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人気のある微分積分 >

integral from 0 to 4 of pi(4x-x^2)^2

解

\int_{0}^{4} π (4 x - x^{2})^{2} d x

解

π \frac{512}{15}

+1

十進法表記

107.23302 \dots

解答ステップ

\int_{0}^{4} π (4 x - x^{2})^{2} d x

定数を除く:

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

= π \cdot \int_{0}^{4} (4 x - x^{2})^{2} d x

拡張

(4 x - x^{2})^{2} : 16 x^{2} - 8 x^{3} + x^{4}

= π \cdot \int_{0}^{4} 16 x^{2} - 8 x^{3} + 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

= π (\int_{0}^{4} 16 x^{2} d x - \int_{0}^{4} 8 x^{3} d x + \int_{0}^{4} x^{4} d x)

\int_{0}^{4} 16 x^{2} d x = \frac{1024}{3}

\int_{0}^{4} 8 x^{3} d x = 512

\int_{0}^{4} x^{4} d x = \frac{1024}{5}

= π (\frac{1024}{3} - 512 + \frac{1024}{5})

簡素化

π (\frac{1024}{3} - 512 + \frac{1024}{5}) : π \frac{512}{15}

= π \frac{512}{15}

グラフ

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