changing limit of this double integral $\int \int _R^{ }f\left(x,y\right)dA$ with region R bounded by { $y=3x$, $x=3y$, $x+y=4$}

Question

we are asked to use the substitution, $x=3u+v$ and $y=u+3v$. OK so this is what i did....first i plot the region R

I made a mistake, @Kavi Rama Murthy you are right, the area needs to be divided into 2 region, Region 1 can be $\ \int _0^1\ \int _{\frac{y}{3}}^{3y}\ f\left(x,y\right)dxdy\ $ and Region 2 being $\ \int _1^3\int _{\frac{y}{3}}^{4-y}\ f\left(x,y\right)dxdy$

Answer 1

The equations for the lines in terms of $u$ and $v$ are $u=0,v=0$ and $u+v=1$. So the limits are $0 < v < 1-u$ and $0<u<1$.