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Zuletzt geändert von Holger Engels am 2026/10/17 14:38
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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 ---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)
20 -
21 21  == Bestandsrekonstruktion ==
16 +
22 22  {{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"}}
23 23  Ein Gefäß sei zu Beginn der Beobachtung mit //10 l// gefüllt. Es wird folgender Zufluss/ Abluss beobachtet:
24 24  
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49 49  1. Start-, Flug- und Bremsphase: In einem weiteren Testflug startet die Rakete aus dem Stand (//v(0) = 0//), beschleunigt 8 Sekunden lang mit einer konstanten Rate, fliegt dann 4 Sekunden lang mit konstanter Geschwindigkeit und bremst schließlich mit einer konstanten negativen Rate wieder 8 Sekunden lang bis zum Stillstand (//v(20)=0//) ab.
50 50  {{/aufgabe}}
51 51  
52 -{{aufgabe id="Streifenmethode" afb="II" kompetenzen="" quelle="Stefanie Walz" zeit=""}}
53 -[[Freier Fall.svg||width=300 class=right]]Das Schaubild zeigt die Höhe über der Zeit für einen Fall aus 5 Metern Höhe, modelliert durch die Funktion {{formula}}h(t)=-5\cdot t^2+5{{/formula}}. Für //n// Streifen und einer Streifenbreite von {{formula}}\frac{b}{n}{{/formula}} ergibt sich die Untersumme folgendermaßen:
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}}. Allgemein bestimmt sich sich die Untersumme für die Streifen folgendermaßen:
54 54  
55 -{{formula}}U_n = \Delta x \cdot h(1\Delta x) + \Delta x \cdot h(2\Delta x) + ... + \Delta x \cdot h(n\Delta x){{/formula}}
56 -{{formula}}U_n = \Delta x \cdot (-5 \cdot (1\cdot Deltax)^2 +5) + \Delta x \cdot (-5 \cdot (2\cdot Deltax)^2 +5) + ... + \Delta x \cdot (-5 \cdot (n\cdot Deltax)^2 +5){{/formula}}
57 -{{formula}}U_n = -5\cdot b^3\frac16(2+\frac3n+\frac1n^2+5b{{/formula}}
50 +{{formula}}
51 +\begin{aligned}
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)^2
54 +\end{aligned}
55 +{{/formula}}
58 58  
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//.
67 +(%class="border slim"%)
68 +|=//n//| 4 | 8 | 100
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.
59 59  {{/aufgabe}}
60 60  
61 61  == Bestimmtes Integral ==
62 62  
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 +
63 63  == 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 +
64 64  {{aufgabe id="Integraleigenschaften" afb="II" kompetenzen="K1" quelle="Debora Kemm, Benjamin Kaiser" zeit="10"}}
65 -Ermittle eine aufgsteigende Reihenfolge für den Wert der folgenden Integrale
66 -(%class=abc%)
67 -1. {{formula}}\int_{-5}^{5} e^{x+4} \, dx{{/formula}}
68 -1. {{formula}}\int_{-5}^{5} e^{x-4} \, dx{{/formula}}
69 -1. {{formula}}\int_{-5}^{5} (-x^5 + 4x^3-2x)\, dx{{/formula}}
70 -1. {{formula}}\int_{-5}^{5} (\cos(\frac{1}{2}x)-3)\, dx {{/formula}}
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 +)))
71 71  {{/aufgabe}}
72 72  
73 73  {{aufgabe id="Bilanz null" afb="II" kompetenzen="K2, K5" quelle="Holger Engels" zeit="8"}}
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63 +
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Streifenmethode.ggb
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1 +xwiki:XWiki.holgerengels
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1 +49.8 KB
Inhalt
Streifenmethode.svg
Author
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1 +xwiki:XWiki.holgerengels
Größe
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1 +12.6 KB
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