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受欢迎的微积分 >

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

解答

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

解答

\frac{80 π}{3}

+1

十进制

83.77580 \dots

求解步骤

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

使用分布积分法

= π [\frac{1}{25} x (20 - 4 x)^{2} - \int - \frac{8}{25} x (20 - 4 x) d x]_{0}^{5}

\int - \frac{8}{25} x (20 - 4 x) d x = - \frac{8}{25} (10 x^{2} - \frac{4 x ^{3}}{3})

= π [\frac{1}{25} x (20 - 4 x)^{2} - (- \frac{8}{25} (10 x^{2} - \frac{4 x ^{3}}{3}))]_{0}^{5}

化简

π [\frac{1}{25} x (20 - 4 x)^{2} - (- \frac{8}{25} (10 x^{2} - \frac{4 x ^{3}}{3}))]_{0}^{5} : π [\frac{1}{75} (16 x^{3} - 240 x^{2} + 1200 x)]_{0}^{5}

= π [\frac{1}{75} (16 x^{3} - 240 x^{2} + 1200 x)]_{0}^{5}

带入上下限计算后得:

\frac{80}{3}

= π \frac{80}{3}

化简

= \frac{80 π}{3}

作图

流行的例子

\int 3 u d u

slopeintercept (8.1)(24.9)

s l o p e in t erce pt (8.1) (24.9)

x^{2} y^{^{'}} = y^{2} + x y

斜率 (3,3),(-3,-1)

s l o p e (3, 3), (- 3, - 1)

limit as x approaches 2-of 2-x

x \to 2 - lim (2 - x)