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A Course in Mathematical Analysis by D. J. H. Garling

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18

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 ...

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