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.