Chapter 10 - Parametric Equations and Polar Coordinates - 10.4 Areas and Lengths in Polar Coordinates - 10.4 Exercises - Page 713: 45

$2\pi$

The exact length is: $$L=\int_{0}^{\pi}\sqrt{4\cos^{2}(\theta)+(-2\sin(\theta))^{2}}d\theta$$ $$L=\int_{0}^{\pi}\sqrt{4\cos^{2}(\theta)+4\sin^{2}(\theta)}d\theta$$ $$L=\int_{0}^{\pi}\sqrt{4}d\theta$$ $$L=\int_{0}^{\pi}2d\theta$$ $$L=[2\theta]_{0}^{\pi}=2\pi-2\cdot 0=2\pi$$