Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
The refinement of the notion of fiber integration in generalized (Eilenberg-Steenrod) cohomology to differential cohomology.
General discussion of fiber integration for Hopkins-Singer coefficients is in (HopkinsSinger, section 4.10) and for integration over circle fibers in (Bunke-Nikolaus-Völkl 13, section 4.5).
In (Bunke-Schick 09) and precursors, existence of fiber integration in differential cohomology over circle fibers was formalized as a separate axiom (Bunke-Schick 09, def. 1.3).
Mike Hopkins, Isadore Singer, section 4.10 in Quadratic Functions in Geometry, Topology, and M-Theory
Ulrich Bunke, Thomas Schick, Uniqueness of smooth extensions of generalized cohomology theories, J. Topol. 3 (2010), no. 1, 110-156 (arXiv:0901.4423)
Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)