Finding the volume of two subsets of R 3 .Define two subsets of R 3 . A = { ( x , y , z ) ∈ R 3 : | x | + | y | + | z | ≤ 1 } B = { ( x , y , z ) ∈ R 3 : max { | x | , | y | , | z | } ≤ 1 }1. Find vol(B)-vol(A)2. Set up an integral which evaluates to vol(A)-vol(B)

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Finding the volume of two subsets of             R        3  .Define two subsets of             R        3  .  A  =  {  (  x  ,  y  ,  z  )  ∈            R        3    :      |    x      |    +      |    y      |    +      |    z      |    ≤  1  }  B  =  {  (  x  ,  y  ,  z  )  ∈            R        3    :  max      {          |        x          |        ,          |        y          |        ,          |        z          |        }    ≤  1  }1. Find vol(B)-vol(A)2. Set up an integral which evaluates to vol(A)-vol(B)

cdtortosadn · Accepted Answer

Step 1For the first, your figure is symmetrical over the octants, so you you&#039;re looking at eight times the volume of the simplex bounded by   x  ,  y  ,  z  ⩾  0,   x  +  y  +  z  =  1. You&#039;re correct that the second figure is a cube.Step 2As further help, if I wanted the area of the two dimensional simplex   x  ,  y  ⩾  0  ,  x  +  y  =  1I would set up   A  =      ∫    0    1        ∫    0          1      −      x        d  y  d  x

Define two subsets of mathbb{R}^3. A={(x, y, z) in mathbb{R}^3:|x|+|y|+|z| leq 1}. B={(x, y, z) in mathbb{R}^3:max{|x|, |y|, |z|} leq 1}. 1. Find vol(B)-vol(A). 2. Set up an integral which evaluates to vol(A)-vol(B).

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