integral from 0 to 3 of 2pi(5-x)(9-x^2)

Lösung

\int_{0}^{3} 2 π (5 - x) (9 - x^{2}) d x

\frac{279 π}{2}

Dezimale

438.25217 \dots

Schritte zur Lösung

\int_{0}^{3} 2 π (5 - x) (9 - x^{2}) d x

Entferne die Konstante:

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

= 2 π \cdot \int_{0}^{3} (5 - x) (9 - x^{2}) d x

Multipliziere aus

(5 - x) (9 - x^{2}) : 45 - 5 x^{2} - 9 x + x^{3}

= 2 π \cdot \int_{0}^{3} 45 - 5 x^{2} - 9 x + x^{3} 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

= 2 π (\int_{0}^{3} 45 d x - \int_{0}^{3} 5 x^{2} d x - \int_{0}^{3} 9 x d x + \int_{0}^{3} x^{3} d x)

\int_{0}^{3} 45 d x = 135

\int_{0}^{3} 5 x^{2} d x = 45

\int_{0}^{3} 9 x d x = \frac{81}{2}

\int_{0}^{3} x^{3} d x = \frac{81}{4}

= 2 π (135 - 45 - \frac{81}{2} + \frac{81}{4})

Vereinfache

2 π (135 - 45 - \frac{81}{2} + \frac{81}{4}) : \frac{279 π}{2}

= \frac{279 π}{2}

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