Let $X,Y,Z$ be i.i.d. $\mathbb{R}$-valued random variables each with the uniform distribution on [0,1]. ie. $X,Y,Z\sim \mathcal{U}[0,1]$.
Determine the probability density function of X+Y+Z.
For this question, Can I consider this equals $\int\limits_{x=0}^1\int\limits_{y=0}^1 \int\limits_{z=0}^1 [yz,1]1 dx dy dz$?
One would have to use convolutions to get the sum of random variables. For example, in the case of sum of two variables, if $X+Y=s$, then you would have to integrate over all $X$ (in the domain) and fix $Y = s-X$. As the number of variables increases, the convolution operation is not straightforward. You can read more here: https://en.wikipedia.org/wiki/Convolution
In the case of sum of uniform RVs, one would use the Irvin Hall distribution, which in your case (for sum of 3 variables) becomes:
$f_{X+Y+Z}(X+Y+Z=s) = \begin{cases} \frac{1}{2}s^2 & s\in[0,1]\\ \frac{1}{2}(-2s^2 + 6s - 3)& s\in[1,2]\\ \frac{1}{2}(s - 3)^2 & s\in[2,3] \end{cases} $