面積 5x, 1/7 x,50-x^2

解

面積

5 x, \frac{1}{7} x, 50 - x^{2}

\frac{271337}{1029}

十進法表記

263.68999 \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)

5 x = \frac{1}{7} x : x = 0

5 x = 50 - x^{2} : x = - 10, x = 5

\frac{1}{7} x = 50 - x^{2} : x = - \frac{50}{7}, x = 7

= \int_{- 10}^{- \frac{50}{7}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{50}{7}}^{0} ∣ f_{2} (x) - f_{1} (x) ∣ d x + \int_{0}^{5} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{5}^{7} ∣ f_{2} (x) - f_{3} (x) ∣ d x

= \int_{- 10}^{- \frac{50}{7}} 5 x - (50 - x^{2}) d x + \int_{- \frac{50}{7}}^{0} \frac{1}{7} x - 5 x d x + \int_{0}^{5} 5 x - \frac{1}{7} x d x + \int_{5}^{7} \frac{1}{7} x - (50 - x^{2}) d x

\int_{- 10}^{- \frac{50}{7}} 5 x - (50 - x^{2}) d x = \frac{55000}{1029}

\int_{- \frac{50}{7}}^{0} \frac{1}{7} x - 5 x d x = \frac{42500}{343}

\int_{0}^{5} 5 x - \frac{1}{7} x d x = \frac{425}{7}

\int_{5}^{7} \frac{1}{7} x - (50 - x^{2}) d x = \frac{538}{21}

= \frac{55000}{1029} + \frac{42500}{343} + \frac{425}{7} + \frac{538}{21}

簡素化

= \frac{271337}{1029}