Integration, average behavior a=π r^2

In this chapter we will derive the formula $A=\pi r^2$ for the area of a circle of radius $r.$ As a matter of fact, we will first have to settle on exactly what is the definition of the area of a region in the plane,and more subtle than that, we must decide what kinds of regions in the plane “have” areas. Before we consider the problem of area, we willdevelop the notion of the integral (or average value) of a function defined on an interval $[a,b],$ which notion we will use later to compute areas, once they have been defined.

The main results of this chapter include:

1. The definition of integrability of a function, and the definition of the integral of an integrable function,
2. The Fundamental Theorem of Calculus ( [link] ),
3. The Integral Form of Taylor's Remainder Theorem ( [link] ),
4. The General Binomial Theorem ( [link] ),
5. The definition of the area of a geometric set,
6.   $A=\pi r^2$ ( [link] ), and
7. The Integral Test ( [link] ).