A Course in Multivariable Calculus and Analysis pp 185-290 | Cite as

# Multiple Integration

## Abstract

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 R^{2}. The general case of double integrals of functions defined on bounded subsets of R^{2} is developed in Section 5.2. This will lead, in particular, to the general concept of area of a bounded region in R^{2}. 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 R^{3}, and discuss the general concept of volume of such subsets.

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