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受欢迎的微积分 >

integral from 0 to 6 of x^2sqrt(36-x^2)

解答

\int_{0}^{6} x^{2} 36 - x^{2} d x

解答

81 π

+1

十进制

254.46900 \dots

求解步骤

\int_{0}^{6} x^{2} 36 - x^{2} d x

使用三角换元法

= \int_{0}^{\frac{π}{2}} 1296 sin^{2} (u) cos^{2} (u) d u

提出常数:

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

= 1296 \cdot \int_{0}^{\frac{π}{2}} sin^{2} (u) cos^{2} (u) d u

使用三角恒等式改写

= 1296 \cdot \int_{0}^{\frac{π}{2}} \frac{1 - cos ( 4 u )}{8} d u

提出常数:

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

= 1296 \cdot \frac{1}{8} \cdot \int_{0}^{\frac{π}{2}} 1 - cos (4 u) d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 1296 \cdot \frac{1}{8} (\int_{0}^{\frac{π}{2}} 1 d u - \int_{0}^{\frac{π}{2}} cos (4 u) d u)

\int_{0}^{\frac{π}{2}} 1 d u = \frac{π}{2}

\int_{0}^{\frac{π}{2}} cos (4 u) d u = 0

= 1296 \cdot \frac{1}{8} (\frac{π}{2} - 0)

化简

1296 \cdot \frac{1}{8} (\frac{π}{2} - 0) : 81 π

= 81 π

作图

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