Use the transformation x = u + v, y = u-v to evaluate \iint_R (x + 2y) \, dA , where R is...

Question:

Use the transformation {eq}x = u + v, y = u-v {/eq} to evaluate {eq}\iint_R (x + 2y) \, dA {/eq}, where {eq}R {/eq} is the square in the xy-plane with vertices at (0,0), (1,1), (2,0) and (1,-1)

Double Integral:

We will set up the integral in terms of u and v form and then we will find the Jacobian where we will find the partial derivatives of x and y with respect to u and v and then solve.

We will solve the problem using a variable change form:

{eq}x=u+v\\ y=u-v\\ u=\frac{x+y}{2}\\ v=\frac{x-y}{2} {/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} \frac{1}{2} &\frac{1}{2} \\ \frac{1}{2} & \frac{-1}{2} \end{bmatrix}\\ =\frac{-1}{2} {/eq}

The integral will be:

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