Answer
$\dfrac{\pi+3}{8}$
Work Step by Step
The length of the curve can be calculated as: $L=\int_{p}^{q}\sqrt{r^2+(\dfrac{dr}{d\theta})^2}d\theta$
Here, we have $L=\int_{0}^{\pi/4} \sqrt{cos^6 (\dfrac{\theta}{3})+cos^4 (\dfrac{\theta}{3}) \sin^2 (\dfrac{\theta}{3})} d \theta=\int_{0}^{\pi/4} \cos^2 (\dfrac{\theta}{3}) d\theta$
This implies that
$L=(\dfrac{1}{2}) [\theta+(\dfrac{3}{2})\sin(\dfrac{2\theta}{3})]_{0}^{\pi/4}=\dfrac{\pi+3}{8}$
