Answer
$\sqrt 2 (e^{2\pi}-1).$
Work Step by Step
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*}