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人気のある微分積分 >

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

解

\int_{0}^{3} π (9 - x^{2})^{2} d x

解

π \frac{648}{5}

+1

十進法表記

407.15040 \dots

解答ステップ

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

拡張

(9 - x^{2})^{2} : 81 - 18 x^{2} + x^{4}

= π \cdot \int_{0}^{3} 81 - 18 x^{2} + 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}^{3} 81 d x - \int_{0}^{3} 18 x^{2} d x + \int_{0}^{3} x^{4} d x)

\int_{0}^{3} 81 d x = 243

\int_{0}^{3} 18 x^{2} d x = 162

\int_{0}^{3} x^{4} d x = \frac{243}{5}

= π (243 - 162 + \frac{243}{5})

簡素化

π (243 - 162 + \frac{243}{5}) : π \frac{648}{5}

= π \frac{648}{5}

グラフ

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