# 7: Integrals of Functions of Several Variables

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IN THIS CHAPTER we study the integral calculus of real-valued functions of several variables.

• SECTION 7.1 defines multiple integrals, first over rectangular parallelepipeds in $$\R^n$$ and then over more general sets. The discussion deals with the multiple integral of a function whose discontinuities form a set of Jordan content zero, over a set whose boundary has Jordan content zero.
• SECTION 7.2 deals with evaluation of multiple integrals by means of iterated integrals.
• SECTION 7.3 begins with the definition of Jordan measurability, followed by a derivation of the rule for change of content under a linear transformation, an intuitive formulation of the rule for change of variables in multiple integrals, and finally a careful statement and proof of the rule. This is a complicated proof.

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