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Date: 1-12-2016
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Date: 24-11-2016
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Date: 3-12-2016
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The development of the theory of integration in RN for N ≥ 2 parallels the one-dimensional case given in Section of (The Darboux Integral for Functions on R1). For functions defined on an interval I in R1, we formed upper and lower sums by dividing I into a number of subintervals. In RN we begin with a domain F, i.e., a bounded region that has volume, and in order to form upper and lower sums, we divide F into a number of subdomains. These subdomains are the generalizations of the subinterval in R1, and the limits of the upper and lower sums as the number of subdomainstends to infinity yield upper and lower integrals. The subdomains can be thought of ashypercubesin RN with the volume of each hypercube just N times the length of a side.
Definition
Let F be a bounded set in RN that is a domain. A subdivision of F is a finite collection of domains {F1,F2,...,Fn} no two of which have common interior points and the union of which is F. That is,