Finding volume of convex polyhedron given verticesI am trying to compute the volume of the convex polyhedron with vertices (0,0,0), (1,0,0), (0,2,0), (0,0,3), and (10,10,10). I am supposed to use a triple integral but am struggling with how to set it up.

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Drew Patel · Accepted Answer

Step 1First we can find the equation for the plane that goes through   A  =  (  1  ,  0  ,  0  ),   B  =  (  0  ,  2  ,  0  ), and   C  =  (  0  ,  0  ,  3  ). It is:   x  +            y      2        +            z      3        =  1 or   6  x  +  3  y  +  2  z  −  6  =  0. Now let   O  =  (  0  ,  0  ,  0  ), and   D  =  (  10  ,  10  ,  10  ). We now find the volume       V    1   of the tetrahedron OABC.      V    1    =      1    6    ⋅  O  A  ⋅  O  B  ⋅  O  C  =                    1        ⋅        2        ⋅        3            6        =  1. Let   d  = distance from O to the plane ABC, and   S  = area of triangle ABC. Then   3      V    1    =  S  ⋅  d, and   d  =      |                      6        ⋅        0        +        3        ⋅        0        +        2        ⋅        0        −        6                              6          2                +                  3          2                +                  2          2                          |    =            6      7      . So   3  ⋅  1  =  S  ⋅            6      7      .Step 2Thus   S  =            7      2      . Next let k be the distance from D to the plane ABC. Then   k  =                    6        ⋅        10        +        3        ⋅        10        +        2        ⋅        10        −        6            7        =            104      7      . So let       V    2   be the volume of the tetrahedron ABCD. So       V    2    =                    S        ⋅        k            3        =            1      3        ⋅            7      2        ⋅            104      7        =            52      3      . So let V be the volume of the polyhedra OABCD, then   V  =      V    1    +      V    2    =  1  +            52      3        =            55      3      .

Compute the volume of the convex polyhedron with vertices (0,0,0), (1,0,0), (0,2,0), (0,0,3), and (10,10,10).

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