Q:

The volume of the region enclosed by $x=y^3$, $x=0$, $y=2$

The volume of the region enclosed by $x=y^{3}$, $x=0$, $y=2$, revolved around the X-axis is? 
My question is, how would that solid be represented? 

A:

Update
Thanks to @ubpdqn for pointing out that it should be {x, 0, 8}, not {x, 0, 2^(1/3)}. 
I think this is what you want:
region = ImplicitRegion[
    y^2 + z^2 < x^(2/3),
    {{x, 0, 8}, {y, -2, 2},{z, -2, 2}}
];

Volume @ DiscretizeRegion @ region

60.1229

The discretization isn't very accurate, so alternatives to produce a more accurate answer are:
Integrate[π x^(2/3), {x, 0, 8}]
%//N

NIntegrate[1, {x,y,z} ∈ region]

96 π/5
60.3186
60.3186

Visualization:
Region[region, Axes->True, AxesLabel->{"x", "y", "z"}]

