Find the area of the region bounded by $x^2=4y, x=4y-2$

The area bounded by the circle $x^2+y^2=8$, the parabola $x^2=2y$ and the line $y=x$ in $y\ge 0$

$\mathrm{I}$. has area $=\left(2\pi +\frac{4}{3}\right)$ sq units

$\mathrm{II}$. has area $=\left(2\pi +\frac{2}{3}\right)$ sq units

$\mathrm{III}$: the point of intersection of circle, parabola and line in $1\mathrm{st}$ quadrant is $\left(2,2\right)$.

Which of the following is true?