Why does the quantity $\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}f(x_i)$ perform poorly for estimating ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(x)dx?$ The quantity is an accurate estimate for the integral. $\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}f(x_i)E(f(x)){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(x)dx$ Since we do not know the distribution of $f(x_i),$ we cannot find any distributional results about the sample average and thus the CLT does not apply and $\frac{1}{n}{\displaystyle \underset{i=1}{\overset{?}{}}}f(x_i).$ We do not know if $f(x_i)$ are independent, so we cannot say anything about the asymptotic behavior of the sample average.