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Activity 12.8.6 .
Consider the solid
\(S'\) defined by the inequalities
\(0 \leq x \leq 2\text{,}\) \(\frac{x}{2} \leq y \leq \frac{x}{2}+1\text{,}\) and
\(0 \leq z \leq 6\text{.}\) Consider the transformation defined by
\(s = \frac{x}{2}\text{,}\) \(t = \frac{x-2y}{2}\text{,}\) and
\(u = \frac{z}{3}\text{.}\) Let
\(f(x,y,x) = x-2y+z\text{.}\)
(a) 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
\(S\text{.}\)
(b) Compute and simplify the Jacobian
\(\frac{\partial(x,y,z)}{\partial(s,t,u)}\text{.}\)
(c)
Use the transformation to perform a change of variables and evaluate \(\iiint_{S'} f(x,y,z) \, dV\) by evaluating
\begin{equation*}
\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{equation*}
