integral from 0 to 2 of pi(8-x^3)^2

解答

\int_{0}^{2} π (8 - x^{3})^{2} d x

π \frac{576}{7}

十进制

258.50819 \dots

求解步骤

\int_{0}^{2} π (8 - x^{3})^{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} (8 - x^{3})^{2} d x

乘开

(8 - x^{3})^{2} : 64 - 16 x^{3} + x^{6}

= π \cdot \int_{0}^{2} 64 - 16 x^{3} + x^{6} 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} 64 d x - \int_{0}^{2} 16 x^{3} d x + \int_{0}^{2} x^{6} d x)

\int_{0}^{2} 64 d x = 128

\int_{0}^{2} 16 x^{3} d x = 64

\int_{0}^{2} x^{6} d x = \frac{128}{7}

= π (128 - 64 + \frac{128}{7})

化简

π (128 - 64 + \frac{128}{7}) : π \frac{576}{7}

= π \frac{576}{7}