Rechenregeln für Integrale

Einführung
[justify]Integralrechnung dient der Ermittlung von Flächenbilanzen oder der Flächenberechnung zwischen einem Graphen einer Funktion und der x-Achse sowie der Strecken- und Massenberechnung.[br][br]In dem folgenden Projekt beschäftigen wir uns mit den [b]Rechenregeln[/b] [b]von Integralen[/b] aus dem "[i]Skript Anallysis 1 und Lineare Algebra für Ingenieurswissenschaften"[/i] (Kap. 30.2, S.238, Oktober 2018). Wir berücksichtigen in diesem Prokjekt nicht die tatsächlichen Flächen, sondern nur die Flächenbilanzen. Das bedeutet, dass negative Flächen mit positiven verrechnet werden, also gilt [b]F (b) - F(a)[/b].[br][br][/justify]
Rechenregel 1: Allgemeine Vereinbarung
[justify][br][img]data:image/png;base64,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[/img][br][br]Das[color=#6aa84f] [b][color=#000000]Applet 1[/color][/b][/color] zeigt uns die Cosinus-Funktion und die dazugehörige Fläche des Integrals in den Intervallgrenzen [a,b] = [-3,3]. Die "Allgemeine Vereinbarung" besagt, wenn wir ein Integral mit gleicher oberer und unterer Integrationsgrenze berechnen, so ergibt der Flächeninhalt Null. [br][/justify]
[b][color=#3c78d8]Forscher*innenauftrag 1 - Allgemeine Vereinbarung[/color][/b][br]Erforsche diesen Sachverhalt und schiebe die Schieberegler a und b an Applet 1 an einer beliebigen Stelle genau übereinander. Was zeigen dir die Flächenbilanzen an den unterschiedlichen Stellen?
Applet 1
[color=#3c78d8][b]Forscher*innenauftrag 2 - Allgemeine Vereinbarung[/b][/color][br]Wir wissen, dass wir die Fläche zwischen unserem Funktionsgraphen und der x-Achse mit Rechtecken approximieren (Riemannsche Summe) und die Summe aller Flächenrechtecke uns nahezu den exakten Wert des [br]Flächeninhaltes liefert. Versuche mit diesem Wissen den Sachverhalt geometrisch zu erklären.
Rechenregel 2: Für a < b
[br][img]data:image/png;base64,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[/img][br][br]Bei einem bestimmten Integral ist die obere Integrationsgrenze [b]b[/b] immer kleiner als die untere[b] a[/b]. Wenn jetzt aber [b]a < b[/b], dann besagt diese Rechenregel, dass wir vor das Integral ein [b]Minus-Zeichen[/b] setzen müssen, um die richtige Flächenbilanz zu erhalten. Allgemein gilt [b]F(a) - F(b)[/b].[br][br]Wir überprüfen das formal:[br][b] [br]      F(a) - F(b)[br]     = -F(b) + F(a)[br]     = -(F(b) - F(a)) = -[/b][math]\int[/math][b]f(x)[br][br][/b]Das Umstellen der Formel zeigt, dass diese Rechenegel rechnerisch korrekt ist. Sie ist sehr wichtig und bildet die Grundlage für die nächste Rechenregel, die Linearität, auch Produktregel gennant.
Rechenregel 3.1: Linearität (Produktregel)
[br][br][img]data:image/png;base64,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[/img][br][br]Wenn eine gegebene Funktion f(x), wie in [b]Applet 2[/b], einen konstanten Faktor [math]\lambda[/math] enthält - hier unsere Funktion [math]\lambda[/math]*g(x) - ist es erlaubt diese Konstante vor das Integral zu ziehen. Man kann dies mit dem Ausklammern einer Konstante Vergleichen. Diese Regel wird auch [b]Produktregel [/b]genannt.[br][br]
[color=#3c78d8][b]Forscher*innenauftrag 1 - Linearität (Produktregel)[/b][/color][br]Schau dir die Funktion f(x) an und ermittle mit den Schiebereglern die Flächenbilanz im Intervall [-3,3]. Jetzt verschiebe den Schieberegler um einzelne positive oder negative Zahlenwerte. Notiere dir die einzelnen Werte [br]und vergleiche sie untereinander. Was fällt dir bei den Werten der Flächenbilanzen auf?
