integral from 0 to 3 of ((30-y)-y^{1/3})

解

\int_{0}^{3} ((30 - y) - y^{\frac{1}{3}}) d y

\frac{171}{2} - \frac{9 \cdot 3 ^{\frac{1}{3}}}{4}

十進法表記

82.25493 \dots

解答ステップ

\int_{0}^{3} 30 - y - y^{\frac{1}{3}} 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

= \int_{0}^{3} 30 d y - \int_{0}^{3} y d y - \int_{0}^{3} y^{\frac{1}{3}} d y

\int_{0}^{3} 30 d y = 90

\int_{0}^{3} y d y = \frac{9}{2}

\int_{0}^{3} y^{\frac{1}{3}} d y = \frac{9 \cdot 3 ^{\frac{1}{3}}}{4}

= 90 - \frac{9}{2} - \frac{9 \cdot 3 ^{\frac{1}{3}}}{4}

分数を組み合わせる

90 - \frac{9}{2} : \frac{171}{2}

= \frac{171}{2} - \frac{9 \cdot 3 ^{\frac{1}{3}}}{4}