### Double Integral Transformation (Parallelogram Region) Example 1

Find $$\int\limits_{}^{} {\int\limits_R^{} {(4x + 8y)dA} }$$, where R is the parallelogram with vertices (-1,3), (1,-3), (3,-1) and (1,5), given the transformation $$x = \frac{{(u + v)}}{4}$$ and $$y = \frac{{(v - 3u)}}{4}$$.

The first thing to do is make equations for the lines that connect the vertices of the parallelogram, so just find those slopes and intercepts. You will get the following 4 equations:$\begin{array}{l}y = 8 - 3x\\y = x + 4\\y = - 3x\\y = x - 4\end{array}$

Then, substitute the transformation equations into each of the four equations above. For example, we will do this for the first equation above.$\begin{array}{l}\frac{{v - 3u}}{4} = 8 - \frac{{3(u + v)}}{4}\\v = 8\end{array}$

Somehow, the u variable cancels out and we obtained that v = 8. You need to do this for the other three line equations and you'll acquire u = -4, v = 0 and u = 4. These serve as the boundaries for the double integral we will be creating.

The next step is to find the Jacobian. I will let "J" represent this.

$\begin{array}{l}J = \left| {\frac{{\partial x}}{{\partial u}} \cdot \frac{{\partial y}}{{\partial v}} - \frac{{\partial x}}{{\partial v}} \cdot \frac{{\partial y}}{{\partial u}}} \right|\\J = \left( {\frac{1}{4}} \right) \cdot \left( {\frac{1}{4}} \right) - \left( {\frac{1}{4}} \right) \cdot \left( { - \frac{3}{4}} \right)\\J = \frac{1}{4}\end{array}$

And we just substitute all this information in to make a double integral to solve. We can take the Jacobian factor out since it's not a variable, it's a constant. Let's call the integral "I".

$\begin{array}{l}I = \frac{1}{4}\int_0^8 {\int_{ - 4}^4 {\left( {3v - 5u} \right)dudv} } \\I = \frac{1}{4}\int_0^8 {\left( 3 \right.(4)v - \frac{5}{2}} \left. {{{(4)}^2}} \right) - \left( {3( - 4)v - \frac{5}{2}{{( - 4)}^2}} \right)dv\\I = \frac{1}{4}\int_0^8 {24vdv} \\I = \frac{1}{4} \cdot \frac{{24}}{2}\left( {{8^2} - {0^2}} \right)\\I = 192\end{array}$

Please see other double integral transformation examples below:

Double Integral Transformation (Ellipse Region) Example 2

Double Integral Transformation (Triangle Region) Example 3