面積 5/x ,y=3x,y= 1/3 x

解

面積

\frac{5}{x}, y = 3 x, y = \frac{1}{3} x

10 ln (3)

十進法表記

10.98612 \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)

3 x = \frac{1}{3} x : x = 0

3 x = \frac{5}{x} : x = \frac{5}{3}, x = - \frac{5}{3}

\frac{1}{3} x = \frac{5}{x} : x = 15, x = - 15

= \int_{- 15}^{- \frac{5}{3}} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{- \frac{5}{3}}^{0} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{0}^{\frac{5}{3}} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{\frac{5}{3}}^{15} ∣ f_{2} (x) - f_{3} (x) ∣ d x

= \int_{- 15}^{- \frac{5}{3}} \frac{1}{3} x - \frac{5}{x} d x + \int_{- \frac{5}{3}}^{0} 3 x - \frac{1}{3} x d x + \int_{0}^{\frac{5}{3}} 3 x - \frac{1}{3} x d x + \int_{\frac{5}{3}}^{15} \frac{1}{3} x - \frac{5}{x} d x

\int_{- 15}^{- \frac{5}{3}} \frac{1}{3} x - \frac{5}{x} d x = - \frac{20}{9} + 5 ln (3)

\int_{- \frac{5}{3}}^{0} 3 x - \frac{1}{3} x d x = \frac{20}{9}

\int_{0}^{\frac{5}{3}} 3 x - \frac{1}{3} x d x = \frac{20}{9}

\int_{\frac{5}{3}}^{15} \frac{1}{3} x - \frac{5}{x} d x = - \frac{20}{9} + 5 ln (3)

= - \frac{20}{9} + 5 ln (3) + \frac{20}{9} + \frac{20}{9} - \frac{20}{9} + 5 ln (3)

簡素化

= 10 ln (3)