In cylindrical coordinates, why do we look at the distance r, which is the length from the origin to the center point of the "cylindrical cuboid's" projection onto the xy plane. In this case, it is to help define the base area $r\Delta\theta$, but I am trying to understand the specific reason for this decision? Why not use "$r - \Delta r /2$" or "$r + \Delta r/2$"?
I am aware that
$\Delta V$
which is defined to be
$ \Delta V = r\Delta \theta \Delta r \Delta z \ $
In the limit $\Delta r\to 0$ the two approaches are identical. When one integrates $r$, you write $$\lim_{\Delta r\to 0}\sum_{r_i=0}^{R-\Delta r}\Delta r$$ in one case, and $$\lim_{\Delta r\to 0}\sum_{r_i0+\Delta r/2}^{R-\Delta r/2}\Delta r$$ Both are equivalent to $$\int_0^R dr$$