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Zuletzt geändert von Holger Engels am 2026/10/17 14:38
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Inhalt
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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 -== 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  
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33 33  {{/aufgabe}}
34 34  
35 35  {{aufgabe id="Test-Rakete" afb="II" kompetenzen="K3,K4" quelle="Holger Engels" zeit="9"}}
36 -Bei der Erprobung eines neuen Raketenantriebs werden verschiedene Flugprofile getestet. Bereits eingezeichnet in das Geschwindigkeits-Zeit-Diagramm ist ein unrealistisches Flugprofil, bei dem die Rakete schlagartig auf ihre Reisegeschwindigkeit kommt und diese die gesamten 20 Sekunden konstant hält.
41 +Bei der Erprobung eines neuen Raketenantriebs werden verschiedene Flugprofile getestet. Die Telemetrie wertet dabei aus, wie sich die Geschwindigkeit //v// (in m/s) über die Zeit //t// (in s) verändert (Geschwindigkeits-Zeit-Diagramm). Der Flächeninhalt unter dem Graphen der Geschwindigkeit entspricht dabei dem zurückgelegten Weg //s// (in m).
37 37  
38 -[[image:Rakete.svg||width=600]]
39 -Der Flächeninhalt unter dem Graphen der Geschwindigkeit entspricht dem zurückgelegten Weg //s// (in m).
43 +Bereits eingezeichnet ist ein unrealistisches Flugprofil, bei dem die Rakete schlagartig auf ihre Reisegeschwindigkeit kommt und diese die gesamten 20 Sekunden konstant hält.
40 40  
41 -Zeichne für die beiden folgenden Flugprofile jeweils einen möglichen Graphen in das Koordinatensystem ein. Allen drei Flugprofilen ist die Dauer von 20 Sekunden und die Strecke von 600 Metern gemein.
45 +[[image:Rakete.svg||width=600]]
46 +Zeichne für die beiden folgenden Flugprofile jeweils einen möglichen Graphen der Geschwindigkeit //v(t)// in das Koordinatensystem ein. Allen drei Flugprofilen ist die Dauer von 20 Sekunden und die Strecke von 600 Metern gemein.
42 42  (%class=abc%)
43 43  1. Konstante Beschleunigung: Die Rakete startet aus dem Stand (//v(0) = 0//) und beschleunigt für 20 Sekunden mit einer konstanten Rate. Die Geschwindigkeit nimmt also linear zu.
44 44  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.
45 45  {{/aufgabe}}
46 46  
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:
49 -
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}}
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//.
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.
73 -{{/aufgabe}}
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 -)))
56 +Ermittle eine aufgsteigende Reihenfolge für den Wert der folgenden Integrale
57 +(%class=abc%)
58 +1. {{formula}}\int_{-5}^{5} e^{x+4} \, dx{{/formula}}
59 +1. {{formula}}\int_{-5}^{5} e^{x-4} \, dx{{/formula}}
60 +1. {{formula}}\int_{-5}^{5} (-x^5 + 4x^3-2x)\, dx{{/formula}}
61 +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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51 - <text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="811" y="396" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">19</text>
52 - <text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="842" y="396" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">20</text>
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54 -
55 - <!-- Y-Achsen-Zahlen -->
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57 - <text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="201" y="470" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">–2</text>
58 - <text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="201" y="428" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">–1</text>
59 - <text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="207" y="343" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">1</text>
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61 - <text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="207" y="258" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">3</text>
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Streifenmethode.ggb
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Streifenmethode.svg
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