Applet 2
[size=150][size=100][size=100][color=#3c78d8][b]Forscher*innenauftrag 2 - Linearität (Produktregel)[/b][/color][br]Überprüfe die Eigenschaft der Linearität von Integralen mit Hilfe der Produktregel rechnerisch. [/size][/size][size=100][size=100] [br]Berechne das unten stehende Integral auf zwei Lösungswegen: Bei der 1. Rechnung integrierst Du das gesamte Integral, bei der 2. Rechnung ziehst du den Faktor vor das Integral und multiplizierst ihn zum Schluss mit Deinem Ergebnis.[br][/size][/size][/size][size=100][size=100][color=#1155Cc][br]Berechne[/color]:[br]   [img]data:image/png;base64,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[/img][/size][/size]
Rechenregel 3.2: Linearität (Summenregel)
[br][img]data:image/png;base64,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[/img][br][br][size=100]Die abgebildete Rechenregel beschreibt die "[i]Linearität[/i]" von bestimmten Integralen. Das bedeutet, dass man ein Integral mit einer Summe in zwei einzelne Integrale aufsplitten und diese nach dem einzelnen Integrieren addieren darf. Gleiches gilt bei einer Subtraktion. Diese Regel wird daher auch als [b]Summenregel [/b]bezeichnet.[/size]
[size=100][color=#3c78d8][b]Forscher*innenauftrag 1 - Linearität (Summenregel)[/b][/color][br]Das [b]Applet 3[/b] zeigt Dir die Funktionen f(x), g(x) und h(x). [size=100]Die Summe aus den Funktionen f(x) und g(x) ist h(x). [/size][br][br]Blende die Funktion h(x) ein und ermittle die Flächenbilanz im Intervall von [-3,3]. [br][br][/size][size=100]Gemäß der Rechenregel, dass die beiden einzeln integrierten Funktionen f(x) und g(x) die gleiche Flächenbilanz ergeben blende nun h(x) aus und ermittle die beiden Flächenbilanzen von f(x) und g(x). Addiere beide Flächenbilanzen und vergleiche das Ergebnis mit der Flächenbilanz von h(x). Was hast du herausgefunden?[/size]
Applet 3
[size=100][size=100][b][color=#3c78d8]Forscher*innenauftrag 2 - Linearität (Summenregel)[/color][/b][br]Überprüfe die Eigenschaft der Linearität von Integralen rechnerisch. Berechne mit Hilfe der Summenregel das unten stehende Integral auf beiden Lösungswegen. D.h. zuerst integrierst Du das gesamte Integral, das zweite Mal splittest du die Summe in zwei Einzelintegrale auf. Du kannst Dich selbst überprüfen, denn beide Ergebnisse müssen gemäß der Summenregel übereinstimmen.[br][br][color=#1155Cc]Berechne:[/color][br]        [img]data:image/png;base64,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[/img][/size][/size]
Rechenregel 4: Monotonie
[br][img]data:image/png;base64,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[/img][br][br]Diese Rechenrgel besagt, wenndie[b] Funktion f(x) kleiner/ gleich g(x)[/b] für alle x Element [a,b], [br]dann ist auch das berechnete Integral bzw. die [b]Fläche von f(x) kleiner/ gleich g(x)[/b].        
[color=#3c78d8][b]Forscher*innenauftrag 1 - Monotonie[/b][/color][br]Um das zu verstehen, betrachte in [b]Applet 4[/b] die Funktion k(x) = ln(x) und die Flächenbilanz im Intervall von [1,8], ln(x) ˃ 0. Checke nun die Flächenbilanz von h(x)=λ [math]\frac{1}{2}x[/math] im gleichen Intervall mit λ ˃1.[br]Wie verhalten sich die Flächenbilanzen beider Funktionen zueinander?[br]
Applet 4
[b][color=#1e84cc]Forscher*innenauftrag 2 - Monotonie[/color][/b][br]Was passiert, wenn Du den Schieberegler für λ in Applet 4 in den negativen Bereich verschiebst?
Rechenregel 5: Dreiecksungleichung
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[/img][br][br]Die Rechenregel "[i]Dreiecksungleichung[/i]" besagt, dass es einen Unterschied macht, ob wir das gesamte Integral oder nur die Funktion in Betrag nehmen. Das in Betrag gesetzte Integral ist immer [math]\le[/math] als die in [br]Betrag gesetzte Funktion.[br][br][img]data:image/png;base64,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[/img][size=50][br]Quelle: Skript Mathematik für Ingenieure[/size] [size=50]Uni Kiel[/size][br][br]Beim Integral auf der linken Seite (Abb. links) werden die beiden äußeren Flächen positiv gezählt und die mittlere Fläche wird abgezogen, und die entstehende Summe ist kleiner als die Summe aller drei positiv gerechneten Flächen auf der rechten Seite (Abb. rechts). Dies begründet eigentlich nur [img]data:image/png;base64,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[/img], aber bei [img]data:image/png;base64,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[/img] werden die beiden äußeren von der mittleren Fläche abgezogen, und wir haben wieder dieselbe Ungleichung. Dies lässt sich formal mit der Rechenregel 2 [b]Monotonie [/b]herleiten.[br][color=#3c78d8][b][br]Forscher*innenauftrag 1 - Dreiecksungleichung[/b][/color][br]Erforsche diesen Sachverhalt der Dreiecksungleichung am [b]Applet 5[/b]. Achte darauf, dass du die Regler der Integrationsgrenzen gemäß oberer und unterer Integrationsgrenze richtig wählst. Beim Applet bedeutet das, dass die untere Integrationsgrenze a immer größer als b sein muss. Schau Dir die Flächenbilanzen von c und d an. Laut Rechenregel muss c kleiner/ gleich d sein. Überprüfe diesen Sachveralt.
Applet 5
[color=#3c78d8][b]Forscher*innenauftrag 2 - Dreiecksungleichung[/b][/color][br]Stimmt das denn in jedem Fall? Was passiert, wenn wir die Integrationsgrenzen vertauschen?[br]Vertausche die Integrationsgrenzen und schau dir das Flächenverhältnis von [color=#BF9000][b]c[/b][/color] zu [color=#0000ff][b]d[/b][/color] an. Was stellst du fest?[br][br]Nach unserer Regel müsste doch [color=#BF9000][b]c[/b][/color] [b]kleiner/ gleich[/b] [color=#0000ff][b]d[/b][/color] sein.[br][br][b]Tipp[/b]: Lässt sich hier vielleicht eine Beziehung zu Integrationsregel 2 [b]für a [/b][b]< [/b][b]b[/b] herstellen?[br][br]
Close

Information: Rechenregeln für Integrale