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Triple Integral using inequalties

  1. Oct 2, 2014 #1
    1. The problem statement, all variables and given/known data

    Evaluate [itex]\iiint z^2 \,dx\,dy\,dz[/itex] over domain V, where V is the solid defined by
    [tex]1 \leq x+y+3z \leq 2[/tex][tex]0 \leq 2y-z \leq 3[/tex][tex]-1 \leq x+y \leq 1[/tex]

    2. Relevant equations

    3. The attempt at a solution

    I know how to do simple triple integrals, but all the variables in the inequalities are tripping me up. I tried fumbling with the inequalities to find [tex]-y-1 \leq x \leq 1-y [/tex][tex]\frac{z}{2} \leq y \leq \frac{3+z}{2}[/tex][tex]\frac{1-x-y}{3} \leq z \leq \frac{2-x-y}{3}[/tex] but quickly realized that if I just did that, my solution would have x and y variables in it. Basically, I'm not sure about what else my first step should be to fully isolate at least one of the variables.
     
  2. jcsd
  3. Oct 2, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    I would consider the substitutions [tex]
    u = x + y + 3z \\
    v = 2y - z \\
    w = x + y [/tex] which give you simple boundaries in the new variables.
     
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