integral from 4 to 8 of (4-x)(x-8)

解

\int_{4}^{8} (4 - x) (x - 8) d x

\frac{32}{3}

十進法表記

10.66666 \dots

解答ステップ

\int_{4}^{8} (4 - x) (x - 8) d x

拡張

(4 - x) (x - 8) : 12 x - x^{2} - 32

= \int_{4}^{8} 12 x - x^{2} - 32 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_{4}^{8} 12 x d x - \int_{4}^{8} x^{2} d x - \int_{4}^{8} 32 d x

\int_{4}^{8} 12 x d x = 288

\int_{4}^{8} x^{2} d x = \frac{448}{3}

\int_{4}^{8} 32 d x = 128

= 288 - \frac{448}{3} - 128

簡素化

288 - \frac{448}{3} - 128 : \frac{32}{3}

= \frac{32}{3}