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
Geometric measure theory and geometric integration theory studies various measures of subsets of Euclidean spaces and possibly of some geometric generalizations) and their geometric properties. Especially, one studies rectifiability of subsets of some lower dimensionality, to define notions like area, arc length etc. and to study distributions and currents on such spaces. Very central questions and motivations belong to the variational problems including the study of minimal surfaces.
wikipedia geometric measure theory
Steven Krantz, Harold Parks, Geometric integration theory (pdf)
Jürgen Jost, The geometric calculus of variations: a short survey and a list of open problems, Exposition. Math. 6 (1988), no. 2, 111–143, MR89h:58036
Herbert Federer, Geometric measure theory, Springer 1969(especially appendices to Russian transl.)
Frederick J., Jr. Almgren, Almgren’s big regularity paper (book form of a 1970s preprint)
F. J. Almgren, Jr., Geometric measure theory and elliptic variational problems, Proc. ICM Nice 1970, vol. 2, djvu:350 K, pdf:649 K
eom: T.C.O’Neil (2001), Geometric measure theory
H. Federer, W. H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 1960 458–520, MR123260, doi
Stephen H. Schanuel, What is the length of a potato? An introduction to geometric measure theory, in: Categories in continuum physics (Buffalo, N.Y., 1982), 118–126, Lecture Notes in Math. 1174, Springer 1986, MR842922,doi
Rodolfo Rios-Zertuche, The variational structure of the space of holonomic measures, arxiv1408.5785