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

The integral from 2 to 5 of 2piy(5-y) is 27pi

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

\int_{2}^{5} 2 π y (5 - y) d y

27 π

Solution steps

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

Expand

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

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

\int_{2}^{5} 5 y d y = \frac{105}{2}

\int_{2}^{5} y^{2} d y = 39

= 2 π (\frac{105}{2} - 39)

Simplify

2 π (\frac{105}{2} - 39) : 27 π

= 27 π

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What is the integral from 2 to 5 of 2piy(5-y) ?
The integral from 2 to 5 of 2piy(5-y) is 27pi