The area bounded by $x^2=4ay$ and $y=2a$ is 

The given curves are $x^2=4ay$ and $y=2a$ 

The rough sketch of the graph of the required region is as below

The required area is the area of the shaded region

We know that the area of the  region bounded by the curve $x=f\left(y\right)$, $y=a,y=b$ and $Y-$axis is ${\int }_a^{b}f\left(y\right)dy$

Hence, the area of the shaded region is ${\int }_0^{2a}2\sqrt{ay}dy$

$\begin{array}{rcl}{\int }_0^{2a}2\sqrt{ay}dy& =& 2\sqrt{a}{\left(\frac{y^{{\displaystyle \frac{3}{2}}}}{{\displaystyle \frac{3}{2}}}\right)}_0^{2a}\\ & =& 2\sqrt{a}\left(\frac{2}{3}\right){\left(2a\right)}^{\frac{3}{2}}\\ & =& \frac{8\sqrt{2}}{3}{a}^2\end{array}$

Therefore the area of the required region is $\begin{array}{rcl} & & \boxed{\frac{8\sqrt{2}}{3}{a}^2}\end{array}$