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受欢迎的微积分 >

integral from 0 to 10 of xsqrt(100-x^2)

解答

\int_{0}^{10} x 100 - x^{2} d x

解答

\frac{1000}{3}

+1

十进制

333.33333 \dots

求解步骤

\int_{0}^{10} x 100 - x^{2} d x

使用换元积分法

= \int_{100}^{0} - \frac{u}{2} d u

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

= - \int_{0}^{100} - \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_{0}^{100} u d u)

使用幂法则

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{100})

化简

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{100}

带入上下限计算后得:

\frac{2000}{3}

= \frac{1}{2} \cdot \frac{2000}{3}

化简

= \frac{1000}{3}

作图

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