What is area 3x,(6x)/7 ,55-x^2 ?

The answer to area 3x,(6x)/7 ,55-x^2 is 83865/686

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area 3x,(6x)/7 ,55-x^2

area

3 x, \frac{6 x}{7}, 55 - x^{2}

\frac{83865}{686}

Solution steps

To find the intersection points solve

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

3 x = \frac{6 x}{7} : x = 0

3 x = 55 - x^{2} : x = - \frac{3 + 229}{2}, x = \frac{229 - 3}{2}

\frac{6 x}{7} = 55 - x^{2} : x = - \frac{55}{7}, x = 7

= \int_{- \frac{3 + 229}{2}}^{- \frac{55}{7}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{55}{7}}^{0} ∣ f_{2} (x) - f_{1} (x) ∣ d x + \int_{0}^{\frac{229 - 3}{2}} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{\frac{229 - 3}{2}}^{7} ∣ f_{2} (x) - f_{3} (x) ∣ d x

= \int_{- \frac{3 + 229}{2}}^{- \frac{55}{7}} 3 x - (55 - x^{2}) d x + \int_{- \frac{55}{7}}^{0} \frac{6 x}{7} - 3 x d x + \int_{0}^{\frac{229 - 3}{2}} 3 x - \frac{6 x}{7} d x + \int_{\frac{229 - 3}{2}}^{7} \frac{6 x}{7} - (55 - x^{2}) d x

\int_{- \frac{3 + 229}{2}}^{- \frac{55}{7}} 3 x - (55 - x^{2}) d x = - \frac{747175}{2058} + \frac{1017 + 229 229}{12}

\int_{- \frac{55}{7}}^{0} \frac{6 x}{7} - 3 x d x = \frac{45375}{686}

\int_{0}^{\frac{229 - 3}{2}} 3 x - \frac{6 x}{7} d x = \frac{15 ( 119 - 3 229 )}{28}

\int_{\frac{229 - 3}{2}}^{7} \frac{6 x}{7} - (55 - x^{2}) d x = - \frac{367 229 + 2401}{21} + 385

= - \frac{747175}{2058} + \frac{1017 + 229 229}{12} + \frac{45375}{686} + \frac{15 ( 119 - 3 229 )}{28} - \frac{367 229 + 2401}{21} + 385

Simplify

= \frac{83865}{686}

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What is area 3x,(6x)/7 ,55-x^2 ?
The answer to area 3x,(6x)/7 ,55-x^2 is 83865/686