The area bounded by the curves $\mathrm{y}=\left|{\mathrm{x}}^2-1\right|\text{ and }\mathrm{y}=1$ is

$\mathrm{y}=\left|{\mathrm{x}}^2-1\right|$

Area = ABCDEA

$\begin{array}{l}=2\left({\int }_0^1 \left(1-\left(1-{\mathrm{x}}^2\right)\right)\mathrm{dx}+{\int }_1^{\sqrt{2}} \left(1-\left({\mathrm{x}}^2-1\right)\right)\mathrm{dx}\right)\\ =2\left({\int }_0^1 {\mathrm{x}}^2\mathrm{dx}+{\int }_1^{\sqrt{2}} \left(2-{\mathrm{x}}^2\right)\mathrm{dx}\right)\\ =2\left({\left(\frac{x^3}{3}\right)}_0^1+{\left(2x-\frac{x^3}{3}\right)}_1^{\sqrt{2}}\right)\\ 2\left(\frac{1}{3}+\left(\left(2\sqrt{2}-\frac{2\sqrt{2}}{3}\right)-\left(2-\frac{1}{3}\right)\right)\right)=2\left(\frac{1}{3}-\frac{5}{3}+\frac{4\sqrt{2}}{3}\right)\\ =\frac{8}{3}(\sqrt{2}-1)\end{array}$