fläche y= 3/x ,y=8x,y= 1/8 x

Lösung

fläche

y = \frac{3}{x}, y = 8 x, y = \frac{1}{8} x

18 ln (2)

Dezimale

12.47664 \dots

Schritte zur Lösung

Um die Schnittpunkte zu finden, löse

f_{i} (x) = f_{j} (x)

\frac{3}{x} = 8 x : x = \frac{3}{2 2}, x = - \frac{3}{2 2}

\frac{3}{x} = \frac{1}{8} x : x = 26, x = - 26

8 x = \frac{1}{8} x : x = 0

= \int_{- 26}^{- \frac{3}{2 2}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{3}{2 2}}^{0} ∣ f_{3} (x) - f_{2} (x) ∣ d x + \int_{0}^{\frac{3}{2 2}} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{\frac{3}{2 2}}^{26} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- 26}^{- \frac{3}{2 2}} \frac{3}{x} - \frac{1}{8} x d x + \int_{- \frac{3}{2 2}}^{0} \frac{1}{8} x - 8 x d x + \int_{0}^{\frac{3}{2 2}} 8 x - \frac{1}{8} x d x + \int_{\frac{3}{2 2}}^{26} \frac{3}{x} - \frac{1}{8} x d x

\int_{- 26}^{- \frac{3}{2 2}} \frac{3}{x} - \frac{1}{8} x d x = 9 ln (2) - \frac{189}{128}

\int_{- \frac{3}{2 2}}^{0} \frac{1}{8} x - 8 x d x = \frac{189}{128}

\int_{0}^{\frac{3}{2 2}} 8 x - \frac{1}{8} x d x = \frac{189}{128}

\int_{\frac{3}{2 2}}^{26} \frac{3}{x} - \frac{1}{8} x d x = 9 ln (2) - \frac{189}{128}

= 9 ln (2) - \frac{189}{128} + \frac{189}{128} + \frac{189}{128} + 9 ln (2) - \frac{189}{128}

Vereinfache

= 18 ln (2)