面积 y=9-x^2,y=0,x

解答

面积

y = 9 - x^{2}, y = 0, x

\frac{55}{6}

十进制

9.16666 \dots

求解步骤

要找到相交点，解

f_{i} (x) = f_{j} (x)

9 - x^{2} = 0 : x = 3, x = - 3

9 - x^{2} = x : x = - \frac{1 + 37}{2}, x = \frac{37 - 1}{2}

0 = x : x = 0

= \int_{- \frac{1 + 37}{2}}^{- 3} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- 3}^{0} ∣ f_{3} (x) - f_{2} (x) ∣ d x + \int_{0}^{\frac{37 - 1}{2}} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{\frac{37 - 1}{2}}^{3} ∣ f_{1} (x) - f_{2} (x) ∣ d x

= \int_{- \frac{1 + 37}{2}}^{- 3} 9 - x^{2} - x d x + \int_{- 3}^{0} ∣ x - 0 ∣ d x + \int_{0}^{\frac{37 - 1}{2}} ∣ 0 - x ∣ d x + \int_{\frac{37 - 1}{2}}^{3} 9 - x^{2} - 0 d x

\int_{- \frac{1 + 37}{2}}^{- 3} 9 - x^{2} - x d x = - \frac{45}{2} + \frac{55 + 37 37}{12}

\int_{- 3}^{0} ∣ x - 0 ∣ d x = \frac{9}{2}

\int_{0}^{\frac{37 - 1}{2}} ∣ 0 - x ∣ d x = \frac{19 - 37}{4}

\int_{\frac{37 - 1}{2}}^{3} 9 - x^{2} - 0 d x = \frac{- 17 37 - 1}{6} + 18

= - \frac{45}{2} + \frac{55 + 37 37}{12} + \frac{9}{2} + \frac{19 - 37}{4} + \frac{- 17 37 - 1}{6} + 18

化简

= \frac{55}{6}

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