integral from 0 to 5r of 2(25r^2-y^2)

解

\int_{0}^{5 r} 2 (25 r^{2} - y^{2}) d y

2 (- \frac{125 r ^{3}}{3} + 125 r^{3})

解答ステップ

\int_{0}^{5 r} 2 (25 r^{2} - y^{2}) d y

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{5 r} - y^{2} + 25 r^{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

= 2 (- \int_{0}^{5 r} y^{2} d y + \int_{0}^{5 r} 25 r^{2} d y)

\int_{0}^{5 r} y^{2} d y = \frac{125 r ^{3}}{3}

\int_{0}^{5 r} 25 r^{2} d y = 125 r^{3}

= 2 (- \frac{125 r ^{3}}{3} + 125 r^{3})