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受欢迎的微积分 >

integral from 1 to 2 of pi(2-x/2)^2

解答

\int_{1}^{2} π (2 - \frac{x}{2})^{2} d x

解答

\frac{19 π}{12}

+1

十进制

4.97418 \dots

求解步骤

\int_{1}^{2} π (2 - \frac{x}{2})^{2} d x

提出常数:

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

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

使用换元积分法

= π \cdot \int_{\frac{3}{2}}^{1} - 2 u^{2} d u

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

= π (- \int_{1}^{\frac{3}{2}} - 2 u^{2} d u)

提出常数:

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

= π (- (- 2 \cdot \int_{1}^{\frac{3}{2}} u^{2} d u))

使用幂法则

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

化简

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

带入上下限计算后得:

\frac{19}{24}

= 2 π \frac{19}{24}

化简

= \frac{19 π}{12}

作图

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