integral from-y to 2y of xy-y^2

解

\int_{- y}^{2 y} x y - y^{2} d x

- y^{3} - \frac{y ^{3}}{2}

解答ステップ

\int_{- y}^{2 y} x y - y^{2} 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_{- y}^{2 y} y^{2} d x + \int_{- y}^{2 y} y x d x

\int_{- y}^{2 y} y^{2} d x = 3 y^{3}

\int_{- y}^{2 y} y x d x = y (2 y^{2} - \frac{y ^{2}}{2})

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

簡素化

- 3 y^{3} + y (2 y^{2} - \frac{y ^{2}}{2}) : - y^{3} - \frac{y ^{3}}{2}

= - y^{3} - \frac{y ^{3}}{2}