Integral calculus in several variables

Abstract

The definite integral of a function of one real variable allowed us, in Vol. I, to define and calculate the area of a sufficiently regular region in the plane. The present chapter extends this notion to multivariable maps by discussing multiple integrals; in particular, we introduce double integrals for dimension 2 and triple integrals for dimension 3. These new tools rely on the notions of a measurable subset of ℝⁿ and the corresponding n-dimensional measure; the latter extends the idea of the area of a plane region (n=2), and the volume of a solid (n=3) to more general situations.