What is the integral from 1 to 5 of 2piy(5-y) ?

The integral from 1 to 5 of 2piy(5-y) is (112pi)/3

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integral from 1 to 5 of 2piy(5-y)

\int_{1}^{5} 2 π y (5 - y) d y

\frac{112 π}{3}

Solution steps

\int_{1}^{5} 2 π y (5 - 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}^{5} y (5 - y) d y

Expand

y (5 - y) : 5 y - y^{2}

= 2 π \cdot \int_{1}^{5} 5 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}^{5} 5 y d y - \int_{1}^{5} y^{2} d y)

\int_{1}^{5} 5 y d y = 60

\int_{1}^{5} y^{2} d y = \frac{124}{3}

= 2 π (60 - \frac{124}{3})

Simplify

2 π (60 - \frac{124}{3}) : \frac{112 π}{3}

= \frac{112 π}{3}

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What is the integral from 1 to 5 of 2piy(5-y) ?
The integral from 1 to 5 of 2piy(5-y) is (112pi)/3