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Use the transformation x = 2u+v, y = u +2v to evaluate the integral \iint_R (x - 3y) \, dA ,...

Question:

Use the transformation {eq}x = 2u+v, y = u +2v {/eq} to evaluate the integral {eq}\iint_R (x - 3y) \, dA {/eq}, where {eq}R {/eq} is the triangular region bounded by {eq}y = 2x, y = -2x, y = -x+3 {/eq}

Double Integral:

We will solve the problem by using the change of variable where we will replace x and y by u and v and then find the equation of the lines in the uv plane and then solve the integral.

Answer and Explanation:

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

{eq}x=2u+v\\ y=u+2v {/eq}

The points of intersection are (0,0), (1,2), (-3,6)

In uv plane the points will become (0,0), (1,1) and (-7,5)

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} 2 & 1\\ 1& 2 \end{bmatrix}\\ =3 {/eq}

The integral will be:

{eq}=-\int_{-7}^{1}\int_{\frac{-5u}{7}}^{u}(u+5v)dvdu\\ =-\int_{-7}^{1}(uv+\frac{5v^{2}}{2})dv\\ =-3\frac{144}{49}\int_{-7}^{1}u^{2}du\\ =\frac{-49536}{49} {/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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