Answer
$-3$
Work Step by Step
Here, we have: $Jacobian =\begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{vmatrix}=\begin{vmatrix} 2&1\\1&2\end{vmatrix}=4-1=3$
In the uv plane , the region can be defined as:
$R=${$(u,v) | 0 \leq v \leq 1-u, 0\leq u \leq 1$}
$\iint_R (x-3y) dA=\int_0^1 [\int_0^{1-u} (2u+v) -3(u+2v)] (3) dv du$
$=3 \int_0^1 \int_0^{1-u} -u-5v dv du$
or, $3 \int_0^1 [-uv-\dfrac{5v^2}{2}]_0^{1-u} du=3 \int_0^1 -u(1-u)-\dfrac{5}{2} (u^2-2u+1) du$
Thus, $\iint_R (x-3y) dA=3 [-\dfrac{5}{2}u-\dfrac{1}{2}u^3+2u^2]_0^1=-3$
