Answer
$\dfrac{19}{3}$
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 $L=\int_{0}^{\sqrt 5} \sqrt{(\theta^4+(2\theta))^2} d \theta=\dfrac{1}{2}\int_{0}^{\sqrt 5} \sqrt {\theta^2+4}(2\theta d\theta)$
Plug $\theta^2+4 =k $ or, $2\theta d\theta= dk$
$L =\dfrac{1}{2}\int_{4}^{9}(k^{1/2}) dk=(\dfrac{1}{2})[(\dfrac{2}{3}) (k^{3/2})]_{4}^{9} $
$\implies L=\dfrac{27-8}{3}=\dfrac{19}{3}$
