What is the integral from 0 to 6 of 2pix(6x-x^2) ?

The integral from 0 to 6 of 2pix(6x-x^2) is 216pi

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integral from 0 to 6 of 2pix(6x-x^2)

\int_{0}^{6} 2 π x (6 x - x^{2}) d x

216 π

Decimal

678.58401 \dots

Solution steps

\int_{0}^{6} 2 π x (6 x - x^{2}) d x

Take the constant out:

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

= 2 π \cdot \int_{0}^{6} x (6 x - x^{2}) d x

Expand

x (6 x - x^{2}) : 6 x^{2} - x^{3}

= 2 π \cdot \int_{0}^{6} 6 x^{2} - x^{3} 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

= 2 π (\int_{0}^{6} 6 x^{2} d x - \int_{0}^{6} x^{3} d x)

\int_{0}^{6} 6 x^{2} d x = 432

\int_{0}^{6} x^{3} d x = 324

= 2 π (432 - 324)

Simplify

= 216 π

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What is the integral from 0 to 6 of 2pix(6x-x^2) ?
The integral from 0 to 6 of 2pix(6x-x^2) is 216pi