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受欢迎的微积分 >

integral from 0 to 2 of (xsqrt(9-x^2))

解答

\int_{0}^{2} (x 9 - x^{2}) d x

解答

\frac{27 - 5 5}{3}

+1

十进制

5.27322 \dots

求解步骤

\int_{0}^{2} x 9 - x^{2} d x

使用换元积分法

= \int_{9}^{5} - \frac{u}{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{5}^{9} - \frac{u}{2} d u

提出常数:

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

= - (- \frac{1}{2} \cdot \int_{5}^{9} u d u)

使用幂法则

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{9})

化简

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{9}

带入上下限计算后得:

18 - \frac{10 5}{3}

= \frac{1}{2} (18 - \frac{10 5}{3})

化简

= \frac{27 - 5 5}{3}

作图

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