  We derive an approximate formula for the mean first-passage time (MFPT) to a
small absorbing target of arbitrary shape inside an elongated domain of a
slowly varying axisymmetric profile. For this purpose, the original Poisson
equation in three dimensions is reduced an effective one-dimensional problem on
an interval with a semi-permeable semi-absorbing membrane. The approximate
formula captures correctly the dependence of the MFPT on the distance to the
target, the radial profile of the domain, and the size and the shape of the
target. This approximation is validated by Monte Carlo simulations.
