Need help understanding rain rate formulas from drop sizes
The first equation does not match up with the units. But first things first... What you're calculating when you calculate rain rate is an accumulation rate, which is volume flux through a plane (mass flux divided by density) divided by the area of that plane. For this reason, it doesn't matter whether it is computed over a square meter or a square kilometer -- since volume flux is proportional to the area, it cancels out

Also it doesn't really matter whether $N(D)$ has units of $mm^{-1} m^{-3}$ or $cm^{-1} m^{-3}$, since that first unit cancels out in integration. 

If the units are as you say they are in the first example, then the expression should read
$$R_1 = 10^{-6}\frac{\pi}{6} \int_0^\infty v(D)D^3N(D)dD$$
to get a result in $mm/s$.

Multiplying this by 3600 to get a result in $mm/hr$ yields the the second expression.