Berechnen Sie die Fläche einer Ellipse, deren Haupt- bzw. Nebenachse die Länge a bzw. b hat.
ist die Ellipsenformel.
Umformen ergibt
Dies ist die Funktion, nach der wir Integrieren.
Wir berechnen die obere Hälfte, danach verdoppeln wir, da beide Hälften die gleiche Fläche haben.
Substitution ![{\displaystyle t={\frac {x}{a}}}](/index.php?title=Spezial:MathShowImage&hash=b71ec10a6b347a3eb28417925bc9f059&mode=mathml)
![{\displaystyle x=ta}](/index.php?title=Spezial:MathShowImage&hash=a17c5d37bbb0236b6c044838fcd0f58e&mode=mathml)
![{\displaystyle dx=a\;dt}](/index.php?title=Spezial:MathShowImage&hash=efe859b0e25904942e35b30ff70fd6f7&mode=mathml)
![{\displaystyle b\int _{-a}^{a}{\sqrt {1-t}}*adt=ab\int _{-a}^{a}{\sqrt {1-t^{2}}}dt}](/index.php?title=Spezial:MathShowImage&hash=7e82f1d4c5707ad1c79038428f03c321&mode=mathml)
Substitution ![{\displaystyle s=\sin ^{-1}{(t)}}](/index.php?title=Spezial:MathShowImage&hash=2faad1061bcde8f2fc50b9a5ec1f6f22&mode=mathml)
![{\displaystyle t=\sin {(s)}}](/index.php?title=Spezial:MathShowImage&hash=a3835d10a1f11456e7ade70f281039c0&mode=mathml)
![{\displaystyle dt=\cos {(s)}ds}](/index.php?title=Spezial:MathShowImage&hash=534337d61e99c34b46f9c3195577bf9f&mode=mathml)
also
Fakt: ![{\displaystyle \cos {(s)}={\sqrt {1-\sin ^{2}{(s)}}}}](/index.php?title=Spezial:MathShowImage&hash=d6e2370df51bb433d6d17f6bc230dca9&mode=mathml)
![{\displaystyle ab\int _{-a}^{a}\cos ^{2}{(s)}=\cos {(s)}*\sin {(s)}-ab\int _{-a}^{a}-\sin {(s)}*\sin {(s)}}](/index.php?title=Spezial:MathShowImage&hash=24abbca559067249c256bc0c0b47ddff&mode=mathml)
![{\displaystyle ab\int _{-a}^{a}\cos ^{2}{(s)}=\cos {(s)}*\sin {(s)}-ab\int _{-a}^{a}-\sin ^{2}{(s)}}](/index.php?title=Spezial:MathShowImage&hash=c496af3d38ca9af228596c8251ee209e&mode=mathml)
![{\displaystyle ab\int _{-a}^{a}\cos ^{2}{(s)}=\cos {(s)}*\sin {(s)}+ab\int _{-a}^{a}\sin ^{2}{(s)}}](/index.php?title=Spezial:MathShowImage&hash=e260ba62de384adb262e18b64e0ae432&mode=mathml)
Fakt: ![{\displaystyle \sin ^{2}{(s)}=1-\cos ^{2}{(s)}}](/index.php?title=Spezial:MathShowImage&hash=6e49b58331731bfdec88d24dbf893d5f&mode=mathml)
![{\displaystyle ab\int _{-a}^{a}\cos ^{2}{(s)}=\cos {(s)}*\sin {(s)}+ab\int _{-a}^{a}1-\cos ^{2}{(s)}}](/index.php?title=Spezial:MathShowImage&hash=f1a13a8edbef2d6dcafe76badbe3b5c0&mode=mathml)
![{\displaystyle ab\int _{-a}^{a}\cos ^{2}{(s)}=\cos {(s)}*\sin {(s)}+ab\int _{-a}^{a}1-ab\int _{-a}^{a}\cos ^{2}{(s)}}](/index.php?title=Spezial:MathShowImage&hash=82161f08f8a2f49b8320c754b4ca5d0d&mode=mathml)
![{\displaystyle 2ab\int _{-a}^{a}\cos ^{2}{(s)}=\cos {(s)}*\sin {(s)}+ab\int _{-a}^{a}1}](/index.php?title=Spezial:MathShowImage&hash=f2b9fc3a344090a128bb429f68edcaf8&mode=mathml)
da wir
berechnen wollten und nicht![{\displaystyle 2ab\int _{-a}^{a}\cos ^{2}{(s)}}](/index.php?title=Spezial:MathShowImage&hash=5cad03847e0fe6478fd1fbc5e9958322&mode=mathml)
Rücksubstitution ![{\displaystyle s=\sin ^{-1}{(t)}}](/index.php?title=Spezial:MathShowImage&hash=2faad1061bcde8f2fc50b9a5ec1f6f22&mode=mathml)
![{\displaystyle \cos {(\sin ^{-1}{(t)})}*\sin {(\sin ^{-1}{(t)})}+ab\sin ^{-1}{(t)}}](/index.php?title=Spezial:MathShowImage&hash=f034926935c517867465b726b44d6e4e&mode=mathml)
Fakt: ![{\displaystyle \cos {(\sin ^{-1}{(t)})}={\sqrt {1-\sin ^{2}{(\sin ^{-1}{(t)})}}}={\sqrt {1-t^{2}}}}](/index.php?title=Spezial:MathShowImage&hash=c04149abcf0e10a74cadf275e770678a&mode=mathml)
![{\displaystyle {\frac {{\sqrt {1-t^{2}}}*t+ab\sin ^{-1}{(t)}}{2}}}](/index.php?title=Spezial:MathShowImage&hash=5a217fc6ddd308e261a18ee7e44d597c&mode=mathml)
Rücksubstitution ![{\displaystyle t={\frac {x}{a}}}](/index.php?title=Spezial:MathShowImage&hash=b71ec10a6b347a3eb28417925bc9f059&mode=mathml)
![{\displaystyle {\frac {{\sqrt {1-{\frac {x^{2}}{a^{2}}}}}*{\frac {x}{a}}+ab\sin ^{-1}{({\frac {x}{a}})}}{2}}\mid _{-a}^{a}={\frac {ab\pi }{4}}-(-{\frac {ab\pi }{4}})={\frac {ab\pi }{2}}}](/index.php?title=Spezial:MathShowImage&hash=284f0694d1c6b49d89c2c4f92a8b9ae6&mode=mathml)
Jetzt verdoppeln wir noch![{\displaystyle 2*{\frac {ab\pi }{2}}=ab\pi }](/index.php?title=Spezial:MathShowImage&hash=d52799399d0a6c05fbd42cf1b29ce2d7&mode=mathml)
--Blueroot 16:26, 15. Mai 2007 (CEST)
Ellipse
Und hier noch die Kurzversion vom Großmeister Karigl:
Substitution in Polarkoordinaten:
mit der Funktionsdeterminante:
(siehe auch Bsp.127b)
--Baccus 12:54, 17. Mai 2007 (CEST)
Wikipädia: