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

The integral from 0 to 1 of pi(y-y^2)^2 is pi/(30)

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

\int_{0}^{1} π (y - y^{2})^{2} d y

\frac{π}{30}

Decimal

0.10471 \dots

Solution steps

\int_{0}^{1} π (y - y^{2})^{2} d y

Take the constant out:

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

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

Expand

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

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

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

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

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

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

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

Simplify

π (\frac{1}{3} - \frac{1}{2} + \frac{1}{5}) : \frac{π}{30}

= \frac{π}{30}

What is the integral from 0 to 1 of pi(y-y^2)^2 ?
The integral from 0 to 1 of pi(y-y^2)^2 is pi/(30)