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Beliebt Rechnen >

integral from 0 to 2 of pi(2x-x^2)^2

Lösung

\int_{0}^{2} π (2 x - x^{2})^{2} d x

Lösung

π \frac{16}{15}

+1

Dezimale

3.35103 \dots

Schritte zur Lösung

\int_{0}^{2} π (2 x - x^{2})^{2} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= π \cdot \int_{0}^{2} (2 x - x^{2})^{2} d x

Multipliziere aus

(2 x - x^{2})^{2} : 4 x^{2} - 4 x^{3} + x^{4}

= π \cdot \int_{0}^{2} 4 x^{2} - 4 x^{3} + x^{4} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= π (\int_{0}^{2} 4 x^{2} d x - \int_{0}^{2} 4 x^{3} d x + \int_{0}^{2} x^{4} d x)

\int_{0}^{2} 4 x^{2} d x = \frac{32}{3}

\int_{0}^{2} 4 x^{3} d x = 16

\int_{0}^{2} x^{4} d x = \frac{32}{5}

= π (\frac{32}{3} - 16 + \frac{32}{5})

Vereinfache

π (\frac{32}{3} - 16 + \frac{32}{5}) : π \frac{16}{15}

= π \frac{16}{15}

Graph

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