Änderungen von Dokument BPE 13.1 Bestandsrekonstruktion und Orientierter Flächeninhalt
Zuletzt geändert von Holger Engels am 2026/10/17 14:38
Von Version 60.1
bearbeitet von Holger Engels
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am 2026/10/12 10:17
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... ... @@ -12,8 +12,13 @@ 12 12 Siehe dazu [[Rekonstruktion einer Größe>>https://kmap.eu/app/browser/Mathematik/Integralrechnung/Rekonstruktion%20einer%20Gr%C3%B6%C3%9Fe]] und [[Obersumme/Untersumme interaktiv>>https://kmap.eu/app/browser/Mathematik/Integralrechnung/Obersumme%20und%20Untersumme#erkunden]] 13 13 {{/lernende}} 14 14 15 -== Bestandsrekonstruktion == 15 +--Deutung des bestimmten Integrals (Bestandsrekonstruktion, z. B. zurückgelegte Strecke bei veränderlicher Geschwindigkeit, Fläche)-- 16 +--gekrümmte Randfunktion-- 17 +--Näherungsweise Berechnung von Integralen (z. B. Kästchenzählen, Ober- und Untersumme)-- 18 +Orientierter Flächeninhalt (z. B. Zu- und Abflussmenge) 19 +Eigenschaften des bestimmten Integrals (Intervalladditivität, Linearität des Integrals, Integralwert ist Null bei ungeraden Funktionen und zu x=0 symmetrischen Integrationsgrenzen) 16 16 21 +== Bestandsrekonstruktion == 17 17 {{aufgabe id="Gefäß" afb="I" kompetenzen="K4" quelle="[[KMap>>https://kmap.eu/app/browser/Mathematik/Integralrechnung/Rekonstruktion%20einer%20Gr%C3%B6%C3%9Fe]]" zeit="5"}} 18 18 Ein Gefäß sei zu Beginn der Beobachtung mit //10 l// gefüllt. Es wird folgender Zufluss/ Abluss beobachtet: 19 19 ... ... @@ -45,68 +45,33 @@ 45 45 {{/aufgabe}} 46 46 47 47 {{aufgabe id="Streifenmethode" afb="II" kompetenzen="" quelle="Holger Engels, Stefanie Walz" zeit=""}} 48 -[[ image:Streifenmethode.svg||width=250 class=right]]Das Schaubild zeigt die Normalparabel {{formula}}f(x)= x^2{{/formula}}und die Streifen der Untersumme für //n=4// Streifen und einer Streifenbreite von {{formula}}\frac{b}{n}=\frac{1}{4}{{/formula}}.Allgemeinbestimmtsichsichdie Untersumme für die Streifen folgendermaßen:53 +[[Streifenmethode.svg||width=300 class=right]]Das Schaubild zeigt die Normalparabel {{formula}}f(x)= x^2{{/formula}}. Für //n// Streifen und einer Streifenbreite von {{formula}}\frac{b}{n}{{/formula}} ergibt sich die Untersumme folgendermaßen: 49 49 50 50 {{formula}} 51 -\begin{align ed}52 -U_n &= \Delta x \cdot f(0\cdot\Delta x) + \Delta x \cdot f(1\cdot\Delta x) + ... + \Delta x \cdot f((n-1)\cdot\Delta x)\\53 - \text{}&= \Delta x \cdot (0\cdot \Delta x)^2 + \Delta x \cdot (1\cdot \Delta x)^2 + ... + \Delta x \cdot ((n-1)\cdot \Delta x)^254 -\end{align ed}56 +\begin{align*} 57 +U_n = \Delta x \cdot f(1\cdot\Delta x) + \Delta x \cdot f(2\cdot\Delta x) + ... + \Delta x \cdot f(n\cdot\Delta x) 58 +&= \Delta x \cdot (1\cdot \Delta x)^2 + \Delta x \cdot (2\cdot \Delta x)^2 + ... + \Delta x \cdot (n\cdot \Delta x)^2 59 +\end{align*} 55 55 {{/formula}} 56 - 57 -.. führt nach einigen Umformungen schließlich zu: 58 - 59 -{{formula}}U_n = b^3\frac16(2-\frac3n+\frac1{n^2}){{/formula}} 60 - 61 -Analog ergibt sich für die Obersumme: 62 - 63 -{{formula}}O_n = b^3\frac16(2+\frac3n+\frac1{n^2}){{/formula}} 64 - 65 -(%class=abc%) 66 -1. (((Berechne den Wert dieser Summen für unterschiedliche //n//. 61 +... führt schließlich zu ... 