Answer
$$\int^{\pi}_0\sqrt{1-\cos^2\theta}d\theta=2$$
Work Step by Step
$$A=\int^{\pi}_0\sqrt{1-\cos^2\theta}d\theta$$
Use the identity $$1-\cos^2\theta=\sin^2\theta$$
That means $$A=\int^{\pi}_0\sqrt{\sin^2\theta}d\theta$$ $$A=\int^{\pi}_0|\sin \theta|d\theta$$
We have $\sin\theta\ge0$ on $[0,\pi]$. Therefore,
$$A=\int^{\pi}_0\sin \theta d\theta$$ $$A=-\cos\theta\Big]^{\pi}_0$$ $$A=-(\cos\pi-\cos0)$$ $$A=-(-1-1)=2$$
