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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 the Jacobian in order that you know how the measure transforms and then just substitute it in...2011-04-20

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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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    i think i understand what you were trying to say in above comment.2011-04-20
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    It's sort of exactly what one might expect naively, except with this Jacobian tossed in as well.2011-04-20
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    Three $\TeX$ nits: 1. `\displaystyle` is superfluous if you have already enclosed your expressions in `$$(something)$$`; 2. `pmatrix` generates parenthesized arrays; 3. use `\partial` for partial derivatives, which are the entities involved in constructing the Jacobian.2011-04-20
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    @mixedmath You mean the determinant of the jacobian matrix?2011-04-20
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    Thank you. My subsequent question is how do I find the limits of integration considering a graph with the following equations: x=1, x=5, xy=1, xy=5. This is what it looks like http://tinyurl.com/3cy2y4r .2011-04-20
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    @J.M.: Thanks for that - I'm still not used to Texing on forums and stacks.2011-04-20
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    Last one: `\iint` is good for double integrals. You're welcome. :)2011-04-20