Chapter 20: The Area under a Curve and its Many Generalizations

Introduction

The area between the curve defined by a positive function f and the x axis between two specific values of y is called the definite integral of f between those values. Starting with the fact that the area of a rectangle is the product of its side lengths, we can give a formal definition of the area under a general curve. The method of doing this used is generalized to define a wide variety of integrals that do not describe area. These include integration on a path in the complex plane, along a path in any Euclidean space, over an area in the plane, over a surface in three dimensional space and over volume.