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

$= 2{∫}_{0}^{\sqrt{2}}\left(1 - \left|{x}^{2} - 1\right|\right)dx$

$= 2\left[{∫}_{0}^{1}\left(1 + \left({x}^{2} - 1\right)\right)dx + {∫}_{1}^{\sqrt{2}}\left(1 - \left({x}^{2} - 1\right)\right)dx\right]$

$= 2{\left[\dfrac{{x}^{3}}{3}\right]}_{0}^{1} + 2{\left[2x - \dfrac{{x}^{3}}{3}\right]}_{1}^{\sqrt{2}}$

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

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