Answer
$$\int_{0}^{\pi / 2} \sqrt{1-\cos \theta} \ \ d \theta =2 \sqrt{2}\left(1-\frac{1}{\sqrt{2}}\right) $$
Work Step by Step
Given $$\int_{0}^{\pi / 2} \sqrt{1-\cos \theta} d \theta $$
So, we have
\begin{aligned} I &=\int_{0}^{\pi / 2} \sqrt{1-\cos \theta} d \theta \\ & .=\int_{0}^{\pi / 2} \sqrt{1-\left(1-2 \sin ^{2} \frac{\theta}{2}\right)} d \theta \\
&= \sqrt{2} \int_{0}^{\pi / 2}\sin \frac{\theta}{2} d \theta \\
&= 2\sqrt{2} \int_{0}^{\pi / 2}\frac{1}{2}\sin \frac{\theta}{2} d \theta \\
&=- 2\sqrt{2} \cos \frac{\theta}{2}|_{0}^{\pi / 2}\\
&=- 2\sqrt{2} \cos \frac{\theta}{4}+ 2\sqrt{2} \cos0 \\
&=2 \sqrt{2}\left(1-\frac{1}{\sqrt{2}}\right) \end{aligned}
