The area bounded by the curves y = x 2 - 1 and y = 1 is

Plotting the diagram of $y=\left|x^2-1\right|$ and $y=1$, we get,

Area of the region between the two curves

 $=2{\int }_0^{\sqrt{2}}\left(1-\left|x^2-1\right|\right)dx$

$=2\left[{\int }_0^1\left(1+\left(x^2-1\right)\right)dx+{\int }_1^{\sqrt{2}}\left(1-\left(x^2-1\right)\right)dx\right]$

$=2{\left[\frac{x^3}{3}\right]}_0^1+2{\left[2x-\frac{x^3}{3}\right]}_1^{\sqrt{2}}$

$=2\left(\frac{1}{3}\right)+2\left(2\sqrt{2}-\frac{2\sqrt{2}}{3}\right)-2\left(2-\frac{1}{3}\right)$

$=\frac{8}{3}\left(\sqrt{2}-1\right)$