In one-variable calculus, we study the theory of Riemann integration. (See, for example, Chapter 6 of ACICARA.) In this chapter, we will extend this theory to functions of several variables. As in the previous chapters, we shall mainly restrict to functions of two variables and briefly show how things work for functions of three variables. Further extension to the case of functions of n variables, where n ≥ 4, is similar.
In Section 5.1 we consider the relatively simpler case of double integrals of functions defined on rectangles in R2. The general case of double integrals of functions defined on bounded subsets of R2 is developed in Section 5.2. This will lead, in particular, to the general concept of area of a bounded region in R2. Next, in Section 5.3, we discuss the change of variables formula for double integrals and prove it in an important special case. Finally, in Section 5.4, we will indicate how the theory of double integrals extends to triple integrals of functions defined on bounded subsets of R3, and discuss the general concept of volume of such subsets.
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