  In [5], Connes and Chamseddine defined a Hochschild cocycle in the general
framework of noncommutative geometry. They computed this Hochschild cocycle for
the Dirac operator on 4-dimensioanl manifolds. We propose a way to study the
Connes-Chamseddine Hochschild cocycle from the viewpoint of the noncommutative
integral on 6-dimensional manifolds in this paper. We compute several
interesting noncommutative integral defined in [8] by the normal coodinated way
on n-dimensional manifolds. As a corollary, the Connes-Chamseddine Hochschild
cocycle on 6-dimensional manifolds is obtained.
