There are $C^{\infty}$ test functions in $L^{1}(0, \infty)$ that make the integral value of $\int_{0}^{\infty}\sinx/x\phi(x) dx$ range from 0 to $\infty$. What does narrowing these test functions down so that the integral falls in the interval $(\pi/2 -  \varepsilon, \pi/2 + \varepsilon)$ tell us distribution theorywise?

What is the transformation $T: R^2 \rightarrow R^2$ that sends the area bounded by the curve sinx/x for the given interval to say half the upper unit disk.