integral from 0 to 2 of pi(x^2-L)^2

解答

\int_{0}^{2} π (x^{2} - L)^{2} d x

π (2 L^{2} - \frac{16 L}{3} + \frac{32}{5})

求解步骤

\int_{0}^{2} π (x^{2} - L)^{2} d x

提出常数:

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

= π \cdot \int_{0}^{2} (- L + x^{2})^{2} d x

乘开

(- L + x^{2})^{2} : L^{2} - 2 L x^{2} + x^{4}

= π \cdot \int_{0}^{2} L^{2} - 2 L x^{2} + x^{4} d x

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

= π (\int_{0}^{2} L^{2} d x - \int_{0}^{2} 2 L x^{2} d x + \int_{0}^{2} x^{4} d x)

\int_{0}^{2} L^{2} d x = 2 L^{2}

\int_{0}^{2} 2 L x^{2} d x = \frac{16 L}{3}

\int_{0}^{2} x^{4} d x = \frac{32}{5}

= π (2 L^{2} - \frac{16 L}{3} + \frac{32}{5})

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