What is the integral from 2 to 7 of 2piy(7-y) ?

The integral from 2 to 7 of 2piy(7-y) is (275pi)/3

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integral from 2 to 7 of 2piy(7-y)

\int_{2}^{7} 2 π y (7 - y) d y

\frac{275 π}{3}

Decimal Notation

287.97932 \dots

Solution steps

\int_{2}^{7} 2 π y (7 - 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}^{7} y (7 - y) d y

Expand

y (7 - y) : 7 y - y^{2}

= 2 π \cdot \int_{2}^{7} 7 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}^{7} 7 y d y - \int_{2}^{7} y^{2} d y)

\int_{2}^{7} 7 y d y = \frac{315}{2}

\int_{2}^{7} y^{2} d y = \frac{335}{3}

= 2 π (\frac{315}{2} - \frac{335}{3})

Simplify

2 π (\frac{315}{2} - \frac{335}{3}) : \frac{275 π}{3}

= \frac{275 π}{3}

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What is the integral from 2 to 7 of 2piy(7-y) ?
The integral from 2 to 7 of 2piy(7-y) is (275pi)/3