answer

Step 1Let&#039;s assume, that hyperplanes are given by formulas      x    n    =      f    1    (      x    1    ,  ⋯  ,      x          n      −      1        )  =      a    1        x    1    +  ⋯  +      a          n      −      1            x          n      −      1            x    n    =      f    2    (      x    1    ,  ⋯  ,      x          n      −      1        )  =      b    1        x    1    +  ⋯  +      b          n      −      1            x          n      −      1      and we know that       f    1    (      x    1    ,  ⋯  ,      x          n      −      1        )  ⩽      f    2    (      x    1    ,  ⋯  ,      x          n      −      1        ).Step 2Then   V  =      ∫          0              1        ⋯      ∫          0              1                  (            f    2    (      x    1    ,  ⋯  ,      x          n      −      1        )  −      f    1    (      x    1    ,  ⋯  ,      x          n      −      1        )            )        d      x    1    ⋯  d      x          n      −      1      .If hyperplanes have intersection inside cube, then we should find it and divide integral in two parts.

Find the volume of this region: ln mathbb(R^n),(x_{1},x_{2}, ..., x_{n}). 0<x_{i}<1 for i=1,2,...,n, and 0<epsilon_{1}<x_{1}+x_{2}+cdots x_{n}<epsilon_{2]<n (two planes).

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