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Use the change of variables

$x = u \quad y = \frac{v}{u}$

to evaluate the double integral

$\iint \frac{x}{1+x^2y^2} \, \mathrm{dA}$

I would like some direction as to how to solve this. Thank you.

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    Calculate norm of det of t$h$e Jacobia$n$ i$n$ order that you know how the measure transforms and then just substitute it in...2011-04-20

1 Answers 1

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So you carry out the simple substitutions. The general change of basis theorem says:

$\iint_{\Omega} f(x,y) dx dy = \iint_{\Gamma} f[x(u,v), y(u,v)] \left| \dfrac{d(x,y)}{d(u,v)} \right| du dv$

Where I use $\dfrac{d(x,y)}{d(u,v)}$ to refer to the Jacobian matrix: $\begin{pmatrix}\partial x/\partial u&\partial x/\partial v\\\partial y/\partial u&\partial y/\partial v\end{pmatrix}$,

And I use $\Omega$ to refer to the original coordinate basis and $\Gamma$ to refer to the new one. So you need only to calculate this matrix, multiply it in, and integrate as normal.

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    Last one: `\iint` is good for double integrals. You're welcome. :)2011-04-20