fläche 2/x ,8x, x/8

Lösung

fläche

\frac{2}{x}, 8 x, \frac{x}{8}

12 ln (2)

Dezimale

8.31776 \dots

Schritte zur Lösung

Um die Schnittpunkte zu finden, löse

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

\frac{2}{x} = 8 x : x = \frac{1}{2}, x = - \frac{1}{2}

\frac{2}{x} = \frac{x}{8} : x = 4, x = - 4

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

= \int_{- 4}^{- \frac{1}{2}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{1}{2}}^{0} ∣ f_{3} (x) - f_{2} (x) ∣ d x + \int_{0}^{\frac{1}{2}} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{\frac{1}{2}}^{4} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- 4}^{- \frac{1}{2}} \frac{2}{x} - \frac{x}{8} d x + \int_{- \frac{1}{2}}^{0} \frac{x}{8} - 8 x d x + \int_{0}^{\frac{1}{2}} 8 x - \frac{x}{8} d x + \int_{\frac{1}{2}}^{4} \frac{2}{x} - \frac{x}{8} d x

\int_{- 4}^{- \frac{1}{2}} \frac{2}{x} - \frac{x}{8} d x = 6 ln (2) - \frac{63}{64}

\int_{- \frac{1}{2}}^{0} \frac{x}{8} - 8 x d x = \frac{63}{64}

\int_{0}^{\frac{1}{2}} 8 x - \frac{x}{8} d x = \frac{63}{64}

\int_{\frac{1}{2}}^{4} \frac{2}{x} - \frac{x}{8} d x = 6 ln (2) - \frac{63}{64}

= 6 ln (2) - \frac{63}{64} + \frac{63}{64} + \frac{63}{64} + 6 ln (2) - \frac{63}{64}

Vereinfache

= 12 ln (2)

x \to 5 lim (\frac{20 - 4 x}{x - 5})

\int_{0}^{\infty} x^{- 2} d x

a re a x = 2 y^{2}, x + y = 1

\int_{0}^{a} sin^{2} (\frac{nπ x}{a}) d x

d er i v a t i v e f (x) = - x^{2} + 2