Limits while calculating area with integrals Let's derive the area of a circle of radius $R$ using integration. Here, we take a random distance $r$ from the center and take an infinitesimal increment $dr$ which gives us an infinitesimal ring.The area of that ring is $2\pi rdr$. Now we integrate these infinitesimals ring to get the exact area. The expression stands out to be $\int_{0}^{R} 2\pi rdr$.
I don't understand why we are taking the upper limit $R$. I mean we go a distance $r$ and then take an increment and calculate the ring's area. But if we take increment after reaching $R$,that makes no sense to me since we are now overcounting an extra ring outside of the circle. Please help clear my doubts.

The extra ring that you are counting is technically an infinitesimally thin ring. So, that doesn't contribute to the area.
Mathematically, you could have said something like

The area is
$$\lim_{h\to 0}\left(\int_{0}^R 2\pi r dr + h\right)=\int_{0}^R 2\pi r dr$$

Could I make myself clear?
