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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?

Ram Keswani
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1 Answers1

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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.

Narasimham
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