fläche y=-x^2+16,y=(-x)/4-1,4(y+6.5)=6.5x

Lösung

fläche

y = - x^{2} + 16, y = \frac{- x}{4} - 1, 4 (y + 6.5) = 6.5 x

\frac{97}{384} + 57.41100 \dots

Dezimale

57.66360 \dots

Schritte zur Lösung

Stelle

y

nach

4 (y + 6.5) = 6.5 x

um:

y = 1.625 x - 6.5

f_{1} (x) = - x^{2} + 16

f_{2} (x) = \frac{- x}{4} - 1

Um die Schnittpunkte zu finden, löse

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

- x^{2} + 16 = \frac{- x}{4} - 1 : x = - 4, x = \frac{17}{4}

- x^{2} + 16 = 1.625 x - 6.5 : x = - \frac{45}{8}, x = 4

\frac{- x}{4} - 1 = 1.625 x - 6.5 : x = \frac{44}{15}

= \int_{- \frac{45}{8}}^{- 4} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- 4}^{\frac{44}{15}} ∣ f_{3} (x) - f_{2} (x) ∣ d x + \int_{\frac{44}{15}}^{4} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{4}^{\frac{17}{4}} ∣ f_{1} (x) - f_{2} (x) ∣ d x

= \int_{- \frac{45}{8}}^{- 4} - x^{2} + 16 - (1.625 x - 6.5) d x + \int_{- 4}^{\frac{44}{15}} 1.625 x - 6.5 - (\frac{- x}{4} - 1) d x + \int_{\frac{44}{15}}^{4} \frac{- x}{4} - 1 - (1.625 x - 6.5) d x + \int_{4}^{\frac{17}{4}} - x^{2} + 16 - (\frac{- x}{4} - 1) d x

\int_{- \frac{45}{8}}^{- 4} - x^{2} + 16 - (1.625 x - 6.5) d x = 11.27766 \dots

\int_{- 4}^{\frac{44}{15}} 1.625 x - 6.5 - (\frac{- x}{4} - 1) d x = 45.06666 \dots

\int_{\frac{44}{15}}^{4} \frac{- x}{4} - 1 - (1.625 x - 6.5) d x = 1.06666 \dots

\int_{4}^{\frac{17}{4}} - x^{2} + 16 - (\frac{- x}{4} - 1) d x = \frac{97}{384}

= 11.27766 \dots + 45.06666 \dots + 1.06666 \dots + \frac{97}{384}

Vereinfache

= \frac{97}{384} + 57.41100 \dots

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