integral from 0 to 1 of y^2(1-y^{3/2})

解

\int_{0}^{1} y^{2} (1 - y^{\frac{3}{2}}) d y

\frac{1}{9}

十進法表記

0.11111 \dots

解答ステップ

\int_{0}^{1} y^{2} (1 - y^{\frac{3}{2}}) d y

拡張

y^{2} (1 - y^{\frac{3}{2}}) : y^{2} - y^{\frac{7}{2}}

= \int_{0}^{1} y^{2} - y^{\frac{7}{2}} d y

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

= \int_{0}^{1} y^{2} d y - \int_{0}^{1} y^{\frac{7}{2}} d y

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

\int_{0}^{1} y^{\frac{7}{2}} d y = \frac{2}{9}

= \frac{1}{3} - \frac{2}{9}

簡素化

\frac{1}{3} - \frac{2}{9} : \frac{1}{9}

= \frac{1}{9}