What is the integral from 0 to 12 of 2piy(144-y^2) ?

The integral from 0 to 12 of 2piy(144-y^2) is 10368pi

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integral from 0 to 12 of 2piy(144-y^2)

\int_{0}^{12} 2 π y (144 - y^{2}) d y

10368 π

Decimal

32572.03263 \dots

Solution steps

\int_{0}^{12} 2 π y (144 - y^{2}) d y

Take the constant out:

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

= 2 π \cdot \int_{0}^{12} y (144 - y^{2}) d y

Expand

y (144 - y^{2}) : 144 y - y^{3}

= 2 π \cdot \int_{0}^{12} 144 y - y^{3} 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_{0}^{12} 144 y d y - \int_{0}^{12} y^{3} d y)

\int_{0}^{12} 144 y d y = 10368

\int_{0}^{12} y^{3} d y = 5184

= 2 π (10368 - 5184)

Simplify

= 10368 π

What is the integral from 0 to 12 of 2piy(144-y^2) ?
The integral from 0 to 12 of 2piy(144-y^2) is 10368pi