# Arc length of $|cos\theta|$ from $\theta=\frac{\pi}{6}$ to $\theta=\frac{\pi}{3}$

An insect is moving along the curve $$r=|cos\theta|$$ such that $$\theta =\frac{\pi t}{6}$$, where $$t$$ is time measured in seconds. What is the distance travelled by the insect in the time interval between $$t=1$$ and $$t=2$$ ?

My attempt: The arc length is given by the formula : $$\int_{a}^{b} \sqrt{1+f'(x)^2}$$. So here in the given region, we have $$r=|\cos\theta|=cos\theta$$. Now, we have $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\sqrt{1+sin^2(x)}$$. This integration is difficult to carry out. I am unable to proceed further.

Any help is appreciated. Thanks in advance.

$$L=\int_{\alpha}^{\beta}\sqrt{r^2+\left(\frac{dr}{dt}\right)^2}d\theta$$.
$$L=\int_{\pi/6}^{\pi/3}\sqrt{cos^2(\theta)+sin^2(\theta)}d\theta=\theta|_{\pi/6}^{\pi/3}=\pi/6$$.