fläche 3/x ,12x, 1/3 x

Lösung

fläche

\frac{3}{x}, 12 x, \frac{1}{3} x

6 ln (6)

Dezimale

10.75055 \dots

Schritte zur Lösung

Um die Schnittpunkte zu finden, löse

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

\frac{3}{x} = 12 x : x = \frac{1}{2}, x = - \frac{1}{2}

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

12 x = \frac{1}{3} x : x = 0

= \int_{- 3}^{- \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}}^{3} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- 3}^{- \frac{1}{2}} \frac{3}{x} - \frac{1}{3} x d x + \int_{- \frac{1}{2}}^{0} \frac{1}{3} x - 12 x d x + \int_{0}^{\frac{1}{2}} 12 x - \frac{1}{3} x d x + \int_{\frac{1}{2}}^{3} \frac{3}{x} - \frac{1}{3} x d x

\int_{- 3}^{- \frac{1}{2}} \frac{3}{x} - \frac{1}{3} x d x = 3 ln (6) - \frac{35}{24}

\int_{- \frac{1}{2}}^{0} \frac{1}{3} x - 12 x d x = \frac{35}{24}

\int_{0}^{\frac{1}{2}} 12 x - \frac{1}{3} x d x = \frac{35}{24}

\int_{\frac{1}{2}}^{3} \frac{3}{x} - \frac{1}{3} x d x = 3 ln (6) - \frac{35}{24}

= 3 ln (6) - \frac{35}{24} + \frac{35}{24} + \frac{35}{24} + 3 ln (6) - \frac{35}{24}

Vereinfache

= 6 ln (6)

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