I want to compute $\int_\Bbb{R}\int_\Bbb{R} f(x/y)\frac{1}{2\pi} e^{-\frac{1}{2}(x^2+y^2)}dxdy$ where $f$ is a measurable bounded function.
I want to use the reparametrization $x=r\sin(t)$ $y=r\cos(t)$. Then I would get $$\int \int f(\tan(t))\frac{1}{2\pi} re^{-\frac{r^2}{2}}drdt$$ now I have some problems from where to where to integral boundarys goes. Could maybe someone help me how to proceed here?
Thanks a lot
This is not the answer but only a hint, but I am not allowed to comment.
In a surface integral you need to ensure that after the change of variables your new coordinates cover the same region that your old coordinates originally covered. Your integration region is the whole $\mathbb{R}^2$ plane. In Cartesian coordinates (your original coordinates), this region is covered when both $x$ and $y$ go from $-\infty$ to $\infty$. Your new coordinates are the polar coordinates ($r$ is the radius or distance from the origin and $t$ is the counter-clockwise angle with respect to the positive $y$ axis). Can you guess the range of these two variables such that the whole $\mathbb{R}^2$ plane is covered? If you have trouble picturing it, a good starting point for a more formal way may be to solve your change of variable for $r$ and $t$. Let me know if you need more help.