The distribution of the product of uniform variates on the interval can be found directly as
(1)
 
(2)

where is a delta function. The distributions are plotted above for (red), (yellow), and so on.
The distribution of the product of uniform variates on the interval can be found directly as
(1)
 
(2)

where is a delta function. The distributions are plotted above for (red), (yellow), and so on.
Weisstein, Eric W. "Uniform Product Distribution." From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/UniformProductDistribution.html