What is the integral from 1 to 6 of 2piy(6-y) ?

The integral from 1 to 6 of 2piy(6-y) is (200pi)/3

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integral from 1 to 6 of 2piy(6-y)

\int_{1}^{6} 2 π y (6 - y) d y

\frac{200 π}{3}

Decimal Notation

209.43951 \dots

Solution steps

\int_{1}^{6} 2 π y (6 - y) d y

Take the constant out:

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

= 2 π \cdot \int_{1}^{6} y (6 - y) d y

Expand

y (6 - y) : 6 y - y^{2}

= 2 π \cdot \int_{1}^{6} 6 y - 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

= 2 π (\int_{1}^{6} 6 y d y - \int_{1}^{6} y^{2} d y)

\int_{1}^{6} 6 y d y = 105

\int_{1}^{6} y^{2} d y = \frac{215}{3}

= 2 π (105 - \frac{215}{3})

Simplify

2 π (105 - \frac{215}{3}) : \frac{200 π}{3}

= \frac{200 π}{3}

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What is the integral from 1 to 6 of 2piy(6-y) ?
The integral from 1 to 6 of 2piy(6-y) is (200pi)/3