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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.
20.1 Area and Notation
20.2 Precise Definition and Riemann Sums
20.3 Always 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