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.

There are three main ways to describe a curve


*

*by a cartesian equation $y=f(x)$,

*by parametric equation(s) $x=x(t), \ y=y(t)$,

*by a polar equation $r=r(\theta)$. We are here in this third case with
$$r(\theta)=\frac{2}{\theta}+1$$
Each "way" has its own formulas for the computation of areas, lengths, etc. which IMHO have not to be established again each time one uses them :
In particular, the length formula (6) for polar equations is 
$$L=\int_{\theta_1}^{\theta_2}\sqrt{r^2+\left(\dfrac{dr}{dθ}\right)^2} d\theta$$
Which here becomes :
$$\int_{0.25}^{2 \pi}\frac{\sqrt{(\theta^2+2\theta)^2+4}}{\theta^2}d\theta$$
which has no simple expression... You must use numerical methods for getting an approximate result. I found $16.325196$ with Matlab, which looks good when seeing the curve below (the equivalent of a circle with perimeter $2 \pi R$,  $R\approx 1.5$ + a "line" from $(1.5, 2.3)$ to $(8.5,2)$). 

