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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.

Topics

20.1 Area and Notation

20.2 Precise Definition and Riemann Sums

20.3 Always Integrable Functions

20.4 Non Integrable Functions

20.5 Special Riemann Sums

20.6 Integration Over Curves in the Complex Plane

20.7 Integration Over Curves in Euclidean Space

20.8 Area Integrals

20.9 Surface Integrals

20.10 Volume Integrals