Riemann Sums and the Integral in Edwards & Penney

presentation of Edwards & Penney's work, by presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

[the following will not stray from Edwards & Penney's procedure, so it is largely quotation.]

We take function f as a positive-valued and increasing function that is defined on a set of real numbers including the interval [a, b]. We previously used inscribed and circumscribed rectangles to set up the sums

that approximate the area A under the graph of y = f (x) from x = a to x = b. Recall that the notation in the above equation is based on a division of the interval [a, b] into n subintervals, all with the same length Δx = (b - a)/n, and that

denotes the ith subinterval.

The approximating sums in the above equation both take the form

where

denotes a selected point of the ith subinterval

seen here in the figure below:

Sums of the form

appear as approximations in a wide range of applications and form the basis for the definition of the integral. Building from our previous efforts, we want to define the integral of f from a to b as some sort of limit, as

or sums such as the one in our equation above. The aim is to begin with a fairly general function f and define a computable real number I (the integral of f) that -- in the special case when f is continuous and positive-valued on [a, b] -- equal the area under the graph of y = f (x).

Riemann Sums

We will start with a function f defined on [a, b] that is not necessarily either continuous or positive valued. A partition P of [a, b] is a collection of subintervals

of [a, b] such that

as in the figure above [the a = the x sub zero by itself, and the x sub n = b by themselves, and the rest are between those two equalities]. We write

for the length of the ith subinterval

.

To obtain a sum such as

we need a point

in the ith subinterval for each

A collection of points

with

in

for each i (as in the figure below

) is called a selection for the partition P.

Definition for Riemann Sum

Let f be a function defined on the interval [a, b]. If P is a partition of [a, b] and S is a selection for P, then the Riemann sum for f determined by P and S is

We also say that this Riemann sum is associated with the partition P.

The point

in the above equation is simply a selected point of the ith subinterval

That is, it can be any point of this subinterval. But when we compute Riemann sums, we usually choose the points of the selection S in some systematic manner, as illustrated below:

Here we show different Riemann sums for the function

on the interval [0, 3]. This figure below

shows rectangles associated with the left-endpoint sum

in which each

is selected to be

the left endpoint of the ith subinterval

of length Δx = (b - a)/n. This figure below

shows rectangles associated with the right endpoint sum

in which each

is selected to be

the right endpoint of

In each figure, some of the rectangles are inscribed and others are circumscribed.

This figure below

shows rectangles associated with the midpoint sum

in which

the midpoint of the ith subinterval

.

The dashed lines in the above figure represent the ordinates of f at these midpoints.

Example 1:

In the example from this entry we calculated left- and right-endpoint sums for

on [0, 3] with n = 10 subintervals. Now we will do this more concisely by using summation notation, and we also calculate the analogous midpoint sum. The figure below

shows a typical approximating rectangle for each of these sums. With a = 0, b = 3, and

we see that the ith subdivision point is

The ith subinterval, as well as its midpoint

are shown in the figure below

With

we obtain the left-endpoint sum in the equation

[The authors seemed to have used the above equation 8 to simply the arithmetic, although they could also have summed all ten squares as well.]

With

we get the right-endpoint sum in this equation

At last, with

We see that the midpoint sum is much closer than either endpoint sums to the actual 9 value (of the area under the graph of

that we found in the example in this entry.

The Integral as a Limit

In the case of a function f that has both positive and negative values on [a, b], it is necessary to consider the signs indicated in the figure below

On each subinterval

we have a rectangle with width Δx and "height"

then this rectangle stands above the x-axis; if

it lies below the x-axis. The Riemann sum R is then the sum of the signed areas of these rectangles -- that is, the sum of the areas of those rectangles that lie above the x-axis minus the sum of the areas of those that lie below the x-axis.

If the widths of

of these rectangles are all very small, then it appears that the corresponding Riemann sum R will closely approximate the area from x = a to x = b under y = f (x) and above the x-axis, minus the area that lies above the graph and below the x-axis. This suggests that the integral of f from a to b should be defined by taking the limit of the Riemann sums as the widths

all approach zero:

The formal definition of the integral is obtained by saying precisely what it means for this limit to exist. The norm of the partition P is the largest of the lengths

of the subintervals in P and is denoted by |P|. Briefly, the equation

means that if |P| is sufficiently small, then all Riemann sums associated with the partition P are close to the number I.

Definition: The Definite Integral

The definite integral of the function f from a to b is the number

provided that this limit exists, in which case we say that f is integrable on [a, b]. The above equation means that, for each number

there exists a number

such that

for every Riemann sum associated with any partition P of [a, b] for which

The customary notation for the integral of f from a to b, from Leibniz, is:

Considering I to be the area under y = f (x) from a to b, Leibniz first thought of a narrow strip with height f (x) and "infinitesimally small" width dx (as in the figure below

), so that its area would be the product f (x) dx. He regarded the integral as a sum of areas of such strips and denoted this sum by the elongated capital S (for summa) that appears in the above equation.

We will notice that this integral notation is not only highly suggestive, but also is exceedingly useful in manipulations with integrals. The numbers a and b are called the lower limit and upper limit, respectively, of the integral; they are the endpoints of the interval of integration. The function f (x) that appears between the integral sign and dx is called the integrand. The symbol dx that follows the integrand in the above equation should, for now, be thought of as simply an indication of what the independent variable is. Like the index of summation, the independent variable x is a "dummy variable" -- it may be replaced with any other variable without affecting the meaning of the above equation. Thus if f is integrable on [a, b], we can write

The definition given for the definite integral applies only if a <>, but it is convenient to include the cases a = b and a <>as well. The integral is defined in these cases as follows:'

provided that the right-hand integral exists. Thus interchanging the limits of integration reverses the sign of the integral.

Just as not all functions are differentiable, not every function is integrable. Suppose that c is a point of [a, b] such that

is the subinterval of the partition P that contains c, then the Riemann sum in the equation

can be made arbitrarily large by choosing

For our purposes, however, we need to know only that every continuous function is integrable.

Theorem 1 Existence of the Integral

If the function f is continuous on [a, b], then f is integrable on [a, b].

Theorem 2 The Integral as a Limit of a Sequence

The function f is integrable on [a, b] with integral I if and only if

for every sequence

of Riemann sums associated with a sequence of partitions

Riemann Sum Computations

The reformulation in Theorem 2 of the definition of the integral is helpful because it is easier to visualize a specific sequence of Riemann sums than to visualize the vast totality of all possible Riemann sums. In the case of a continuous function f, the situation can be simplified even more by using only Riemann sums associated with partitions consisting of subintervals all with the same length

Such a partition of [a, b] into equal-length subintervals is called a regular partition of [a, b].

Any Riemann sum associated with a regular partition can be written in the form

where the absence of a subscript in Δx signifies that the sum is associated with a regular partition. In such a case the conditions

are equivalent, so the integral of a continuous function can be defined quite simply:

Consequently, we henceforth will use only regular partitions; the subintervals will thus have length and endpoints given by

for i = 0, 1, 2, 3, . . . , n. If we select

then

gives

Example 3

Use Riemann sums to evaluate

With a = 0 and b = 4 in

we have

Thus

[Recall summation rules 1 and 2].

We now use equations

and

to convert each of the last two sums to closed form:

[The 1/n's subtract each other out].

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.298a-304d.