The area of the region bounded by $x^2+y^2-2x-3=0$ and $y=|x|+1$ is

$x^2+y^2-2x-3=0$

$\Rightarrow (x-1)^2+y^2=4$

A= $\displaystyle \int_{1-\sqrt{2}}^{0} ( \sqrt{4-(x-1)^2} - (-x+1)dx + \displaystyle \int_{0}^{1} ( \sqrt{4-(x-1)^2} - (x+1))dx$

$= \dfrac{x-1}{2} \sqrt{4-(x-1)^{2}}+ \dfrac{4}{2} \sin^{-1} \dfrac{x-1}{2}+ \dfrac{x^{2}}{2}-x\left |_{1-\sqrt{2}}^{0}+ \dfrac{x-1}{2} \sqrt{4-(x-1)^2}+ \dfrac{4}{2} \sin^{-1} \dfrac{x-1}{2}- \dfrac{x^{2}}{2}-x \right |_{0}^{1}$

$-\left ( -1- \dfrac{\pi}{2}+\dfrac{3}{2}- \sqrt{2}-1+\sqrt{2} \right )- \dfrac{3}{2}=\dfrac{\pi}{2}-1 \, sq.units$