Triple Integral using inequalties

1. Oct 2, 2014

1up20x6

1. The problem statement, all variables and given/known data

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

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 $$-y-1 \leq x \leq 1-y$$$$\frac{z}{2} \leq y \leq \frac{3+z}{2}$$$$\frac{1-x-y}{3} \leq z \leq \frac{2-x-y}{3}$$ 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. Oct 2, 2014

pasmith

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