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人気のある微分積分 >

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

解

\int_{0}^{3} 2 π x (3 x - x^{2}) d x

解

\frac{27 π}{2}

+1

十進法表記

42.41150 \dots

解答ステップ

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

拡張

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

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

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

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

= 2 π (27 - \frac{81}{4})

簡素化

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

= \frac{27 π}{2}

グラフ

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