fläche y=3sqrt(x),x+y=4,y=1

Lösung

fläche

y = 3 x, x + y = 4, y = 1

\frac{82}{27}

Dezimale

3.03703 \dots

Schritte zur Lösung

Stelle

y

nach

x + y = 4

um:

y = 4 - x

f_{1} (x) = 3 x

f_{2} (x) = 4 - x

Um die Schnittpunkte zu finden, löse

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

3 x = 4 - x : x = 1

3 x = 1 : x = \frac{1}{9}

4 - x = 1 : x = 3

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

= \int_{\frac{1}{9}}^{1} 3 x - 1 d x + \int_{1}^{3} ∣ 1 - (4 - x) ∣ d x

\int_{\frac{1}{9}}^{1} 3 x - 1 d x = \frac{28}{27}

\int_{1}^{3} ∣ 1 - (4 - x) ∣ d x = 2

= \frac{28}{27} + 2

Vereinfache

= \frac{82}{27}