f ( x ) = − 9 25 x 2 + 81 {\displaystyle \mathrm {f} (x)={\sqrt {-{\frac {9}{25}}x^{2}+81}}}
R o t a t i o n s v o l u m e n d e r F u n k t i o n u m d i e y − A c h s e i m I n t e r v a l l I = [ 0 ; y 0 ] y = − 9 25 x 2 + 81 y 2 = − 9 25 x 2 + 81 y 2 − 81 = − 9 25 x 2 x 2 = − 25 9 y 2 + 225 x = − 25 9 y 2 + 225 f ( y ) = − 25 9 y 2 + 225 0 = − 25 9 y 2 + 225 0 = − 25 9 y 2 + 225 y 0 , 1 = 9 y 0 , 2 = − 9 ( e n t f a ¨ l l t , d a − 9 ∉ [ 0 , y 0 ] ) V Y = π ∫ 0 9 f ( y ) d y V Y = 4241 m 3 {\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&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\\0&=-{\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)\,\mathrm {d} y\\V_{Y}&=4241\ m^{3}\end{aligned}}}
D e r A n s t i e g a n d e r S t e l l e x 0 i s t d i e A b l e i t u n g v o n f ( x ) a n x 0 . y = m x + n m = f ′ ( x 0 ) = − 9 25 x 0 ⋅ ( − 1 − 9 25 x 0 2 + 81 ) = 9 25 x 0 − 9 25 x 0 2 + 81 y = ( 9 25 x 0 − 9 25 x 0 2 + 81 ) x + n n = y − ( 9 25 x 0 − 9 25 x 0 2 + 81 ) x n = − 9 25 x 0 2 + 81 − ( 9 25 x 0 − 9 25 x 0 2 + 81 ) x n = − 9 25 x 0 2 + 81 + 9 25 x 0 2 − 9 25 x 0 2 + 81 n = 81 − 9 25 x 0 2 + 81 t x 0 ( x ) = ( 9 25 x 0 − 9 25 x 0 2 + 81 ) x + 81 − 9 25 x 0 2 + 81 {\displaystyle {\begin{aligned}&\mathrm {Der\ Anstieg\ an\ der\ Stelle\ x_{0}\ ist\ die\ Ableitung\ von\ f(x)\ an\ x_{0}.} \\y&=mx+n\\m&=\mathrm {f} '(x_{0})\\&=-{\frac {9}{25}}x_{0}\cdot (-{\frac {1}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}})\\&={\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\\\ &\ \\y&=\left({\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\right)x+n\\n&=y-\left({\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\right)x\\n&={\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}-\left({\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\right)x\\n&={\frac {-{\frac {9}{25}}{x_{0}}^{2}+81+{\frac {9}{25}}{x_{0}}^{2}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\\n&={\frac {81}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\\\ &\ \\\mathrm {t} _{x_{0}}(x)&=\left({\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\right)x+{\frac {81}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\end{aligned}}}
tan α 1 = 2 9 tan α 2 = m t = f ′ ( 9 ) g ( x ) = m x + n m = tan α A n s t i e g m = tan ( 90 ∘ − α − α 2 ) m = tan ( 90 ∘ − 70 , 9 ∘ − arctan f ′ ( 9 ) ) [ m ] = 0,369 3 n = y 0 = 5,494 4 {\displaystyle {\begin{aligned}\tan \alpha _{1}&={\frac {2}{9}}\\\tan \alpha _{2}&=m_{t}=\mathrm {f} '(9)\\\ &\ \\\mathrm {g} (x)&=mx+n\\m&=\tan \alpha _{Anstieg}\\m&=\tan \left(90^{\circ }-\alpha -\alpha _{2}\right)\\m&=\tan \left(90^{\circ }-70{,}9^{\circ }-\arctan \mathrm {f} '(9)\right)\\\left[m\right]&=0{,}3693\\\ &\ \\n&=y_{0}=5{,}4944\end{aligned}}}
α = 90 ∘ {\displaystyle \alpha =90^{\circ }}
![{\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&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\\0&=-{\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)\,\mathrm {d} y\\V_{Y}&=4241\ m^{3}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6109b79fc8f9704c406550234fcb8448f9a200)
![{\displaystyle {\begin{aligned}\tan \alpha _{1}&={\frac {2}{9}}\\\tan \alpha _{2}&=m_{t}=\mathrm {f} '(9)\\\ &\ \\\mathrm {g} (x)&=mx+n\\m&=\tan \alpha _{Anstieg}\\m&=\tan \left(90^{\circ }-\alpha -\alpha _{2}\right)\\m&=\tan \left(90^{\circ }-70{,}9^{\circ }-\arctan \mathrm {f} '(9)\right)\\\left[m\right]&=0{,}3693\\\ &\ \\n&=y_{0}=5{,}4944\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b7205a81e1f26e642957a68e0b908e8bb48666)
