Wegintegrale
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Exercise:
Berechne mittels Wegegral die Länge der folg angegebenen Kurven im gegebenen Intervall. abcliste abc Kreis: displaystyle ktleftcos t sin tright für tinpi. abc Zykloide: displaystyle ztleftrt-sin tr-cos tright für tinpi. abc Archimedische Spirale: displaystyle at leftt cos t t sin t right abc Schraubenlinie/Helix: displaystyle htleftrcospi t rsinpi thtright für tinpi. abcliste
Solution:
abclist abc kt cos t sin t quad t in pi k't -sin t cos t |k't| sqrtsin^ t + cos^ t L _^pi dd t pi abc zt leftrt - sin t r - cos tright z't leftr - cos t r sin t right |z't| r sqrt - cos t^ + sin^ t r sqrt - cos t r sinleftfractright L _^pi r sinleftfractright dd t -r cosleftfractrightBig|_^pi -rcospi - cos -r- - r abc at t cos t t sin t a't cos t - t sin t sin t + t cos t |a't| sqrtcos t - t sin t^ + sin t + t cos t^ sqrt + t^ L _^pi sqrt + t^ ddt &quad textkeine elementare Stammfunktion numerisch lösbar abc ht leftr cospi t r sinpi t ht right h't left-pi r sinpi t pi r cospi t h right |h't| sqrtpi r^ + h^ quad textkonstant L _^pi sqrtpi r^ + h^ dt pi sqrtpi r^ + h^ abclist
Berechne mittels Wegegral die Länge der folg angegebenen Kurven im gegebenen Intervall. abcliste abc Kreis: displaystyle ktleftcos t sin tright für tinpi. abc Zykloide: displaystyle ztleftrt-sin tr-cos tright für tinpi. abc Archimedische Spirale: displaystyle at leftt cos t t sin t right abc Schraubenlinie/Helix: displaystyle htleftrcospi t rsinpi thtright für tinpi. abcliste
Solution:
abclist abc kt cos t sin t quad t in pi k't -sin t cos t |k't| sqrtsin^ t + cos^ t L _^pi dd t pi abc zt leftrt - sin t r - cos tright z't leftr - cos t r sin t right |z't| r sqrt - cos t^ + sin^ t r sqrt - cos t r sinleftfractright L _^pi r sinleftfractright dd t -r cosleftfractrightBig|_^pi -rcospi - cos -r- - r abc at t cos t t sin t a't cos t - t sin t sin t + t cos t |a't| sqrtcos t - t sin t^ + sin t + t cos t^ sqrt + t^ L _^pi sqrt + t^ ddt &quad textkeine elementare Stammfunktion numerisch lösbar abc ht leftr cospi t r sinpi t ht right h't left-pi r sinpi t pi r cospi t h right |h't| sqrtpi r^ + h^ quad textkonstant L _^pi sqrtpi r^ + h^ dt pi sqrtpi r^ + h^ abclist