面積 y=27-x^2,y=6x,y= 2/5 x

解

面積

y = 27 - x^{2}, y = 6 x, y = \frac{2}{5} x

\frac{70196}{375}

十進法表記

187.18933 \dots

解答ステップ

交点を見つけるには以下を解く:

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

27 - x^{2} = 6 x : x = - 9, x = 3

27 - x^{2} = \frac{2}{5} x : x = - \frac{27}{5}, x = 5

6 x = \frac{2}{5} x : x = 0

= \int_{- 9}^{- \frac{27}{5}} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{- \frac{27}{5}}^{0} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{0}^{3} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{3}^{5} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- 9}^{- \frac{27}{5}} 27 - x^{2} - 6 x d x + \int_{- \frac{27}{5}}^{0} 6 x - \frac{2}{5} x d x + \int_{0}^{3} 6 x - \frac{2}{5} x d x + \int_{3}^{5} 27 - x^{2} - \frac{2}{5} x d x

\int_{- 9}^{- \frac{27}{5}} 27 - x^{2} - 6 x d x = \frac{7776}{125}

\int_{- \frac{27}{5}}^{0} 6 x - \frac{2}{5} x d x = \frac{10206}{125}

\int_{0}^{3} 6 x - \frac{2}{5} x d x = \frac{126}{5}

\int_{3}^{5} 27 - x^{2} - \frac{2}{5} x d x = \frac{272}{15}

= \frac{7776}{125} + \frac{10206}{125} + \frac{126}{5} + \frac{272}{15}

簡素化

= \frac{70196}{375}