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Intelligent reflecting surface (IRS) has emerged as an appealing solution to enhance the wireless communication performance by reconfiguring the wireless propagation environment. In this paper, we propose to apply IRS to the physical-layer service integration (PHY-SI) system, where a single-antenna access point (AP) integrates two sorts of service messages, i.e., multicast message and confidential message, via superposition coding to serve multiple single-antenna users. Our goal is to optimize the power allocation (for transmitting different messages) at the AP and the passive beamforming at the IRS to maximize the achievable secrecy rate region. To this end, we formulate this problem as a bi-objective optimization problem, which is shown equivalent to a secrecy rate maximization problem subject to the constraints on the quality of multicast service. Due to the non-convexity of this problem, we propose two customized algorithms to obtain its high-quality suboptimal solutions, thereby approximately characterizing the secrecy rate region. The resulting performance gap with the globally optimal solution is analyzed. Furthermore, we provide theoretical analysis to unveil the impact of IRS beamforming on the performance of PHY-SI. Numerical results demonstrate the advantages of leveraging IRS in improving the performance of PHY-SI and also validate our theoretical analysis.