Question: Let $U\_1$ and $U\_2$ are iid continuous random variable over $(0,1)$ and $S\_1$ and $S\_2$ are iid continuous random variable over $(-\infty,+\infty)$. Then what is $$P(U\_2S\_2<U\_1S\_1)?$$

I think it is $1/2$, because $U\_1S\_1$ and $U\_2S\_2$ are identically distributed. My main confusion is: am I right in assuming $U\_1S\_1$ and $U\_2S\_2$ are identically distributed?

Answer: 
I understand what you're asking now. If you want to integrate a function $f\colon [0,\infty) \to [0,\infty)$, then you can write $\int\_{0}^{\pi/2} \frac12 r^2(\theta) d\theta$. You're asking if $\int\_{0}^{\infty} \frac12 r^2(\theta) d\theta=\infty$. Well unless $r=0$ then yes of course. You're adding up a positive number infinitely many times. You are correct.
If you want your integral to have the meaning of an area, then you should not integrate the polar function itself but its square:

$$\frac12\int\_{\phi=0}^{2\pi}r(\phi)^2d\phi$$

Of course, if you decide to evaluate that integral over a larger interval then you are running around the origin multiple times. For a closed curve and an integration interval that spans an integer multiple of $2\pi$ the integral would be the same integer multiple of the original area.