# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 18

In the following lectures, we will be concerned with *integration theory*, i.e. we want to study and to compute the area of a surface which is bounded by the graph of a function
(the *integrand*)

and the -axis. At the same time, there is a direct relation with finding a *primitive function* of , these are functions such that their derivative equals . The concept of the area of a surface itself is problematic, which is understood thoroughly within *measure theory*. However, this concept can be understood from an intuitive perspective, and we will only use some basic facts. These are only used for motivation, and not as arguments. The starting point is that the area of a rectangle is just the product of the side lengths, and that the area of a surface, which one can *exhaust* with rectangles, equals the sum of the areas of these rectangles. We will work with the *Riemann integral*, which provides a satisfactory theory for continuous functions. Here, all rectangles are parallel to the coordinate system, their width
(on the -axis)
can vary and their height
(length)
is in relation to the value of the function over the base. By this method, the functions are approximated by so-called *step functions*.

*Step functions*

Let be a real interval with endpoints . Then a function

is called a *step function*, if there exists a partition

of such that is constant

on every open interval .This definition does not require a certain value at the partition points. We call the interval the -th interval of the partition, and is called the length of this interval. If the lengths of all intervals are constant, then the partition is called an *equidistant partition*.

Let be a real interval with endpoints , and let

denote a step function for the partition , with the values , . Then

*step integral*of on .

We denote the step integral also by . If we have an equidistant partition of interval length , then the step integral equals . The step integral does not depend on the partition chosen. As long as we have a step function with respect to the partition, one can pass to a refinement of the partition.

Let denote a bounded interval, and let

denote a function. Then a step function

is called a *step function from above* for , if
holds for all
.
A step function

is called a *step function from below* for , if
holds for all

A step function from above (below) for exists if and only if is bounded from above (from below).

Let denote a bounded interval, and let

denote a function. For a step function from above

of , with respect to the partition , , and values , , the step integral

*step integral from above*for on .

Let denote a bounded interval, and let

denote a function. For a step function from below

of , with respect to the partition , , and values , , the step integral

*step integral from below*for on .

Different step functions from above yield different step integrals from above.

For further integration concepts, we need the following definitions which refer to arbitrary subsets of the real numbers.

For a nonempty subset
,
an
upper bound
of is called the *supremum* of , if

For a nonempty subset
,
a
lower bound
of is called the *infimum* of , if

The existence of infimum and supremum follows from the completeness of the real numbers.

Every nonempty subset of the real numbers, which is bounded from above, has a supremum

in .### Proof

Let denote a bounded interval, and let

denote a function, which is bounded from above. Then the infimum of all step integrals of step functions from above

of is called the*upper integral*of .

Let denote a bounded interval, and let

denote a function, which is bounded from below. Then the supremum of all step integrals of step functions from below

of is called the*lower integral*of .

The boundedness from below makes sure that there exists at all a step function from below, so that the set of step integrals from below is not empty. This condition alone does not guarantee that a supremum exists. However, if the function is bounded from both sides, then the upper integral and the lower integral exist. If a partition is given, then there exists a smallest step function from above (a largest from below) which is given by the suprema (infima) of the function on the intervals of the partition. For a continuous function on a closed interval, these are maxima and minima. To compute the integral, we have to look at all step functions for all partitions.

*Riemann-integrable functions*

In the following, we will talk about compact interval, which is just a bounded and closed interval, hence of the form with .

Let denote a compact interval and let

denote a
function.
Then is called *Riemann-integrable* if the
upper integral
and the
lower integral

It might by historically more adequate to call this *Darboux-integrable*.

Let denote a compact interval. For a Riemann-integrable function

we call the
upper integral
of
(which by definition coincides with the
lower integral)
the *definite integral* of over . It is denoted by

The computation of such integrals is called to *integrate*. Don't think too much about the symbol . It expresses that we want to integrate with respect to this variable. The name of the variable is not relevant, we have

Let denote a compact interval, and let

denote a function. Suppose that there exists a sequence of lower step functions with and a sequence of upper step functions with . Suppose furthermore that the corresponding sequences of step integrals converge to the same real number. Then is Riemann-integrable, and the definite integral equals this limit, so

### Proof

We consider the function

which is strictly increasing in this interval. Hence, for a subinterval , the value is the minimum, and is the maximum of the function on this subinterval. Let be a positive natural number. We partition the interval into the subintervals , , of length . The step integral for the corresponding lower step function is

(see Exercise 2.10 for the formula for the sum of the squares). Since the sequences and converge to , the limit for of these step integrals equals . The step integral for the corresponding step function from above is

The limit of this sequence is again . By Lemma 18.13 , the upper integral and the lower integral coincide, hence the function is Riemann-integrable, and for the definite integral we get

Let be a compact interval, and let

- The function is Riemann-integrable.
- There exists a partition , such that the restrictions are Riemann-integrable.
- For every partition , the restrictions are Riemann-integrable.

### Proof

Let
be a function on a
real interval.
Then is called *Riemann-integrable*, if the
restriction
of to every
compact
interval
is

Due to this lemma, both definitions coincide for a compact interval . The integrability of a function does not mean that has a meaning or exists.

*Riemann-integrability of continuous functions*

Let denote a continuous function. Then is

Riemann-integrable.### Proof

Let denote a compact interval, and let denote Riemann-integrable

functions. Then the following statements hold.- If holds for all , then holds.
- If holds for all , then holds.
- The sum is Riemann-integrable, and the identity holds.
- For we have .
- The functions and are Riemann-integrable.
- The function is Riemann-integrable.
- The product is Riemann-integrable.

For (1) to (4) see Exercise 18.14 . For (5) see Exercise 18.17 . (6) follows directly from (5), because of . For (7), see Exercise 18.18 .

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