What is the integral from 0 to 1 of 3y^2(-y+1)^2 ?

The integral from 0 to 1 of 3y^2(-y+1)^2 is 1/10

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integral from 0 to 1 of 3y^2(-y+1)^2

\int_{0}^{1} 3 y^{2} (- y + 1)^{2} d y

\frac{1}{10}

Decimal

0.1

Solution steps

\int_{0}^{1} 3 y^{2} (- y + 1)^{2} d y

Take the constant out:

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

= 3 \cdot \int_{0}^{1} y^{2} (- y + 1)^{2} d y

Expand

y^{2} (- y + 1)^{2} : y^{4} - 2 y^{3} + y^{2}

= 3 \cdot \int_{0}^{1} y^{4} - 2 y^{3} + y^{2} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (\int_{0}^{1} y^{4} d y - \int_{0}^{1} 2 y^{3} d y + \int_{0}^{1} y^{2} d y)

\int_{0}^{1} y^{4} d y = \frac{1}{5}

\int_{0}^{1} 2 y^{3} d y = \frac{1}{2}

\int_{0}^{1} y^{2} d y = \frac{1}{3}

= 3 (\frac{1}{5} - \frac{1}{2} + \frac{1}{3})

Simplify

3 (\frac{1}{5} - \frac{1}{2} + \frac{1}{3}) : \frac{1}{10}

= \frac{1}{10}

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What is the integral from 0 to 1 of 3y^2(-y+1)^2 ?
The integral from 0 to 1 of 3y^2(-y+1)^2 is 1/10