An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the non-negative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are defined.
Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)