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.
No. $$\int 0 = 0$$ no matter of the domain.