面積 y= 3/x ,y=12x,y= x/3

解

面積

y = \frac{3}{x}, y = 12 x, y = \frac{x}{3}

6 ln (6)

十進法表記

10.75055 \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)

\frac{3}{x} = 12 x : x = \frac{1}{2}, x = - \frac{1}{2}

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

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

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

= \int_{- 3}^{- \frac{1}{2}} \frac{3}{x} - \frac{x}{3} d x + \int_{- \frac{1}{2}}^{0} \frac{x}{3} - 12 x d x + \int_{0}^{\frac{1}{2}} 12 x - \frac{x}{3} d x + \int_{\frac{1}{2}}^{3} \frac{3}{x} - \frac{x}{3} d x

\int_{- 3}^{- \frac{1}{2}} \frac{3}{x} - \frac{x}{3} d x = 3 ln (6) - \frac{35}{24}

\int_{- \frac{1}{2}}^{0} \frac{x}{3} - 12 x d x = \frac{35}{24}

\int_{0}^{\frac{1}{2}} 12 x - \frac{x}{3} d x = \frac{35}{24}

\int_{\frac{1}{2}}^{3} \frac{3}{x} - \frac{x}{3} d x = 3 ln (6) - \frac{35}{24}

= 3 ln (6) - \frac{35}{24} + \frac{35}{24} + \frac{35}{24} + 3 ln (6) - \frac{35}{24}

簡素化

= 6 ln (6)