fläche y=3x,x=8-y^2

Lösung

fläche

y = 3 x, x = 8 - y^{2}

\frac{4913}{162}

Dezimale

30.32716 \dots

Schritte zur Lösung

Stelle

y

nach

x = 8 - y^{2}

um:

y = - x + 8, y = - - x + 8

f_{1} (x) = 3 x

f_{2} (x) = - x + 8

Um die Schnittpunkte zu finden, löse

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

3 x = - x + 8 : x = \frac{8}{9}

3 x = - - x + 8 : x = - 1

- x + 8 = - - x + 8 : x = 8

= \int_{- 1}^{\frac{8}{9}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{\frac{8}{9}}^{8} ∣ f_{3} (x) - f_{2} (x) ∣ d x

= \int_{- 1}^{\frac{8}{9}} 3 x - (- - x + 8) d x + \int_{\frac{8}{9}}^{8} - - x + 8 - - x + 8 d x

\int_{- 1}^{\frac{8}{9}} 3 x - (- - x + 8) d x = \frac{817}{162}

\int_{\frac{8}{9}}^{8} - - x + 8 - - x + 8 d x = \frac{2048}{81}

= \frac{817}{162} + \frac{2048}{81}

Vereinfache

= \frac{4913}{162}

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