Answer
The exact length of the polar curve:
$$ r=2 \cos \theta \quad \text {for } \theta =0 \quad \text {to} \quad \theta= \pi $$
is equal to
$$
\begin{aligned} L &=\int_{a}^{b} \sqrt{r^{2}+(d r / d \theta)^{2}} d \theta \\
&=2 \pi \end{aligned}
$$
Work Step by Step
The exact length of the polar curve:
$$ r=2 \cos \theta \quad \text {for } \theta =0 \quad \text {to} \quad \theta= \pi $$
is equal to
$$
\begin{aligned} L &=\int_{a}^{b} \sqrt{r^{2}+(d r / d \theta)^{2}} d \theta \\
&=\int_{0}^{\pi} \sqrt{(2 \cos \theta)^{2}+(-2 \sin \theta)^{2}} d \theta \\ &=\int_{0}^{\pi} \sqrt{4\left(\cos ^{2} \theta+\sin ^{2} \theta\right)} d \theta \\
&=\int_{0}^{\pi} \sqrt{4} d \theta \\
&=[2 \theta]_{0}^{\pi} \\
&=2 \pi \end{aligned}
$$
