Finding a volume of a region defined by | x − y + z | + | y − z + x | + | z − x + y | = 1.I'm having trouble approaching this problem. Could someone maybe give me a hint or a solution, it would be so helpful.

Question

Finding a volume of a region defined by       |    x  −  y  +  z      |    +      |    y  −  z  +  x      |    +      |    z  −  x  +  y      |    =  1.I&#039;m having trouble approaching this problem. Could someone maybe give me a hint or a solution, it would be so helpful.

Malcolm Mcbride · Accepted Answer

Step 1Let&#039;s find the volume of the related region       |    x      |    +      |    y      |    +      |    z      |    ≤  1.By the symmetry of the region we can reduce it to an integral in the first octant only      ∭    E    d  V  =  8      ∭          E            ∩            First Octant        d  VStep 2Then setting up and doing the integral is not that hard  =  8      ∫    0    1        ∫    0          1      −      x            ∫    0          1      −      x      −      y        d  z    d  y    d  x  =  8      ∫    0    1        1    2    −  x  +      1    2        x    2      d  x  =      4    3  Now how does this help us with this problem? We can use the substitution      {                            u          =          x          −          y          +          z                                      v          =          x          +          y          −          z                                      w          =          −          x          +          y          +          z                            ⟹        J          −      1        =      |                            1                          −          1                          1                                      1                          1                          −          1                                      −          1                          1                          1                      |    =  4Thus with this change of variables we get that      ∭                  |            x      −      y      +      z              |            +              |            y      −      z      +      x              |            +              |            z      −      x      +      y              |            ≤      1        d  V  =      ∭                  |            u              |            +              |            v              |            +              |            w              |            ≤      1            1    4      d      V    ′    =      1    3

Finding a volume of a region defined by |x-y+z|+|y-z+x|+|z-x+y|=1

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