integral from 0 to 2a of sqrt(2ay)

解答

\int_{0}^{2 a} 2 a y d y

\frac{8 a a ^{\frac{3}{2}}}{3}

求解步骤

\int_{0}^{2 a} 2 a y d y

使用根式运算法则:

n ab = n a n b,

假定

a \geq 0, b \geq 0

= \int_{0}^{2 a} 2 a y d y

提出常数:

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

= 2 \cdot \int_{0}^{2 a} a y d y

a y = a y

= 2 \cdot \int_{0}^{2 a} a y d y

提出常数:

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

= 2 a \cdot \int_{0}^{2 a} y d y

使用幂法则

= 2 a [\frac{2}{3} y^{\frac{3}{2}}]_{0}^{2 a}

带入上下限计算后得:

\frac{4 2}{3} a^{\frac{3}{2}}

= 2 a \frac{4 2}{3} a^{\frac{3}{2}}

化简

= \frac{8 a a ^{\frac{3}{2}}}{3}

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