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: 
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.
Hint:

I'm not sure to interpret correctly your question, but, for $r=e^{-\theta^2}$ we have:

$$
\int\_0^\infty r d\theta=\frac{\sqrt{\pi}}{2}
$$
and also

$$
\int\_0^\infty r^2 d\theta=\frac{1}{2}\sqrt{\frac{\pi}{2}}
$$