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人気のある微分積分 >

integral from 0 to 5 of pi(-3/5 x+3)^2

解

\int_{0}^{5} π (- \frac{3}{5} x + 3)^{2} d x

解

15 π

+1

十進法表記

47.12388 \dots

解答ステップ

\int_{0}^{5} π (- \frac{3}{5} x + 3)^{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_{0}^{5} (- \frac{3}{5} x + 3)^{2} d x

拡張

(- \frac{3}{5} x + 3)^{2} : \frac{9 x ^{2}}{25} - \frac{18 x}{5} + 9

= π \cdot \int_{0}^{5} \frac{9 x ^{2}}{25} - \frac{18 x}{5} + 9 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}^{5} \frac{9 x ^{2}}{25} d x - \int_{0}^{5} \frac{18 x}{5} d x + \int_{0}^{5} 9 d x)

\int_{0}^{5} \frac{9 x ^{2}}{25} d x = 15

\int_{0}^{5} \frac{18 x}{5} d x = 45

\int_{0}^{5} 9 d x = 45

= π (15 - 45 + 45)

簡素化

= 15 π

グラフ

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