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.

Answer and Explanation:

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}


Learn more about this topic:

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Double Integrals: Applications & Examples

from AP Calculus AB & BC: Help and Review

Chapter 12 / Lesson 14
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