Integrating functions of several variables
18.1 Elementary vector-valued integrals
We now consider the problem of integrating vector-valued functions of several variables. We begin by considering dissections, step functions and elementary integrals, as in the case of real-valued functions of a single variable. A cell C in Rd is a subset of Rd of the form I1 × · · · × Id, where I1, . . . , Id are intervals (open, closed, or neither) in R. Thus a one-dimensional cell is an interval and a two-dimensional cell is a rectangle. The d-dimensional volume or content vd(C) is defined to be .
Suppose that C = I1 × · · · × Id is a compact cell, and suppose ...