In polar coordinates with $r$ as radius and $\theta$ as azimuth, the arc length between the points corresponding to $\theta =a$ and $\theta =b$ is given by $$\int_a^b \sqrt{r^2(\theta )+r'^2(\theta )}\, d\theta.$$ if $0\lt b-a\lt 2\pi$.
On the other hand, the area enclosed by $r(\theta )$ and the rays $\theta =a$ and $\theta =b$ is $$\frac{1}{2}\int_a^b r^2(\theta )\, d\theta$$ if $0\lt b-a\le 2\pi$.
In Cartesian coordinates $(x,y)$, $$\int_a^b y(x)\, dx$$ represents the familiar area under a curve from $x=a$ to $x=b$.
But what is the geometric representation of $$\int_a^b r(\theta )\, d\theta$$ in polar coordinates?