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
A step function. In the statistic context, it is also called a histogram or a bar chart.

Definition  

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.


Definition  

Let be a real interval with endpoints , and let

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

is called the 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 Failed to parse (syntax error): {\displaystyle {{}} \frac{b-a}{n} { \left( \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 18]] __NOINDEX__ 1}^n t_i \right) }} . 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.


Definition  

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).


Definition  

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

is called a step integral from above for on .

Definition  

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

is called a 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.


Definition  

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

holds for all upper bounds of .

Definition  

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

holds for all lower bounds of .

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


Theorem

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

Proof

This proof was not presented in the lecture.

Definition  

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 .

Definition  

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 .

A step function from above and one from below. The green area is a step integral from below and the yellow area (partly covered) is a step function from above.

Definition  

Let denote a compact interval and let

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

of exist and coincide.

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


Definition  

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

Failed to parse (syntax error): {\displaystyle f \colon I <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 18]] __NOINDEX__ [a,b] \longrightarrow \R , t \longmapsto f(t) , }

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


Lemma

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



Example

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

Failed to parse (syntax error): {\displaystyle {{}} \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 18]] __NOINDEX__ 0}^{n-1} \frac{1}{n} { \left(i \frac{1}{n}\right) }^2 = \frac{1}{n^3} \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 18]] __NOINDEX__ 0}^{n-1} i^2 = \frac{1}{n^3} { \left( \frac{1}{3} n^3 - \frac{1}{2}n^2 + \frac{1}{6} n\right) } = \frac{1}{3} - \frac{1}{2n} + \frac{1}{6n^2} \, }

(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

Failed to parse (syntax error): {\displaystyle {{}} \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 18]] __NOINDEX__ 0}^{n-1} \frac{1}{n} { \left((i+1) \frac{1}{n}\right) }^2 = \frac{1}{n^3} \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 18]] __NOINDEX__ 0}^{n-1} (i+1)^2 = \frac{1}{n^3} \sum_{j <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 18]] __NOINDEX__ 1}^{n} j^2 = \frac{1}{n^3} { \left(\frac{1}{3} n^3 + \frac{1}{2}n^2 + \frac{1}{6} n\right) } = \frac{1}{3} + \frac{1}{2n} + \frac{1}{6n^2} \, . }

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


Lemma

Let be a compact interval, and let

be a function. Then the following statements are equivalent.
  1. The function is Riemann-integrable.
  2. There exists a partition , such that the restrictions are Riemann-integrable.
  3. For every partition , the restrictions are Riemann-integrable.
In this situation, the equation
Failed to parse (syntax error): {\displaystyle {{}} \int_{ a }^{ b } f ( t) \, d t = \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 18]] __NOINDEX__ 1}^n \int_{ a_{i-1} }^{ a_i } f_i ( t) \, d t \, }

holds.

Proof



Definition  

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

Riemann-integrable.

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

Theorem

Let denote a continuous function. Then is Riemann-integrable.

Proof

This proof was not presented in the lecture.



Lemma

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

functions. Then the following statements hold.
  1. If holds for all , then holds.
  2. If holds for all , then holds.
  3. The sum is Riemann-integrable, and the identity holds.
  4. For we have .
  5. The functions and are Riemann-integrable.
  6. The function is Riemann-integrable.
  7. The product is Riemann-integrable.

Proof  

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 .



<< | Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I | >>
PDF-version of this lecture
Exercise sheet for this lecture (PDF)