Answer
$192$
Work Step by Step
Here, we have: $Jacobian =\begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{vmatrix}=\begin{vmatrix} \dfrac{1}{4} \cdot \dfrac{1}{4}-\dfrac{1}{4} \cdot \dfrac{-3}{4}\end{vmatrix}=\dfrac{1}{4}$
$\iint_R (4x+8y) dA=\int_0^8 \int_{-4}^{4} (4x+8y) dA$
$=\int_0^8 \int_{-4}^{4} [4 \dfrac{1}{4}(u+v)]+[8 \dfrac{1}{4}(v-3u)] du dv $
or, $\int_0^8 \int_{-4}^{4} [4 \dfrac{1}{4}(u+v)]+[8 \dfrac{1}{4}(v-3u)] du dv=\int_0^8 \int_{-4}^{4}(3v-5u) \cdot du dv$
Thus, $\iint_R (4x+8y) dA=\dfrac{1}{4} \int_0^8[3uv-2.5u^2]_{-4}^4 dv=\dfrac{1}{4} \int_0^8 24 v dv=\dfrac{1}{4} [12v^2]_0^8=192$
