integral from 0 to 1 of pi(x^2-x^6)

解

\int_{0}^{1} π (x^{2} - x^{6}) d x

π \frac{4}{21}

十進法表記

0.59839 \dots

解答ステップ

\int_{0}^{1} π (x^{2} - x^{6}) 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^{2} - x^{6} 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^{2} d x - \int_{0}^{1} x^{6} d x)

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

\int_{0}^{1} x^{6} d x = \frac{1}{7}

= π (\frac{1}{3} - \frac{1}{7})

結合

\frac{1}{3} - \frac{1}{7} : \frac{4}{21}

= π \frac{4}{21}