In this chapter we will derive the formula A=πr2 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 will develop 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:
The definition of integrability of a function, and the definition of the integral of an integrable function,
The Fundamental Theorem of Calculus (Theorem 5.9.),
The Integral Form of Taylor's Remainder Theorem (Theorem 5.12.),
The General Binomial Theorem (Theorem 5.13.),
The definition of the area of a geometric set,
A=πr2 (Theorem 5.15.), and
The Integral Test (Theorem 5.17.).
We begin by defining the integral of certain (but not all) bounded, real-valued functions whose domains are closed bounded intervals. Later, we will extend the definition of integral to certain kinds of unbounded complex-valued functions whose domains are still intervals, but which need not be either closed or bounded. First, we recall from Section 3.1 the following definitions.
Let [a,b] be a closed bounded interval of real numbers.
By a partition of [a,b] we mean a finite set
of n+1 points,
where x0=a and xn=b.
The n intervals 
are called the closed subintervals
of the partition P, and
the n intervals 
are called the open subintervals
or elements of P.
We write ∥P∥ for the maximum of the numbers (lengths of the subintervals) 
and call ∥P∥ the mesh size of the partition P.
If a partition 
is contained in another partition 
i.e., each xi equals some yj,
then we say that Q is finer than P.
Let f be a function on an interval [a,b], and let 
be a
partition of [a,b].
Physicists often consider
sums of the form
where yi is a point in the subinterval 
These sums (called Riemann sums) are approximations of physical quantities,
and the limit of these sums, as the mesh of the partition becomes smaller and smaller,
should represent a precise value of the physical quantity.
What precisely is meant by the “ limit” of such sums is already a subtle question,
but even having decided on what that definition should be, it is
as important and difficult to determine whether or not such a limit exists for many (or even any) functions f.
We approach this question from a slightly different point of view,
but we will revisit Riemann sums in the end.
Again we recall from Section 3.1 the following.
Let [a,b] be a closed bounded interval in R.
A real-valued function h:[a,b]→R is called a step function
if there exists a partition 
of [a,b] such that
for each 1≤i≤n there exists a number ai such that
h(x)=ai for all 
REMARK A step function h is constant on the open subintervals (or elements) of a certain partition. Of course, the partition is not unique. Indeed, if P is such a partition, we may add more points to it, making a larger partition having more subintervals, and the function h will still be constant on these new open subintervals. That is, a given step function can be described using various distinct partitions.
Also, the values of a step function at the partition points themselves is irrelevant. We only require that it be constant on the open subintervals.
Let h be a step function on [a,b],
and let 
be a partition of [a,b] such that h(x)=ai
on the subinterval 
determined by P.
Prove that the range of h is a finite set. What is an upper bound on the cardinality of this range?
Prove that h is differentiable at all but a finite number of points in [a,b]. What is the value of h' at such a point?
Let f be a function on [a,b]. Prove that f is a step function if and only if f'(x) exists and =0 for every x∈(a,b) except possibly for a finite number of points.
What can be said about the values of h at the endpoints 
of the subintervals of P?
(e) Let h be a step function on [a,b], and let j be a function on [a,b] for which h(x)=j(x) for all x∈[a,b] except for one point c. Show that j is also a step function.
If k is a function on [a,b] that agrees with a step function h except at a finite number of points c1,c2,...,cN, show that k is also a step function.
Let [a,b] be a fixed closed bounded interval in R, and let H([a,b]) denote the set of all step functions on [a,b].
Using Part (c) of Exercise 1., prove that the set H([a,b]) is a vector space of functions; i.e., it is closed under addition and scalar multiplication.
Show that H([a,b]) is closed under multiplication; i.e., if h1,h2∈H([a,b]), then h1h2∈H([a,b]).
Show that H([a,b]) is closed under taking maximum and minimum and that it contains all the real-valued constant functions.
We call a function χ an indicator function if it equals 1 on an interval (c,d) and is 0 outside [c,d]. To be precise, we will denote this indicator function by χ(c,d). Prove that every indicator function is a step function, and show also that every step function h is a linear combination of indicator functions:
Define a function k on [0,1] by setting k(x)=0 if x is a rational number and k(x)=1 if x is an irrational number. Prove that the range of k is a finite set, but that k is not a step function.
Our first theorem in this chapter is a fundamental consistency result
about the “area under the graph” of a step function.
Of course, the graph of a step function looks like a collection of horizontal line segments,
and the region under this graph is just a collection of rectangles.
Actually, in this remark, we are
implicitly thinking that the values 
of the step function
are positive. If some of these values are negative, then
we must re-think what we mean by the area under the graph.
We first introduce the following bit of notation.
Let h be a step function on the closed interval [a,b].
Suppose 
is a partition of [a,b] such that h(x)=ai on the interval
Define the weighted average of hrelative toP
to be the number SP(h) defined by
REMARK Notice the similarity between the formula for a weighted average and the formula for a Riemann sum. Note also that if the interval is a single point, i.e., a=b, then the only partition P of the interval consists of the single point x0=a, and every weighted average SP(h)=0.
The next theorem is not a surprise, although its proof takes some careful thinking. It is simply the assertion that the weighted averages are independent of the choice of partition.
Let h be a step function on the closed interval [a,b].
Suppose 
is a partition of [a,b] such that h(x)=ai on the interval
and suppose 
is another partition of [a,b] such that h(x)=bj on the
interval 
Then the weighted average of h relative to P
is the same as the weighted average of h relative to Q. That is,
SP(h)=SQ(h).
Suppose first that the partition Q is obtained from the partition P by adding one additional point. Then m=n+1, and there exists an i0 between 1 and n–1 such that
for 0≤i≤i0 we have yi=xi.
xi0<yi0+1<xi0+1.
For i0<i≤n we have xi=y