integral from-1.6 to 0 of pi((6x-x^2))^2

解

\int_{- 1.6}^{0} π ((6 x - x^{2}))^{2} d x

222.77018 \dots

解答ステップ

\int_{- 1.6}^{0} π (6 x - x^{2})^{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_{- 1.6}^{0} (6 x - x^{2})^{2} d x

拡張

(6 x - x^{2})^{2} : 36 x^{2} - 12 x^{3} + x^{4}

= π \cdot \int_{- 1.6}^{0} 36 x^{2} - 12 x^{3} + x^{4} 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_{- 1.6}^{0} 36 x^{2} d x - \int_{- 1.6}^{0} 12 x^{3} d x + \int_{- 1.6}^{0} x^{4} d x)

\int_{- 1.6}^{0} 36 x^{2} d x = 49.152

\int_{- 1.6}^{0} 12 x^{3} d x = - 19.6608

\int_{- 1.6}^{0} x^{4} d x = 2.097152

= π (49.152 - (- 19.6608) + 2.097152)

簡素化

= 222.77018 \dots