Distribution of multivariate uniform random variable in R

I have k>2uniform independent random variables with different upper and lower for example: X~U[-1,1], Y~U[0,1],Z~U[-1,0] and so on.

I have to calculate for a RV given by RV=X+Y+Z the probability that it is greater than some scalar x for example:x=0.2 and I have to calculate Pr(RV>x). Is there any easier way I can calculate using R. I have an array of >1000 random variables and I need to calculate this for each possible combinations of these variables. Therefore I am trying to avoid sampling route.

Thanks for help!