This paper proposes a passive methodology for detecting a class of stealthy intermittent integrity attacks in cyber-physical systems subject to process disturbances and measurement noise. A stealthy intermittent integrity attack strategy is first proposed by modifying a zero-dynamics attack model. The stealthiness of the generated attacks is rigorously investigated under the condition that the adversary does not know precisely the system state values. In order to help detect such attacks, a backward-in-time detection residual is proposed based on an equivalent quantity of the system state change, due to the attack, at a time prior to the attack occurrence time. A key characteristic of this residual is that its magnitude increases every time a new attack occurs. To estimate this unknown residual, an optimal fixed-point smoother is proposed by minimizing a piece-wise linear quadratic cost function with a set of specifically designed weighting matrices. The smoother design guarantees robustness with respect to process disturbances and measurement noise, and is also able to maintain sensitivity as time progresses to intermittent integrity attack by resetting the covariance matrix based on the weighting matrices. The adaptive threshold is designed based on the estimated backward-in-time residual, and the attack detectability analysis is rigorously investigated to characterize quantitatively the class of attacks that can be detected by the proposed methodology. Finally, a simulation example is used to demonstrate the effectiveness of the developed methodology.