  Active Brownian particles (ABPs) are a model for nonequilibrium systems in
which the constituent particles are self-propelled in addition to their
Brownian motion. Compared to the well-studied mean first passage time (MFPT) of
passive Brownian particles, the MFPT of ABPs is much less developed. In this
paper, we study the MFPT for ABPs in a 1-D domain with absorbing boundary
conditions at both ends of the domain. To reveal the effect of swimming on the
MFPT, we consider an asymptotic analysis in the weak-swimming or small P\'eclet
(Pe) number limit. In particular, analytical expressions for the survival
probability and the MFPT are developed up to O(Pe$^2$). We explore the effects
of the starting positions and starting orientations on the MFPT. Our analysis
shows that if the starting orientations are biased towards one side of the
domain, the MFPT as a function of the starting position becomes asymmetric
about the center of the domain. The analytical results were confirmed by the
numerical solutions of the full PDE model.
