Below, we see a region bounded by two curves and the $x-$axis.

The region shaded in light blue is bounded on the left by the curve $y=x^2($in dark blue$),$ on the right by the curve $y=-\frac{1}{3}x\frac{4}{3}($in dark red$),$ and below by the $x-$axis,

Part $1.$

${\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(-\frac{1}{3}x\frac{4}{3})dx$

Part $2.$

$=\sqrt[\phantom{2}]{}$

Part $3.$

Finally, after evaluating the integrals above, we find that the area of the shaded region equals $$