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

解答

\int_{0}^{2} 2 π (x - 3) x^{2} d x

- 8 π

十进制

- 25.13274 \dots

求解步骤

\int_{0}^{2} 2 π (x - 3) x^{2} d x

提出常数:

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

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

乘开

(x - 3) x^{2} : x^{3} - 3 x^{2}

= 2 π \cdot \int_{0}^{2} x^{3} - 3 x^{2} 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

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

\int_{0}^{2} x^{3} d x = 4

\int_{0}^{2} 3 x^{2} d x = 8

= 2 π (4 - 8)

化简

= - 8 π

\int_{- 1}^{1} (x^{3} - 3 x^{2} - x + 3) d x

\int_{0}^{1} \frac{2}{x ^{2}} d x

\int_{2}^{6} - x^{2} + 8 d x

\int_{0}^{2 π} 1 + sin (θ) d θ

\int_{- 2}^{0} 5 x^{4} - 3 x^{2} + x - 3 d x