Activity 12.9.6

Consider the solid $S'$ defined by the inequalities $0 \leq x \leq 2$, $\frac{x}{2} \leq y \leq \frac{x}{2}+1$, and $0 \leq z \leq 6$. Consider the transformation defined by $s = \frac{x}{2}$, $t = \frac{x-2y}{2}$, and $u = \frac{z}{3}$. Let $f(x,y,x) = x-2y+z$.

1. The transformation turns the solid $S'$ in $xyz$-coordinates into a box $S$ in $stu$-coordinates. Apply the transformation to the boundries of the solid $S'$ to find $stu$-coordinate descriptions of the box $S$.

2. Compute and simplify the Jacobian $\frac{\partial(x,y,z)}{\partial(s,t,u)}$.

3. Use the transformation to perform a change of variables and evaluate $\iiint_{S'} f(x,y,z) \, dV$ by evaluating $$\begin{gathered} \iiint_{S} f(x(s,t,u),y(s,t,u),z(s,t,u)) \ \left| \frac{\partial(x,y,z)}{\partial(s,t,u)} \right| \, ds \, dt \, du \end{gathered}$$