Advanced Engineering Mathematics, 7th Edition

9.15 Triple Integrals

INTRODUCTION

The steps leading to the definition of the three-dimensional definite integral or triple integral are quite similar to the steps leading to the definition of the double integral. Obvious differences: Instead of a function of two variables we are integrating a function f of three variables, not over a region R in a coordinate plane, but over a region D of 3-space.

1. Let w = F(x, y, z) be defined over a closed and bounded region D of space.
2. By means of a three-dimensional grid of vertical and horizontal planes parallel to the coordinate planes, form a partition P of D into n subregions (boxes) D_k of volumes ΔV_k that lie entirely in D.
3. Let be the norm of the partition or the length of the longest diagonal of ...