  We present an ab initio approach to integration theory for nonunital spectral
triples. This is done without reference to local units and in the full
generality of semifinite noncommutative geometry. The main result is an
equality between the Dixmier trace and generalised residue of the zeta function
and heat kernel of suitable operators. We also examine definitions for
integrable bounded elements of a spectral triple based on zeta function, heat
kernel and Dixmier trace techniques. We show that zeta functions and heat
kernels yield equivalent notions of integrability, which imply Dixmier
traceability.
