面积 y=7x,y= 7/4 x,y=78-x^2

解答

面积

y = 7 x, y = \frac{7}{4} x, y = 78 - x^{2}

\frac{59619}{128}

十进制

465.77343 \dots

求解步骤

要找到相交点，解

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

7 x = \frac{7}{4} x : x = 0

7 x = 78 - x^{2} : x = - 13, x = 6

\frac{7}{4} x = 78 - x^{2} : x = - \frac{39}{4}, x = 8

= \int_{- 13}^{- \frac{39}{4}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{39}{4}}^{0} ∣ f_{2} (x) - f_{1} (x) ∣ d x + \int_{0}^{6} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{6}^{8} ∣ f_{2} (x) - f_{3} (x) ∣ d x

= \int_{- 13}^{- \frac{39}{4}} 7 x - (78 - x^{2}) d x + \int_{- \frac{39}{4}}^{0} \frac{7}{4} x - 7 x d x + \int_{0}^{6} 7 x - \frac{7}{4} x d x + \int_{6}^{8} \frac{7}{4} x - (78 - x^{2}) d x

\int_{- 13}^{- \frac{39}{4}} 7 x - (78 - x^{2}) d x = \frac{17069}{192}

\int_{- \frac{39}{4}}^{0} \frac{7}{4} x - 7 x d x = \frac{31941}{128}

\int_{0}^{6} 7 x - \frac{7}{4} x d x = \frac{189}{2}

\int_{6}^{8} \frac{7}{4} x - (78 - x^{2}) d x = \frac{197}{6}

= \frac{17069}{192} + \frac{31941}{128} + \frac{189}{2} + \frac{197}{6}

化简

= \frac{59619}{128}

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