# Evaluate the integral \iint_R e^{x-2y} dA where R is the parallelogram ABCD with vertices...

## Question:

Evaluate the integral

{eq}\iint_R e^{x-2y} dA {/eq}

where R is the parallelogram ABCD with vertices A=(0,0),B=(4,1),C=(7,4),and D=(3,3) using the transformation x= 4u+3v Andy= u +3v.

## Double Integral:

We will change the integral where we set up the square matrix of second-order and then find the Jacobian and then times the integral by it and then using the standard result to solve the integral.

To solve the problem we will use the change of variable:

{eq}\int \int e^{x-2y}dxdy {/eq}

Now let us find the Jacobian:

{eq}\begin{bmatrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}\\ =\begin{bmatrix} 4 &3 \\ 1 & 3 \end{bmatrix}\\ =9\\ u=\frac{x-y}{3}\\ v=\frac{-x+4y}{9} {/eq}

The integral will be:

{eq}=9\int_{0}^{1}\int_{0}^{1}e^{2u}e^{-3v}dudv {/eq}

Integrating we get:

{eq}=\frac{9}{2}\int_{0}^{1}e^{2u}dv\\ =\frac{-3}{2}(e^{-3}-1)(e^{2}-1) {/eq}