I have $X_1,X_2,\dots$ independent and uniformly distributed on $[0,1]$, with $$S_n = \frac{1}{n}\sum_{k=1}^nf(X_k),\qquad S=\int_0^1fdx$$ and $f$ Borel measurable on $[0,1]$ and integrable, and $\int_0^1f(x)^2dx<\infty$. I want to use Chebyshev's inequality to estimate $$P(|S_n-S|>\alpha n^\frac{-1}{2})$$.
I have shown that $S_n$ converges to $S$, and I know that by Chebyshev's inequality $$P(|S_n-S|>\alpha n^\frac{-1}{2})\leq\frac{E(S_n-S)^2}{\alpha n^{-1}}$$ I'm struggling to compute this. Can anyone help?