Solution:

The line, $x=a$, divides the area bounded by the parabola and $x=4$ into two equal parts.
$\therefore$ Area $O A D=$ Area $A B C D$

It is observed that the given area is symmetrical about $x$-axis.

$\Rightarrow$ Area $\mathrm{OED}=$ Area EFCD
$\begin{aligned}
\text { Area } O E D &=\int_{0}^{v} y d x \\
&=\int^{n} \sqrt{x} d x \\
&=\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]_{0}^{a} \\
&=\frac{2}{3}(a)^{\frac{3}{2}}
\end{aligned}$
The area of $E F C D=\int_{0}^{4} \sqrt{x} d x$
$\begin{array}{l}
=\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]_{0}^{4} \\
=\frac{2}{3}\left[8-a^{\frac{3}{2}}\right]
\end{array}$
From eq.(1) and eq.(2), we get
$\begin{array}{l}
\frac{2}{3}(a)^{\frac{3}{2}}=\frac{2}{3}\left[8-(a)^{\frac{3}{2}}\right] \\
\Rightarrow 2 \cdot(a)^{\frac{3}{2}}=8 \\
\Rightarrow(a)^{\frac{3}{2}}=4 \\
\Rightarrow a=(4)^{\frac{2}{3}}
\end{array}$
As a result, the value of a is $(4)^{\frac{2}{3}}$