面積 66x, 2/3 x,40-x^2

解

面積

66 x, \frac{2}{3} x, 40 - x^{2}

\frac{4092872}{81}

十進法表記

50529.28395 \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)

66 x = \frac{2}{3} x : x = 0

66 x = 40 - x^{2} : x = - 33 - 1129, x = 1129 - 33

\frac{2}{3} x = 40 - x^{2} : x = - \frac{20}{3}, x = 6

= \int_{- 33 - 1129}^{- \frac{20}{3}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{20}{3}}^{0} ∣ f_{2} (x) - f_{1} (x) ∣ d x + \int_{0}^{1129 - 33} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{1129 - 33}^{6} ∣ f_{2} (x) - f_{3} (x) ∣ d x

= \int_{- 33 - 1129}^{- \frac{20}{3}} 66 x - (40 - x^{2}) d x + \int_{- \frac{20}{3}}^{0} \frac{2}{3} x - 66 x d x + \int_{0}^{1129 - 33} 66 x - \frac{2}{3} x d x + \int_{1129 - 33}^{6} \frac{2}{3} x - (40 - x^{2}) d x

\int_{- 33 - 1129}^{- \frac{20}{3}} 66 x - (40 - x^{2}) d x = \frac{1915118 + 60966 1129}{81}

\int_{- \frac{20}{3}}^{0} \frac{2}{3} x - 66 x d x = \frac{39200}{27}

\int_{0}^{1129 - 33} 66 x - \frac{2}{3} x d x = 66 (1109 - 331129) - \frac{2 ( 1109 - 33 1129 )}{3}

\int_{1129 - 33}^{6} \frac{2}{3} x - (40 - x^{2}) d x = - 401129 + \frac{4330 1129 - 145526}{3} + 1488

= \frac{1915118 + 60966 1129}{81} + \frac{39200}{27} + 66 (1109 - 331129) - \frac{2 ( 1109 - 33 1129 )}{3} - 401129 + \frac{4330 1129 - 145526}{3} + 1488

簡素化

= \frac{4092872}{81}