  The Poincare series of a multi-index filtration on the ring of germs of
functions can be written as a certain integral with respect to the Euler
characteristic over the projectivization of the ring. Here this integral is
considered with respect to the generalized Euler characteristic with values in
the Grothendieck ring of varieties. For the filtration defined by orders of
functions on the components of a plane curve singularity $C$ and for the so
called divisorial filtration for a modification of $(\C^2,0)$ by a sequence of
blowing-ups there are given formulae for this integral in terms of an embedded
resolution of the germ $C$ or in terms of the modification respectively. The
generalized Euler characteristic of the extended semigroup corresponding to the
divisorial filtration is computed.
