[the data is modified so that it keeps the main principle of the problem I'm working on but it's not identical]

I have a $k$ number of boxes manufactured and the length of each side of the box is determined by a random var with a density function

$f(x) = \frac{5x}{3},$ for $0 \leq x \leq 1$.

I need to find the mean of the volume of a box.

So, consider the box is a cube and its volume is another random variable, define it $g(x) = V = x^3$

I know that I can find the cumulative distribution function of $g(x)$ by letting $Y = g(x)$, then

$F_y(y) = P(g(x) \leq y) = p(x \in g^{-1}(-\infty, y] )$, so I assume that the next step is

$ = \int_{-infty}^{y} x^{1/3}dx = ...$, but it's kind of weird since the improper integral I get will not converge...

Then the p.d.f is the derivative of c.d.f, but I am a bit confused about how in general I should deal with random values like this, when once comes from the other.

Would greatly appreciate some help,

Thanks