integral from 0 to 11 of 2piy(121-y^2)

解答

\int_{0}^{11} 2 π y (121 - y^{2}) d y

\frac{14641 π}{2}

十进制

22998.02902 \dots

求解步骤

\int_{0}^{11} 2 π y (121 - y^{2}) d y

提出常数:

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

= 2 π \cdot \int_{0}^{11} y (121 - y^{2}) d y

乘开

y (121 - y^{2}) : 121 y - y^{3}

= 2 π \cdot \int_{0}^{11} 121 y - y^{3} d y

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 π (\int_{0}^{11} 121 y d y - \int_{0}^{11} y^{3} d y)

\int_{0}^{11} 121 y d y = \frac{14641}{2}

\int_{0}^{11} y^{3} d y = \frac{14641}{4}

= 2 π (\frac{14641}{2} - \frac{14641}{4})

化简

2 π (\frac{14641}{2} - \frac{14641}{4}) : \frac{14641 π}{2}

= \frac{14641 π}{2}

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