integral from 0 to 3 of pi(3x-x^2)^2

解答

\int_{0}^{3} π (3 x - x^{2})^{2} d x

π \frac{81}{10}

十进制

25.44690 \dots

求解步骤

\int_{0}^{3} π (3 x - x^{2})^{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}^{3} (3 x - x^{2})^{2} d x

乘开

(3 x - x^{2})^{2} : 9 x^{2} - 6 x^{3} + x^{4}

= π \cdot \int_{0}^{3} 9 x^{2} - 6 x^{3} + 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}^{3} 9 x^{2} d x - \int_{0}^{3} 6 x^{3} d x + \int_{0}^{3} x^{4} d x)

\int_{0}^{3} 9 x^{2} d x = 81

\int_{0}^{3} 6 x^{3} d x = \frac{243}{2}

\int_{0}^{3} x^{4} d x = \frac{243}{5}

= π (81 - \frac{243}{2} + \frac{243}{5})

化简

81 - \frac{243}{2} + \frac{243}{5} : \frac{81}{10}

= π \frac{81}{10}

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