## Calculus 8th Edition

$-3$
The region $R$ in the $uv$ plane is defined as follows: $R=${$(u,v) | 0 \leq v \leq 1-u, 0\leq u \leq 1$} $J(u,v)=\begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{vmatrix}=\begin{vmatrix} 2&1\\1&2\end{vmatrix}=4-1=3$ $\iint_R (x-3y) dA=\int_0^1 [\int_0^{1-u} (2u+v) -3(u+2v)] (3) dv du$ or, $=3 \int_0^1 \int_0^{1-u} -u-5v dv du$ or, $=3 \int_0^1 [-uv-\dfrac{5v^2}{2}]_0^{1-u} du$ or, $=3 \int_0^1 -u(1-u)-\dfrac{5}{2} (u^2-2u+1) du$ or, $3 [-\dfrac{5}{2}u-\dfrac{1}{2}u^3+2u^2]_0^1=-3$