Multivariate Calculus

Chapter 6 Multiple Integrals

6.1 Introduction

In this chapter we shall study integrals of functions of several variables involving multiple integrals. The motivations connected with these integrals are much the same as with simple integrals of functions of one variable which we have studied earlier. The chapter deals mainly with two types of integrals in ℝ² and ℝ³, viz., (i) Double integrals and (ii) Triple integrals. These integrals have important applications to geometry and mechanics.

6.2 Double Integrals

The definite integral ${\displaystyle {\int }_a^{b}f\left(x\right)}dx$is defined as the limit of the sum ${\displaystyle \sum _{i=1}^{n}f\left(x_i\right)}\delta x_i$ where n→∞ and each of the lengthsδ_x1 δ_x2,...tends to zero[ref.§0.3.4]. A double integral is its counterpart in two dimensions.

Let f(x, y) be a function of ...