There's a method of computing the area of a circle by dividing it in concentric rings with infinitesimal width. Let $R$ be the radius of the circle and $r$ be the radius of the rings. The area of the circle is
$$\int_0^R 2 \pi r \,\mathrm d r$$
My questions are:
I do not understand, though, how to justify the $2 \pi r \,\mathrm d r$ approximation for the area of each ring. Its actual area would be $$\pi (r + \mathrm d r)^2 - \pi r^2 = 2 \pi r \,\mathrm d r + \pi \left( \mathrm d r \right)^2$$ right? Could I use this more precise formula if I wanted to? How?
The area on the integral above looks more like the lateral area of a cylinder of height $\mathrm d r$, which is different from the actual area between two concentric circles. So why does that work?
Would the lateral area of a truncated cone (which seems to be the intermediate between the ring area and the cylinder lateral area) also work as an approximation?
Also, how do you come up with such an idea for an approximation that makes the calculation so beautifully simple (i.e. adopting the lateral area of a cylinder as the area of the rings)? It is considered a trivial integral, but there is a huge and mostly ignored step to be taken there.