Answer
$$
\text { The length }
=\int_{0}^{\pi/2} \sqrt{2e^{2\theta}+2e^\theta +1} d \theta.
$$
Work Step by Step
Since $r=f(\theta)=e^\theta+1$, then $f'(\theta)=e^\theta$
The length is given by
$$
\text { The length }=\int_{0}^{\pi/2} \sqrt{f(\theta)^{2}+f^{\prime}(\theta)^{2}} d \theta\\
=\int_{0}^{\pi/2} \sqrt{2e^{2\theta}+2e^\theta +1} d \theta.
$$
