What is the integral from-y to 2y of xy-y^2 ?

The integral from-y to 2y of xy-y^2 is -y^3-(y^3)/2

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

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

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

Solution steps

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

Apply the Sum Rule:

\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})

Simplify

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

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

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What is the integral from-y to 2y of xy-y^2 ?
The integral from-y to 2y of xy-y^2 is -y^3-(y^3)/2