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

The integral from 0 to 1 of pi(x-x^2) is pi/6

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

\int_{0}^{1} π (x - x^{2}) d x

\frac{π}{6}

Decimal

0.52359 \dots

Solution steps

\int_{0}^{1} π (x - x^{2}) d x

Take the constant out:

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

= π \cdot \int_{0}^{1} x - x^{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

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

\int_{0}^{1} x d x = \frac{1}{2}

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

= π (\frac{1}{2} - \frac{1}{3})

Simplify

π (\frac{1}{2} - \frac{1}{3}) : \frac{π}{6}

= \frac{π}{6}

What is the integral from 0 to 1 of pi(x-x^2) ?
The integral from 0 to 1 of pi(x-x^2) is pi/6