面積 y= 2/x ,y= 1/2 x,y=8x

解

面積

y = \frac{2}{x}, y = \frac{1}{2} x, y = 8 x

8 ln (2)

十進法表記

5.54517 \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{2}{x} = \frac{1}{2} x : x = 2, x = - 2

\frac{2}{x} = 8 x : x = \frac{1}{2}, x = - \frac{1}{2}

\frac{1}{2} x = 8 x : x = 0

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

= \int_{- 2}^{- \frac{1}{2}} \frac{2}{x} - \frac{1}{2} x d x + \int_{- \frac{1}{2}}^{0} \frac{1}{2} x - 8 x d x + \int_{0}^{\frac{1}{2}} \frac{1}{2} x - 8 x d x + \int_{\frac{1}{2}}^{2} \frac{2}{x} - \frac{1}{2} x d x

\int_{- 2}^{- \frac{1}{2}} \frac{2}{x} - \frac{1}{2} x d x = 4 ln (2) - \frac{15}{16}

\int_{- \frac{1}{2}}^{0} \frac{1}{2} x - 8 x d x = \frac{15}{16}

\int_{0}^{\frac{1}{2}} \frac{1}{2} x - 8 x d x = \frac{15}{16}

\int_{\frac{1}{2}}^{2} \frac{2}{x} - \frac{1}{2} x d x = 4 ln (2) - \frac{15}{16}

= 4 ln (2) - \frac{15}{16} + \frac{15}{16} + \frac{15}{16} + 4 ln (2) - \frac{15}{16}

簡素化

= 8 ln (2)