integral from 0 to a of rsqrt(a^2-r^2)

解答

\int_{0}^{a} r a^{2} - r^{2} d r

\frac{( a ^{2} ) ^{\frac{3}{2}}}{3}

求解步骤

\int_{0}^{a} r a^{2} - r^{2} d r

使用换元积分法

= \int_{a^{2}}^{0} - \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_{a^{2}}^{0} u d u

使用幂法则

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

带入上下限计算后得:

- \frac{2}{3} (a^{2})^{\frac{3}{2}}

= - \frac{1}{2} (- \frac{2}{3} (a^{2})^{\frac{3}{2}})

化简

= \frac{( a ^{2} ) ^{\frac{3}{2}}}{3}

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