Witten's top Chern class is a particular cohomology class on the
moduli space of Riemann surfaces endowed with r–spin structures.
It plays a key role in Witten's conjecture relating to the intersection
theory on these moduli spaces.
Our first goal is to compute the integral of Witten's class over
the so-called double ramification cycles in genus 1. We obtain
a simple closed formula for these integrals.
This allows us, using the methods of the first author
[Int. Math. Res. Not. 38 (2003) 2051-2094], to find an algorithm for
computing the intersection numbers of the Witten class with powers of
the ψ–classes over any moduli space of r–spin structures, in
short, all numbers involved in Witten's conjecture.