I'm trying to evaluate $\iiint_E(3 z-x-y) d A(x, y)$, where $E$ is the parallelepiped with vertices (0, 0, 0), (2, 1, 1), (3, 3, 2), (1, 2, 1), (2, 3, 3), (1, 1, 2), (3, 2, 3), (4,4,4).
I suspect that I should employ the Jacobian to make a change of variables from $x,y,z$ to $u,v,w$, where the new set of variables will equal the equations of the three set of planes that make up the 6 faces of the parallelepiped. To find the equations of the planes, I can create vectors using the given points and find the cross product. As for the scale factor, I presume I would calculate the 3D determinant $$\begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{bmatrix}$$
However, I'm not sure if the above reasoning is correct and if so, how I can express the integrand $3z-x-y$ in terms of $u,v,w$. I would greatly appreciate any suggestions.