  This paper deal with some questions regarding the notion of integral in the
framework of Connes's noncommutative geometry. First, we present a purely
spectral theoretic construction of Connes' integral. This answers a question of
Alain Connes. We also deal with the compatibility of Dixmier traces with
Lebesgue's integral. This answers another question of Alain Connes. We further
clarify the relationship of Connes' integration with Weyl's laws for compact
operators and Birman-Solomyak's perturbation theory. We also give a "soft
proof" of Birman-Solomyak's Weyl's law for negative order pseudodifferential
operators on closed manifold. This Weyl's law yields a stronger form of Connes'
trace theorem. Finally, we explain the relationship between Connes' integral
and semiclassical Weyl's law for Schroedinger operators. This is an easy
consequence of the Birman-Schwinger principle. We thus get a neat link between
noncommutative geometry and semiclassical analysis.
