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受欢迎的微积分 >

integral from-10 to 8 of sqrt(80-2x-x^2)

解答

\int_{- 10}^{8} 80 - 2 x - x^{2} d x

解答

\frac{81 π}{2}

+1

十进制

127.23450 \dots

求解步骤

\int_{- 10}^{8} 80 - 2 x - x^{2} d x

对

80 - 2 x - x^{2}

配方:

- (x + 1)^{2} + 81

= \int_{- 10}^{8} - (x + 1)^{2} + 81 d x

使用换元积分法

= \int_{- 9}^{9} - u^{2} + 81 d u

使用三角换元法

= \int_{- \frac{π}{2}}^{\frac{π}{2}} 81 cos^{2} (v) d v

提出常数:

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

= 81 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{2} (v) d v

使用三角恒等式改写

= 81 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + cos ( 2 v )}{2} d v

提出常数:

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

= 81 \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + cos (2 v) d v

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

= 81 \cdot \frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d v + \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 v) d v)

\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d v = π

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 v) d v = 0

= 81 \cdot \frac{1}{2} (π + 0)

化简

81 \cdot \frac{1}{2} (π + 0) : \frac{81 π}{2}

= \frac{81 π}{2}

作图

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