Calculate the shaded area as given below :

The correct answer is option (1) → $\frac{8}{3}$ sq.units

area is symmetric about x

so computing one sides area and multiplying by 2

area = $2×\left(\int\limits_{-2}^2\frac{x}{2}+1dx-\int\limits_0^2\sqrt{2x}dx\right)$

$=2\left[\left[\frac{x^2}{4}+x\right]_{-2}^2-\left[\frac{2}{3}\sqrt{2}×\sqrt{x}\right]_0^1\right]$

$=2\left[4-\frac{2}{3}\sqrt{2}×2\sqrt{2}\right]$

$=2[\frac{4}{3}]=\frac{8}{3}$ sq. units