The area of the region bounded by the curve $x{y}^2=1$ and the lines $x=1$, $x=2$ is

The given curves are $x{y}^2=1$ and the lines are  $x=1$, $x=2$

The given curves are $y=\frac{1}{\sqrt{x}},y=-\frac{1}{\sqrt{x}}$

The rough sketch for the required area is 

Obserivng the rough skecth, the area is symmetric about X- axis

Hence, the area of the shaded region is $A=2{\int }_1^2\frac{1}{\sqrt{x}}dx$

$$\begin{gathered} A=2{\int }_1^2\frac{1}{\sqrt{x}}dx \\ =2\left[\frac{x^{{\displaystyle \frac{1}{2}}}}{{\displaystyle \frac{1}{2}}}\right] \\ ={\left[4\sqrt{x}\right]}_1^2 \\ =4\sqrt{2}-4 \\ =4\left(\sqrt{2}-1\right) \end{gathered}$$

Therefore, the area of regired region is $\boxed{4\left(\sqrt{2}-1\right)}$