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

Lösung

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

\frac{135 π}{2}

Dezimale

212.05750 \dots

Schritte zur Lösung

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

Multipliziere aus

(3 - x) (9 - x^{2}) : 27 - 3 x^{2} - 9 x + x^{3}

= 2 π \cdot \int_{0}^{3} 27 - 3 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} 27 d x - \int_{0}^{3} 3 x^{2} d x - \int_{0}^{3} 9 x d x + \int_{0}^{3} x^{3} d x)

\int_{0}^{3} 27 d x = 81

\int_{0}^{3} 3 x^{2} d x = 27

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

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

= 2 π (81 - 27 - \frac{81}{2} + \frac{81}{4})

Vereinfache

2 π (81 - 27 - \frac{81}{2} + \frac{81}{4}) : \frac{135 π}{2}

= \frac{135 π}{2}

d er i v a t i v e - 4 e^{- x} + 1

\int 8 x e^{x} d x

t an g e n t f (x) = (1 + 2 x)^{2}, at x = 3

l a pl a ce t r an s f or m g

d er i v a t i v e t^{2} + 5