Integraler Ma3C: Skillnad mellan sidversioner
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Versionen från 18 april 2016 kl. 21.57
Derivatan av y = e^x
Naturliga logaritmer
Derivatan av 2^x
Problemlösning exponentialfunktioner
Primitiva funktioner
Beräkna integraler
Arean av ett område mellan två kurvor
Mer om integraler
Tillämpningar av integraler
Intro - Riemannsumma
Kan man tänka sig någon trevlig frågeställning som ingång till integralerna?
Börja med att visa Riemannsumman för att ta reda på arean under en graf.
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Övning Riemannsumma i GGb
Uppgift |
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laborera själv i Geogebra
Denna GGB ger dig möjlighet att flytta stapeln och att testa olika funktioner. Du kan ändra på antalet staplar och se hur det påverkar beräkningen. Vad lärde du dig av denna övning? |
uppg 2
Testa denna: http://www.geogebratube.org/student/m11330
Hur hanteras negativa areor?
Uppg 3
Man kan skapa Riemannsummor mellan två funktioner:
Newtons Integralbevis
Mer om integraler
Mekaniken
Jämför med mekaniken, sträckan är arean under en vt-graf.
Tillämpningar - exempel på cirkelns area
Det finns många praktiska tillämpningar av integraler och nedanstående exempel är snarare ett sätt att visa att formeln stämmer. Men tillvägagångssättet är lätt att kopiera till andra områden därför passar det här.
Beräkning av cirkelskivans area med koncentriska skal
![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Circle-calc-area.svg/langsv-200px-Circle-calc-area.svg.png)
Om cirkelskivan delas upp i koncentriska ringar med omkretsen [math]\displaystyle{ 2\pi t }[/math] kan arean beräknas med integralen
- [math]\displaystyle{ A = \int_0^{r} 2 \pi t \, dt = \left[ 2\pi \frac{t^2}{2} \right]_{0}^{r} = \pi r^2 }[/math]