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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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50 50  {{/aufgabe}}
51 51  
52 52  {{aufgabe id="Streifenmethode" afb="II" kompetenzen="" quelle="Holger Engels, Stefanie Walz" zeit=""}}
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:
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 55  {{formula}}
56 56  \begin{aligned}
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
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
59 59  \end{aligned}
60 60  {{/formula}}
61 61  
62 -... führt schließlich zu ...
57 +.. führt nach einigen Umformungen schließlich zu:
63 63  
64 -{{formula}}U_n = b^3\frac16(2+\frac3n+\frac1{n^2}){{/formula}}
59 +{{formula}}U_n = b^3\frac16(2-\frac3n+\frac1{n^2}){{/formula}}
65 65  
66 -Berechne den Wert der Summe für unterschiedliche //n//.
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 67  (%class="border slim"%)
68 -|=n| 4 | 8 | 100
68 +|=//n//| 4 | 8 | 100
69 69  |={{formula}}U_n{{/formula}}|||
70 -Betrachte den Term und begründe, gegen welchen Wert er für {{formula}}n\rightarrow\infty{{/formula}} strebt.
70 +|={{formula}}O_n{{/formula}}|||
71 +)))
72 +1. Betrachte die Terme und begründe, gegen welchen Wert beide für {{formula}}n\rightarrow\infty{{/formula}} streben.
71 71  {{/aufgabe}}
72 72  
73 73  == Bestimmtes Integral ==
74 74  
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 +
75 75  == 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 +
76 76  {{aufgabe id="Integraleigenschaften" afb="II" kompetenzen="K1" quelle="Debora Kemm, Benjamin Kaiser" zeit="10"}}
77 -Ermittle eine aufgsteigende Reihenfolge für den Wert der folgenden Integrale
78 -(%class=abc%)
79 -1. {{formula}}\int_{-5}^{5} e^{x+4} \, dx{{/formula}}
80 -1. {{formula}}\int_{-5}^{5} e^{x-4} \, dx{{/formula}}
81 -1. {{formula}}\int_{-5}^{5} (-x^5 + 4x^3-2x)\, dx{{/formula}}
82 -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 +)))
83 83  {{/aufgabe}}
84 84  
85 85  {{aufgabe id="Bilanz null" afb="II" kompetenzen="K2, K5" quelle="Holger Engels" zeit="8"}}
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50 + <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="780" y="396" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">18</text>
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>
53 + <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="874" y="396" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">21</text>
54 +
55 + <!-- Y-Achsen-Zahlen -->
56 + <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="512" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">–3</text>
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>
60 + <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="301" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">2</text>
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>
62 + <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="396" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">0</text>
63 +
64 + <!-- Funktionsgraph -->
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69 + <!-- Achsenbeschriftungen -->
70 + <text fill="rgb(0,0,0)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="16px" font-style="normal" font-weight="normal" text-decoration="normal" x="227" y="238" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">Q [l/min]</text>
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75 +