Solution:
The area in the first quadrant is bounded by $y=4 x^{2}, x=0, y=1$, and $y=4$ is represented by the shaded area ABCDA as

$\begin{aligned}
\therefore \operatorname{Area} A B C D &=\int^{1} x d x \\
&=\int^{4} \frac{\sqrt{y}}{2} d x \\
&=\frac{1}{2}\left[\frac{y^{\frac{3}{2}}}{3}\right.\\
&=\frac{1}{3}\left[(4)^{\frac{3}{2}}-1\right]
\end{aligned}$
$=\frac13\mathrm[8-1]$
$=\frac73$ sq. units.