${\left(\frac{\mathrm{dx}}{d\mathrm{\θ}}\right)}^2\+{\left(\frac{\mathrm{dy}}{d\mathrm{\θ}}\right)}^2$

$\={\left(r\prime \cos \left(\mathrm{\θ}\right)-r \sin \left(\mathrm{\θ}\right)\right)}^2\+{\left(r\prime \sin \left(\mathrm{\θ}\right)\+r \cos \left(\mathrm{\θ}\right)\right)}^2$

$$\begin{gathered} \= r{\prime }^2{\cos }^2\left(\mathrm{\θ}\right)-2 \mathrm{rr}\prime \cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+r^2{\sin }^2\left(\mathrm{\θ}\right) \\ \+r{\prime }^2{\sin }^2\left(\mathrm{\θ}\right)\+2 \mathrm{rr}\prime \cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+r^2{\cos }^2\left(\mathrm{\θ}\right) \end{gathered}$$

$\=r{\prime }^2\left({\cos }^2\left(\mathrm{\θ}\right)\+{\sin }^2\left(\mathrm{\θ}\right)\right)\+0\+r^2\left({\cos }^2\left(\mathrm{\θ}\right)\+{\sin }^2\left(\mathrm{\θ}\right)\right)$

$\=r{\prime }^2\+r^2$

Define the parametric equations $x\=x\left(\mathrm{\θ}\right)$ and $y\=y\left(\mathrm{\θ}\right)$ 

$x\left(\mathrm{\theta }\right)\=r\left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right)$$\stackrel{\text{assign as function}}{\to }$$x$

$y\left(\mathrm{\theta }\right)\=r\left(\mathrm{\theta }\right)\sin \left(\mathrm{\theta }\right)$$\stackrel{\text{assign as function}}{\to }$$y$

Obtain the radicand of the integrand for the arc-length integral

$x\prime {\left(\mathrm{\theta }\right)}^2\+y\prime {\left(\mathrm{\theta }\right)}^2$

${\left(\left(\frac{\ⅆ}{\ⅆ\mathrm{\θ}}r\left(\mathrm{\θ}\right)\right)\cos \left(\mathrm{\θ}\right)-r\left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\right)}^2\+{\left(\left(\frac{\ⅆ}{\ⅆ\mathrm{\θ}}r\left(\mathrm{\θ}\right)\right)\sin \left(\mathrm{\θ}\right)\+r\left(\mathrm{\θ}\right)\cos \left(\mathrm{\θ}\right)\right)}^2$

$\stackrel{\text{simplify}}{\=}$

${\left(\frac{\ⅆ}{\ⅆ\mathrm{\θ}}r\left(\mathrm{\θ}\right)\right)}^2\+{r\left(\mathrm{\θ}\right)}^2$