  In this paper, we study bounds of expected $L_2-$discrepancy to give mean
square error of uniform integration approximation for functions in Sobolev
space $\mathcal{H}^{\mathbf{1}}(K)$, where $\mathcal{H}$ is a reproducing
Hilbert space with kernel $K$. Better order $O(N^{-1-\frac{1}{d}})$ of
approximation error is obtained, comparing with previously known rate
$O(N^{-1})$ using crude Monte Carlo method. Secondly, we use expected
$L_{p}-$discrepancy bound($p\ge 1$) of stratified samples to give several upper
bounds of $p$-moment of integral approximation error in general Sobolev space
$F_{d,q}^{*}$.
