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人気のある微分積分 >

integral from 0 to 2 of 2pi(3-x)(8-x^3)

解

\int_{0}^{2} 2 π (3 - x) (8 - x^{3}) d x

解

\frac{264 π}{5}

+1

十進法表記

165.87609 \dots

解答ステップ

\int_{0}^{2} 2 π (3 - x) (8 - x^{3}) 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}^{2} (3 - x) (8 - x^{3}) d x

拡張

(3 - x) (8 - x^{3}) : 24 - 3 x^{3} - 8 x + x^{4}

= 2 π \cdot \int_{0}^{2} 24 - 3 x^{3} - 8 x + x^{4} 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}^{2} 24 d x - \int_{0}^{2} 3 x^{3} d x - \int_{0}^{2} 8 x d x + \int_{0}^{2} x^{4} d x)

\int_{0}^{2} 24 d x = 48

\int_{0}^{2} 3 x^{3} d x = 12

\int_{0}^{2} 8 x d x = 16

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

= 2 π (48 - 12 - 16 + \frac{32}{5})

簡素化

2 π (48 - 12 - 16 + \frac{32}{5}) : \frac{264 π}{5}

= \frac{264 π}{5}

グラフ

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