Explanation of $E(\eta)=\int_0^{1}\min\left(\xi,\frac23\right)p(\xi)\,d\xi=\int_0^{2/3}\xi\,p(\xi)\,d\xi+\int_{2/3}^1\frac23\,p(\xi)\,d\xi.$

Question

Also how would this differ if $\eta = min(\xi, \phi)$ and $\eta = min(\xi, \phi, \rho)$ where $ \eta, \phi, \rho$ are random variables.