Finding a distribution function of random variable sum Let $\xi_1, \xi_2, \xi_3$ independent random variables in $(\Omega, \mathcal{F},\mathbb{P}).$ Also, they are evenly distributed  in $[0,1]$. I need to find a distribution function of sum $\xi_1+ \xi_2+ \xi_3$.
So, I know that $F_{\xi}(x)=\mathbb{P}(\omega : \xi(\omega \leq x)=\mathbb{P}(\xi \leq x)=\mathbb{P}(\xi^{-1}(-\infty;x])$. Also I know that $\mathbb{P}(\xi_1,\xi_2,\xi_3)=\mathbb{P}(\xi_1)* \mathbb{P}(\xi_1)* \mathbb{P}(\xi_1)$.
But how to use it and find a distribution function?

One way is (I used $Z$ instead of $\xi$)$$P[Z_1+Z_2+Z_3 \leq z] = \int_{z_3=0}^{\min(z,\,1)}\int_{z_2=0}^{\min(z-z_3,\,1)}\int_{z_1=0}^{\min(z-z_2-z_3,\,1)}dz_1\,dz_2\,dz_3$$
