integral from 0 to 9 of pi(sqrt(x-1))^2

解答

\int_{0}^{9} π (x - 1)^{2} d x

π \frac{63}{2}

十进制

98.96016 \dots

求解步骤

\int_{0}^{9} π (x - 1)^{2} d x

提出常数:

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

= π \cdot \int_{0}^{9} (x - 1)^{2} d x

化简

(x - 1)^{2} : x - 1

= π \cdot \int_{0}^{9} x - 1 d x

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

= π (\int_{0}^{9} x d x - \int_{0}^{9} 1 d x)

\int_{0}^{9} x d x = \frac{81}{2}

\int_{0}^{9} 1 d x = 9

= π (\frac{81}{2} - 9)

化简

\frac{81}{2} - 9 : \frac{63}{2}

= π \frac{63}{2}

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