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