integral from 0 to 4 of x(4-x)

解

\int_{0}^{4} x (4 - x) d x

\frac{32}{3}

十進法表記

10.66666 \dots

解答ステップ

\int_{0}^{4} x (4 - x) d x

拡張

x (4 - x) : 4 x - x^{2}

= \int_{0}^{4} 4 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

= \int_{0}^{4} 4 x d x - \int_{0}^{4} x^{2} d x

\int_{0}^{4} 4 x d x = 32

\int_{0}^{4} x^{2} d x = \frac{64}{3}

= 32 - \frac{64}{3}

簡素化

32 - \frac{64}{3} : \frac{32}{3}

= \frac{32}{3}