Triple Integral using inequalties

[1up20x6]

Homework Statement

Evaluate \iiint z^2 \,dx\,dy\,dz over domain V, where V is the solid defined by
1 \leq x+y+3z \leq 20 \leq 2y-z \leq 3-1 \leq x+y \leq 1

Homework Equations

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.

 

[pasmith]

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