What is the integral from 0 to 2/3 of 4x-6x^2 ?

The integral from 0 to 2/3 of 4x-6x^2 is 8/27

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integral from 0 to 2/3 of 4x-6x^2

\int_{0}^{\frac{2}{3}} 4 x - 6 x^{2} d x

\frac{8}{27}

Decimal

0.29629 \dots

Solution steps

\int_{0}^{\frac{2}{3}} 4 x - 6 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

= \int_{0}^{\frac{2}{3}} 4 x d x - \int_{0}^{\frac{2}{3}} 6 x^{2} d x

\int_{0}^{\frac{2}{3}} 4 x d x = \frac{8}{9}

\int_{0}^{\frac{2}{3}} 6 x^{2} d x = \frac{16}{27}

= \frac{8}{9} - \frac{16}{27}

Simplify

\frac{8}{9} - \frac{16}{27} : \frac{8}{27}

= \frac{8}{27}

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What is the integral from 0 to 2/3 of 4x-6x^2 ?
The integral from 0 to 2/3 of 4x-6x^2 is 8/27