Integrating in Many Dimensions

Abstract

The present chapter covers the problems connected with the integration in more than one dimension. Our concern involves the case of $\mathbb {R}^N$ and the integral is understood in the Riemann sense, which most often appears in physics or engineering. Its construction is a simple generalization of that formulated in Chap. 1: in $\mathbb {R}^2$, instead of intervals one has to divide the domain into rectangles, in $\mathbb {R}^3$ into cuboids, and so on. Then one calculates the upper and lower sums just as in Problem 1 in Sect. 1.1 and compare their limits as the appropriate largest diagonal tends to zero. Such integral is called the Riemann or Darboux integral. The question of the Lebesgue integral remains beyond the scope of the present problem set.