Why can we consider expectation in Gibbs Sampling

Suppose we are doing Gaussian Mixture (1D). The histogram of posterior distribution is (we choose a new $z_i$ from this histogram),
%
Each integral shows posterior predictive distribution of $x_i$ and $z_i$, respectively. We can consider the expectation of $\mu_k$ for the first term instead of calculating everything. $\mu_k$ can take various values, but it follows Normal distribution. Enough amount of data makes the posterior distribution of $\mu_k$ sharp. The expectation can be a good approximation.
%