Benutzer:Hourssales/HA7

${\displaystyle \mathrm {f} (x)={\sqrt {-{\frac {9}{25}}x^{2}+81}}}${\displaystyle \mathrm {f} (x)={\sqrt {-{\frac {9}{25}}x^{2}+81}}}

${\displaystyle {\begin{aligned}&\mathrm {Rotationsvolumen\ der\ Funktion\ um\ die\ y{-}Achse\ im\ Intervall\ I=[0;y_{0}]} \\y&={\sqrt {-{\frac {9}{25}}x^{2}+81}}\\y^{2}&=-{\frac {9}{25}}x^{2}+81\\y^{2}-81&=-{\frac {9}{25}}x^{2}\\x^{2}&=-{\frac {25}{9}}y^{2}+225\\x&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\\\mathrm {f} (y)&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\0円&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\0円&=-{\frac {25}{9}}y^{2}+225\\y_{0,1}&=9\\y_{0,2}&=-9\mathrm {(entf{\ddot {a}}llt,da\ {-}9\notin [0,y_{0}])} \\V_{Y}&=\pi \int \limits _{0}^{9}\mathrm {f} (y),円\mathrm {d} y\\V_{Y}&=4241\ m^{3}\end{aligned}}}${\displaystyle {\begin{aligned}&\mathrm {Rotationsvolumen\ der\ Funktion\ um\ die\ y{-}Achse\ im\ Intervall\ I=[0;y_{0}]} \\y&={\sqrt {-{\frac {9}{25}}x^{2}+81}}\\y^{2}&=-{\frac {9}{25}}x^{2}+81\\y^{2}-81&=-{\frac {9}{25}}x^{2}\\x^{2}&=-{\frac {25}{9}}y^{2}+225\\x&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\\\mathrm {f} (y)&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\0円&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\0円&=-{\frac {25}{9}}y^{2}+225\\y_{0,1}&=9\\y_{0,2}&=-9\mathrm {(entf{\ddot {a}}llt,da\ {-}9\notin [0,y_{0}])} \\V_{Y}&=\pi \int \limits _{0}^{9}\mathrm {f} (y),円\mathrm {d} y\\V_{Y}&=4241\ m^{3}\end{aligned}}}