91Ó°ÊÓ

The arc length is the distance measured along the curved line that makes up the arc.

A polar curve's arc length is defined as follows.

Let $r=f\left(\mathrm{θ}\right)$is any differentiable function of $\mathrm{θ}$such that $f^{\text{'}}\left(\mathrm{θ}\right)$is continuous for all $\mathrm{θ}∈[\mathrm{Î\pm },\mathrm{β}]$

Moreover, consider that $r=f$is a one-one function from $[\mathrm{Î\pm },\mathrm{β}]$to the function graph. then in the interval $[\mathrm{Î\pm },\mathrm{β}]$the arc length of the polar graph,

$\text{Arc length}={∫}_{\mathrm{Î\pm }}^{\mathrm{β}}\sqrt{{\left(f^{\text{'}}\left(\mathrm{θ}\right)\right)}^2+\left(f\right(\mathrm{θ}){)}^2}d\mathrm{θ}$

To determine the length of the arc for the curve $r=f\left(\mathrm{θ}\right)$the function $f\left(\mathrm{θ}\right)$must be differentiable and one to one.

The derivative $f\left(\mathrm{θ}\right)$of that is $f^{\text{'}}\left(\mathrm{θ}\right)$is continuous for all $\mathrm{θ}∈[\mathrm{Î\pm },\mathrm{β}]$.

As a result, for this formula to work, $f\left(\mathrm{θ}\right)$must be differentiable, continuous, and one to one.

Therefore the answer is:

${∫}_a^{\mathrm{β}}\sqrt{{\left(f^{\text{'}}\left(\mathrm{θ}\right)\right)}^2+\left(f\right(\mathrm{θ}){)}^2}d\mathrm{θ}$