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人気のある微分積分 >

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

解

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

解

\frac{279 π}{2}

+1

十進法表記

438.25217 \dots

解答ステップ

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

定数を除く:

\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

拡張

(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

総和規則を適用する:

\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})

簡素化

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

= \frac{279 π}{2}

グラフ

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