At a small distance r from a long cylindrical light source, the...

Intensity of light is defined as the power transmitted per unit area.

For a long cylindrical light source of length $L$, the light spreads out uniformly over a cylindrical surface of radius $r$ coaxial with the source.

The surface area of this imaginary cylinder is $A = 2\pi r L$.

If $P$ is the power emitted by the cylindrical source of length $L$, the intensity $I$ at a distance $r$ is given by:

$I = \dfrac{P}{A} = \dfrac{P}{2\pi r L}$

Since $P$ and $L$ are constant for a given source, we obtain:

$I \propto \dfrac{1}{r}$