Calculate the perimeter of a circle with a continuously increasing radius I have a circle (if that would even be the correct name for this shape), with a radius function equal to $R=\frac 2\theta + 1$, where $\theta$ is the angle in radians. The domain is between $\theta = 0.25$ and $\theta = 2\pi$.
How do I calculate the total outside perimeter of this shape? I tried to divide it into individual sectors, and find the arc length, but couldn't find a way due to the two radii (in each sector) being different.
The main question is what is a generalised method that could be used to calculate the perimeter of the shape, with an increasing radius (which increases with the angle relative to a starting direction)? The exact function isn't too important.

Since $x=r\cos\theta$ and $y=r\sin\theta$, using $\theta$ as the parameter, one have $x=(\frac 2 {\theta} +1)\cos \theta$ and $y=(\frac 2 {\theta} +1)\sin \theta$. Therefore $$ds=\sqrt{(\frac {dx} {d\theta})^2+(\frac {dy} {d\theta})^2}d\theta=\sqrt{(-2\theta^{-2}\cos\theta-(\frac 2 \theta+1)\sin\theta)^2+(-2\theta^{-2}\sin\theta+(\frac 2 {\theta}+1)\cos\theta)^2}d\theta$$.
Now evaluate the integral $$\int _{0.25} ^{2\pi} ds$$.
