Area bounded by |x-1|≤2 and x^2-y^2 =1, is

We have,

$|x-1|≤2⇔-1≤x≤3$

∴ Required area A is the area of the shaded region and is given by

$A=2\int\limits_{1}^{3}\sqrt{x^2-1}dx$

$⇒A=\left[x\sqrt{x^2-1}-\log_e(x+\sqrt{x^2-1})\right]_1^3$

$⇒A=\left\{3\sqrt{8}-\log_e(3+\sqrt{8})\right\}=6\sqrt{2}-\log_e(3+2\sqrt{2})$