Answer
$\dfrac{\pi+3}{8}$
Work Step by Step
The length of the curve is given as: $L=\int_{p}^{q}\sqrt{r^2+(\dfrac{dr}{d\theta})^2}d\theta$
Thus, $L=\int_{0}^{\pi/4} \sqrt{cos^6 (\theta/3)+cos^4 (\theta/3) \sin^2 (\theta/3)} d \theta$
Then, we have $L=\int_{0}^{\pi/4} cos^2 (\theta/3) \theta$
or, $L=(1/2) [\theta+(3/2)sin(2\theta/3)]_{0}^{\pi/4}$
Thus, $L=\dfrac{\pi+3}{8}$
