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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

Integral - 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
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.

Här är GGB:n:

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

ProofWiki

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

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]

Wikipedia skriver om cirkel