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人気のある微分積分 >

integral from 0 to 1 of pi(1-sqrt(x))^2

解

\int_{0}^{1} π (1 - x)^{2} d x

解

\frac{π}{6}

+1

十進法表記

0.52359 \dots

解答ステップ

\int_{0}^{1} π (1 - x)^{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}^{1} (1 - x)^{2} d x

拡張

(1 - x)^{2} : 1 - 2 x + x

= π \cdot \int_{0}^{1} 1 - 2 x + x 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}^{1} 1 d x - \int_{0}^{1} 2 x d x + \int_{0}^{1} x d x)

\int_{0}^{1} 1 d x = 1

\int_{0}^{1} 2 x d x = \frac{4}{3}

\int_{0}^{1} x d x = \frac{1}{2}

= π (1 - \frac{4}{3} + \frac{1}{2})

簡素化

π (1 - \frac{4}{3} + \frac{1}{2}) : \frac{π}{6}

= \frac{π}{6}

グラフ

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