面積 3x, 3/4 x,70-x^2

解

面積

3 x, \frac{3}{4} x, 70 - x^{2}

\frac{20727}{128}

十進法表記

161.92968 \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{3}{4} x : x = 0

3 x = 70 - x^{2} : x = - 10, x = 7

\frac{3}{4} x = 70 - x^{2} : x = - \frac{35}{4}, x = 8

= \int_{- 10}^{- \frac{35}{4}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{35}{4}}^{0} ∣ f_{2} (x) - f_{1} (x) ∣ d x + \int_{0}^{7} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{7}^{8} ∣ f_{2} (x) - f_{3} (x) ∣ d x

= \int_{- 10}^{- \frac{35}{4}} 3 x - (70 - x^{2}) d x + \int_{- \frac{35}{4}}^{0} \frac{3}{4} x - 3 x d x + \int_{0}^{7} 3 x - \frac{3}{4} x d x + \int_{7}^{8} \frac{3}{4} x - (70 - x^{2}) d x

\int_{- 10}^{- \frac{35}{4}} 3 x - (70 - x^{2}) d x = \frac{2425}{192}

\int_{- \frac{35}{4}}^{0} \frac{3}{4} x - 3 x d x = \frac{11025}{128}

\int_{0}^{7} 3 x - \frac{3}{4} x d x = \frac{441}{8}

\int_{7}^{8} \frac{3}{4} x - (70 - x^{2}) d x = \frac{193}{24}

= \frac{2425}{192} + \frac{11025}{128} + \frac{441}{8} + \frac{193}{24}

簡素化

= \frac{20727}{128}