Polar coordinate approximation

Question

This is the note:

I don't understand why the average $r$ can be replaced by $r_i$, I mean, if $r_1$ and $r_2$ are very close therefore $(r_1+r_2)/2$ can be considered as $r_i$, then also we can consider $\Delta r_i$ is $0$, how could it choose to do something on bias?

Answer 1

Note that

$$r_2=r_1+\Delta r \implies \frac12(r_1+r_2)=r_1+\frac12 \Delta r$$

Therefore, the area $A$ can be written

$$\begin{align} A&=\left(r_1+\frac12 \Delta r\right)\,\Delta r\,\Delta \theta\\\\ &=r_1\,\Delta r\,\Delta\theta+\frac12 (\Delta r)^2\,\Delta \theta \end{align}$$

Retaining terms of order $\Delta r\,\Delta \theta$ only and denoting $r=r_1$, we obtain the coveted answer.