integral from 0 to 4 of pi((20-5x)/4)^2

解答

\int_{0}^{4} π (\frac{20 - 5 x}{4})^{2} d x

\frac{100 π}{3}

十进制

104.71975 \dots

求解步骤

\int_{0}^{4} π (\frac{20 - 5 x}{4})^{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}^{4} (\frac{20 - 5 x}{4})^{2} d x

使用分布积分法

= π [\frac{1}{16} x (20 - 5 x)^{2} - \int - \frac{5}{8} x (- 5 x + 20) d x]_{0}^{4}

\int - \frac{5}{8} x (- 5 x + 20) d x = - \frac{5}{8} (- \frac{5 x ^{3}}{3} + 10 x^{2})

= π [\frac{1}{16} x (20 - 5 x)^{2} - (- \frac{5}{8} (- \frac{5 x ^{3}}{3} + 10 x^{2}))]_{0}^{4}

化简

π [\frac{1}{16} x (20 - 5 x)^{2} - (- \frac{5}{8} (- \frac{5 x ^{3}}{3} + 10 x^{2}))]_{0}^{4} : π [\frac{1}{48} (25 x^{3} - 300 x^{2} + 1200 x)]_{0}^{4}

= π [\frac{1}{48} (25 x^{3} - 300 x^{2} + 1200 x)]_{0}^{4}

带入上下限计算后得:

\frac{100}{3}

= π \frac{100}{3}

化简

= \frac{100 π}{3}

d er i v a t i v e y = 4 x^{2} + 1

\int 2 x^{3} e^{x^{2}} d x

\int (\frac{1}{2 x}) d x

a re a y = 3 x, x = 0, x = 4

x d y - y d x = 2 x^{2} y^{2} d y, y (1) = 2