Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.4 Area and Arc Length in Polar - Exercises - Page 625: 28

$\sqrt 2 (e^{2\pi}-1).$

Since $r=f(\theta)=e^\theta$, then $f'(\theta)=e^\theta$. Thus, the length is given by \begin{align*} \text { Length }&=\int_{0}^{2\pi} \sqrt{f(\theta)^{2}+f^{\prime}(\theta)^{2}} d \theta\\ &=\int_{0}^{2\pi}\sqrt{e^{2\theta}+e^{2\theta}} d \theta\\ &=\int_{0}^{2\pi}\sqrt 2 e^\theta d \theta\\ &=\sqrt 2 e^\theta |_{0}^{2\pi}\\ &=\sqrt 2 (e^{2\pi}-1). \end{align*}