62 +{{formula}}U_n = b^3\frac16(2+\frac3n+\frac1{n^2}){{/formula}} 63 +Berechne den Wert der Summe für unterschiedliche //n//. 67 67 (%class="border slim"%) 68 -|= //n//|4|8| 10065 +|=n| 4 | 8 | 100 69 69 |={{formula}}U_n{{/formula}}||| 70 -|={{formula}}O_n{{/formula}}||| 71 -))) 72 -1. Betrachte die Terme und begründe, gegen welchen Wert beide für {{formula}}n\rightarrow\infty{{/formula}} streben. 67 +Betrachte den Term und begründe, gegen welchen Wert er für {{formula}}n\rightarrow\infty{{/formula}} strebt. 73 73 {{/aufgabe}} 74 74 75 75 == Bestimmtes Integral == 76 76 77 -{{aufgabe id="Bestimmtes Integral" afb="I" kompetenzen="K5" quelle="Holger Engels" zeit="7"}} 78 -Bestimme den Wert des Integrals: 79 -(%class="abc horiz"%) 80 -1. ((({{formula}}\int_0^\pi \sin(x) \, dx{{/formula}}))) 81 -1. ((({{formula}}\int_0^{2\pi} \sin(x) \, dx{{/formula}}))) 82 -1. ((({{formula}}\int_0^1 e^x \, dx{{/formula}}))) 83 -{{/aufgabe}} 84 - 85 -{{aufgabe id="Bestimmtes Integral vs Fläche" afb="I" kompetenzen="K1, K6" quelle="Holger Engels" zeit="5"}} 86 -Gegeben ist der Graph einer Funktion //f//, der die x-Achse bei {{formula}}x = 2{{/formula}} schneidet. Die Teilfläche unterhalb der x-Achse im Intervall {{formula}}[0; 2]{{/formula}} beträgt 3 FE, die Teilfläche oberhalb im Intervall {{formula}}[2; 5]{{/formula}} beträgt 7 FE. 87 -(%class=abc%) 88 -1. Bestimme den Wert des Integrals {{formula}}\int_{0}^{5} f(x)\,dx{{/formula}}. 89 -1. Bestimme den geometrischen Flächeninhalt, den der Graph von f mit der x-Achse über {{formula}}[0; 5]{{/formula}} einschließt. 90 -1. Erkläre den Unterschied zwischen dem Wert des bestimmten Integrals und dem Flächeninhalt zwischen Graph und x-Achse. 91 -{{/aufgabe}} 92 - 93 93 == Eigenschaften == 94 - 95 -{{aufgabe id="Eigenschaften erläutern" afb="II" kompetenzen="K6" quelle="" zeit="6"}} 96 -Veranschauliche und begründe anhand einer Skizze oder anhand der Flächenbilanz, warum folgende Rechenregeln für stetige Funktionen gelten: 97 -(%class="abc horiz"%) 98 -1. ((({{formula}}\int_{a}^{b} f(x)\,dx + \int_{b}^{c} f(x)\,dx = \int_{a}^{c} f(x)\,dx{{/formula}}))) 99 -1. ((({{formula}}\int_{a}^{b} c \cdot f(x)\,dx = c \cdot \int_{a}^{b} f(x)\,dx{{/formula}}))) 100 -{{/aufgabe}} 101 - 102 102 {{aufgabe id="Integraleigenschaften" afb="II" kompetenzen="K1" quelle="Debora Kemm, Benjamin Kaiser" zeit="10"}} 103 -Ermittle eine aufsteigende Reihenfolge für den Wert der folgenden Integrale 104 -(% style="display:flex"%)((( 105 -((({{formula}}\int_{-5}^{5} e^{x+4} \, dx{{/formula}}))) 106 -((({{formula}}\int_{-5}^{5} e^{x-4} \, dx{{/formula}}))) 107 -((({{formula}}\int_{-5}^{5} (-x^5 + 4x^3-2x)\, dx{{/formula}}))) 108 -((({{formula}}\int_{-5}^{5} (\cos(\frac{1}{2}x)-3)\, dx{{/formula}}))) 109 -))) 74 +Ermittle eine aufgsteigende Reihenfolge für den Wert der folgenden Integrale 75 +(%class=abc%) 76 +1. {{formula}}\int_{-5}^{5} e^{x+4} \, dx{{/formula}} 77 +1. {{formula}}\int_{-5}^{5} e^{x-4} \, dx{{/formula}} 78 +1. {{formula}}\int_{-5}^{5} (-x^5 + 4x^3-2x)\, dx{{/formula}} 79 +1. {{formula}}\int_{-5}^{5} (\cos(\frac{1}{2}x)-3)\, dx {{/formula}} 110 110 {{/aufgabe}} 111 111 112 112 {{aufgabe id="Bilanz null" afb="II" kompetenzen="K2, K5" quelle="Holger Engels" zeit="8"}}
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