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I want to solve the following double integral:

$$\int_0^{\infty}dx\int_{-\infty}^{\infty}dy\,f(x,y).$$

And for example I made a conversion to the polar coordinates, $x=r\cos{\theta}$ and $y=r\sin{\theta}$ and get a new integral where I plugged in new variables and multiplied the Integral with the Jacobian factor, in this case $r$. How could I find new integration limits in terms of $r$ and $\theta$ based on the integration limits in terms of $x$ and $y$?

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$$r:0 \to \infty \\\theta : - \pi/2 \to \pi/2$$

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  • $\begingroup$ thanks, Could you please explain how one can get those integration limits? $\endgroup$ – user113891 Feb 3 '14 at 12:57
  • $\begingroup$ Your region is the right half plane. You have to cover it using the new limits.For $r$, think of a line starting from the origin and enters your region and leave it. For $\theta$, this line should rotate to cover the region, the angle $\theta$ measured with positive x-axis in positive direction $\endgroup$ – Semsem Feb 3 '14 at 13:20
  • $\begingroup$ is this clear @user113891 $\endgroup$ – Semsem Feb 3 '14 at 17:55
  • $\begingroup$ thanks, I understood. The point is that has to be represented graphically, rather than that the integration limits has to be calculated by the lim, as I thought. $\endgroup$ – user113891 Feb 3 '14 at 22:03
  • $\begingroup$ So, do you accept it. Any if you face any problem just let me know $\endgroup$ – Semsem Feb 3 '14 at 22:44

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