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人気のある微分積分 >

面積 y=2x^2,y=8x^2,5x+y=3

解

面積

y = 2 x^{2}, y = 8 x^{2}, 5 x + y = 3

解

\frac{4157}{384}

+1

十進法表記

10.82552 \dots

解答ステップ

以下のために

y

を分離:

5 x + y = 3 : y = 3 - 5 x

f_{1} (x) = 2 x^{2}

f_{2} (x) = 8 x^{2}

交点を見つけるには以下を解く:

f_{i} (x) = f_{j} (x)

2 x^{2} = 8 x^{2} : x = 0

2 x^{2} = 3 - 5 x : x = \frac{1}{2}, x = - 3

8 x^{2} = 3 - 5 x : x = \frac{3}{8}, x = - 1

= \int_{- 3}^{- 1} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- 1}^{0} ∣ f_{2} (x) - f_{1} (x) ∣ d x + \int_{0}^{\frac{3}{8}} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{\frac{3}{8}}^{\frac{1}{2}} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- 3}^{- 1} 2 x^{2} - (3 - 5 x) d x + \int_{- 1}^{0} 8 x^{2} - 2 x^{2} d x + \int_{0}^{\frac{3}{8}} 2 x^{2} - 8 x^{2} d x + \int_{\frac{3}{8}}^{\frac{1}{2}} 2 x^{2} - (3 - 5 x) d x

\int_{- 3}^{- 1} 2 x^{2} - (3 - 5 x) d x = \frac{26}{3}

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

\int_{0}^{\frac{3}{8}} 2 x^{2} - 8 x^{2} d x = \frac{27}{256}

\int_{\frac{3}{8}}^{\frac{1}{2}} 2 x^{2} - (3 - 5 x) d x = \frac{41}{768}

= \frac{26}{3} + 2 + \frac{27}{256} + \frac{41}{768}

簡素化

= \frac{4157}{384}

グラフ

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