The area of the region x , y : x - 1 ≤ y ≤ 5 - x 2 is equal to

Plotting region for $\left|x-1\right|\le y\le \sqrt{5-x^2}$, we get, 

The intersection points of the curves $x^2+y^2=5$ and $y=\left|x-1\right|$ will be $\left(-1,2\right) \& \left(2,1\right)$

Here chord $AB$ subtends a right angle at the centre of the circle.
So, required area $=$ area of $∆ABC+$ area of segment of circle on chord $AB$

$=\frac{1}{2}AC·BC+$ [area of quarter circle $-$ area of $∆AOB$]

$=\frac{1}{2}\times \sqrt{2}\times 2\sqrt{2}+\left(\frac{90°}{360°}\times \mathrm{\pi }{\left(\sqrt{5}\right)}^2-\frac{1}{2}\times {\left(\sqrt{5}\right)}^2\right)$

$=\frac{5\pi }{4}-\frac{1}{2}$