Rewrite the equations of the given boundary lines:
y = -x + 1 ==> x + y = 1
y = -x + 4 ==> x + y = 4
y = 2x + 2 ==> -2x + y = 2
y = 2x + 5 ==> -2x + y = 5
This tells us the parallelogram in the x-y plane corresponds to the rectangle in the u-v plane with 1 ≤ u ≤ 4 and 2 ≤ v ≤ 5.
Compute the Jacobian determinant for this change of coordinates:
[tex]J=\begin{bmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{bmatrix}=\begin{bmatrix}1&1\\-2&1\end{bmatrix}\implies|\det J|=3[/tex]
Rewrite the integrand:
[tex]-3x+4y=-3\cdot\dfrac{u-v}3+4\cdot\dfrac{2u+v}3=\dfrac{5u+7v}3[/tex]
The integral is then
[tex]\displaystyle\iint_R(-3x+4y)\,\mathrm dx\,\mathrm dy=3\iint_{R'}\frac{5u+7v}3\,\mathrm du\,\mathrm dv=\int_2^5\int_1^45u+7v\,\mathrm du\,\mathrm dv=\boxed{333}[/tex]
