After a short historical overview of integration and an
introductory example indicating the basic idea of several types of
integrals (Choquet, Shilkret, PAN, concave, convex), we introduce
admissible sub- and superdecompositions and construction of
integrals recently proposed by Even & Lehrer and by Mesiar, Li &
Pap, namely decomposition integrals and superdecomposition
integrals. Moreover, we recall the concept of universal integrals,
including an interesting class of integrals based on copulas. When
requiring that the considered integral is both a universal integral
and a (super)decomposition integral, then particular ordered
parametric families of integrals are obtained. After short
discussion on possible modification of standard arithmetical
operations, some examples related to bibliometric indices are
added.