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.

  • $\begingroup$ Calculate norm of det of the Jacobian in order that you know how the measure transforms and then just substitute it in... $\endgroup$ – Fabian Apr 20 '11 at 7:08

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.

  • $\begingroup$ i think i understand what you were trying to say in above comment. $\endgroup$ – Santosh Linkha Apr 20 '11 at 7:54
  • $\begingroup$ It's sort of exactly what one might expect naively, except with this Jacobian tossed in as well. $\endgroup$ – davidlowryduda Apr 20 '11 at 8:11
  • $\begingroup$ 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. $\endgroup$ – J. M. is a poor mathematician Apr 20 '11 at 8:45
  • $\begingroup$ @mixedmath You mean the determinant of the jacobian matrix? $\endgroup$ – user38268 Apr 20 '11 at 13:15
  • $\begingroup$ 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 tinyurl.com/3cy2y4r . $\endgroup$ – user7814 Apr 20 '11 at 15:57

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