$\text{ solve the equations }y=\sqrt{x}\text{ and }y=x-2\text{ simultaneously for }x\text{ : }$

$$\begin{gathered} \sqrt{x}=x-2 \\ x=(x-2{)}^2=x^2-4x+4 \\ {x}^2-5x+4=0 \\ (x-1\left)\right(x-4)=0 \\ x=1, x=4 \end{gathered}$$

$\text{ Total area }=\underset{\text{area of }A}{\underbrace{{\int }_0^2\sqrt{x}dx}}+\underset{\text{area of }B}{\underbrace{{\int }_2^4(\sqrt{x}-x+2)dx}}$

$={\left[\frac{2}{3}{x}^{3/2}\right]}_0^2+{\left[\frac{2}{3}{x}^{3/2}-\frac{x^2}{2}+2x\right]}_2^4$

$=\frac{2}{3}(2{)}^{3/2}-0+\left(\frac{2}{3}(4{)}^{3/2}-8+8\right)-\left(\frac{2}{3}(2{)}^{3/2}-2+4\right)$

$=\frac{2}{3}\left(8\right)-2=\frac{10}{3}\text{. }$