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

解

\int_{0}^{1} π (x - x^{2}) d x

\frac{π}{6}

十進法表記

0.52359 \dots

解答ステップ

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

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

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

= π (\frac{1}{2} - \frac{1}{3})

簡素化

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

= \frac{π}{6}