how to find Jordan-measure of this set? It is 
$ M= \left\{ \begin{pmatrix} -r +2 t -t\\ 3r+t \\2r+2t \end{pmatrix} : 1\leq r \leq 3, 0 <s<1, -1<t\leq 2 \right\} $
I tried to integrate this set, I don't know how to deal with $<$ .
I searched for examples, couldn't find anything helpful thought :/
Any help very appreciated !
 A: You could use a parametrization and the change of variables theorem. Denote by $\mathrm m$ the Jordan measure on $\mathbb R^3$. Consider the function $\unicode{x1D7D9}:\mathbb R^3 \to \mathbb R$ constant and equal to $1$. Then for any Jordan-measurable set $E$,
$
\mathrm m(E) = \displaystyle\int_E \unicode{x1D7D9} \mathrm {dm}
$.
Define $\Phi:(1,3)\times(0,1)\times(-1,2) \to \mathbb R^3$ by $$\Phi(r,s,t)=(-r+2s-t, 3r+t, 2r+2t).$$
We have that $\mbox{Im}\Phi \subset M$ and $\mathrm m(\mbox{Im}\Phi) =\mathrm  m(M)$. Moreover,$$\mbox{d}\Phi(r,s,t)=\left[\begin{array}{ccc}-1 & 2 & -1 \\ 3 & 0 & 1 \\ 2 & 0 & 2\end{array}\right],\ \ \forall (r,s,t)\in (1,3)\times(0,1)\times(-1,2),$$
then 
$$\det\left(\mbox{d}\Phi(r,s,t)\right)=-8\neq0,\ \ \forall (r,s,t)\in (1,3)\times(0,1)\times(-1,2),$$
so it is a diffeomorphism. Applying the theorem of change of variable,
$$
\begin{array}{rl}\displaystyle
\mathrm {m}(M) & = \mathrm{m}(\mbox{Im}\Phi)=\displaystyle\int_{\mbox{Im}\Phi} \unicode{x1D7D9}\mathrm {dm} 
 = \int_{(1,3)\times(0,1)\times(-1,2)} \left(\unicode{x1D7D9}\circ\Phi\right)\left|\det\left(\mbox{d}\Phi(r,s,t)\right)\right|\mathrm {dm}\\
& =  \int_1^3\int_0^1\int_{-1}^2 8 \mathrm {dr}\mathrm {ds}\mathrm {dt} = 8(3-1)(1-0)(2-(-1))= 48.
\end{array}
$$
