For finding the area under a polar curve, we divide the area into small sectors of circles, as shown in image.
Area of polar curve using sectors
Suppose I do the same for finding the arc length.
I divide the curve into small sectors of many circles.
Let dΩ be the small angle subtended by a sector.
Then using the formula of a circumference of a circle,
Circumference = (dΩ/2π)(2πr)= rdΩ
And so to find the total arc length between two angles a and b, we take the limit of the sum of the circumferences of the sectors as dΩ tends to 0, which in other words is the integral from a to b of r*dΩ.
But this method is wrong. Where am I wrong?
Taking $ \Omega= \theta $, a symbol more usually used as also in the ma.utaxas.edu link.
When you compute sector area $$ A= \int \frac12 \cdot r d\theta\cdot r=\int \frac{r^2}2 \cdot d\theta $$ there is a scope for change of $r$ with $\theta$ as $f(\theta)$
To find infinitesimal arc length you have to use Pythagoras thm at infinitesimal level aka differentials.
$$ ds^2= dr^2 + ( r\cdot d\theta)^2 $$
$$ s= \int \sqrt{dr^2 + ( r\cdot d\theta)^2} $$
providing a scope for change of $r$ with $\theta$ here also.