Rad 27:
Rad 27: {{clear}}
{{clear}}
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GeoGebra om Riemannsumma in här
=== Övning Riemannsumma i GGb===
=== Övning Riemannsumma i GGb===
Rad 59:
Rad 59: <script type='text/javascript' src='http://demonstrations.wolfram.com/javascript/embed.js' ></script><script type='text/javascript'>var demoObj = new DEMOEMBED(); demoObj.run('NewtonsIntegrabilityProof', '', '549', '620');</script><div id='DEMO_NewtonsIntegrabilityProof'><a class='demonstrationHyperlink' href='http://demonstrations.wolfram.com/NewtonsIntegrabilityProof/' target='_blank'>Newton's Integrability Proof</a> from the <a class='demonstrationHyperlink' href='http://demonstrations.wolfram.com/' target='_blank'>Wolfram Demonstrations Project</a> by Michael Rogers</div>
<script type='text/javascript' src='http://demonstrations.wolfram.com/javascript/embed.js' ></script><script type='text/javascript'>var demoObj = new DEMOEMBED(); demoObj.run('NewtonsIntegrabilityProof', '', '549', '620');</script><div id='DEMO_NewtonsIntegrabilityProof'><a class='demonstrationHyperlink' href='http://demonstrations.wolfram.com/NewtonsIntegrabilityProof/' target='_blank'>Newton's Integrability Proof</a> from the <a class='demonstrationHyperlink' href='http://demonstrations.wolfram.com/' target='_blank'>Wolfram Demonstrations Project</a> by Michael Rogers</div>
</html>
</html>
== Mer om integraler ==
{{#ev:youtube|OAN8qa-pnIo|340|left}} {{#ev:youtube|i8JPiQ3Ujyc|340|right}}
{{clear}}
[http://www.proofwiki.org/wiki/Fundamental_Theorem_of_Calculus 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 ===
[[File:Circle-calc-area.svg|left|200px]]{{clear|left}}
Om cirkelskivan delas upp i koncentriska ringar med omkretsen <math>2\pi t</math> kan arean beräknas med integralen
:<math>A = \int_0^{r} 2 \pi t \, dt = \left[ 2\pi \frac{t^2}{2} \right]_{0}^{r} = \pi r^2</math>
{{svwp | cirkel}}
Versionen från 18 april 2016 kl. 22.01
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.
GeoGebra om Riemannsumma in här
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