  Integral formulae for foliated Riemannian manifolds provide obstructions for
existence of foliations or compact leaves of them with given geometric
properties. Recently, we associated a new Riemannian metric to a
codimension-one foliated Finsler space and proved integral formulae for general
and for Randers spaces. In the paper, we study this metric for a wider class of
codimension-one foliated $(\alpha,\beta)$-spaces and embody it in a set of
metrics that can be viewed as a perturbed metric associated with $\alpha$. For
such metrics we calculate the Weingarten operator of the leaves and deduce the
Reeb type integral formula, which can be used for $(\alpha,\beta)$-spaces, e.g.
Randers and Kropina spaces.
