Explanations would be much appreciated! Find the area of the hounded region encl

Question

Explanations would be much appreciated!

Find the area of the hounded region enclosed by the parabola y = 10 + 3x - x2 and the straight line y = 4x + 4. Area = 113/6 sq. units Area = 137/6 sq. units Area = 131/6 sq. units Area = 119/6 sq. units Area = 125/6 sq. units

Explanation / Answer

First solve the two equations to get the 2 points (say A and B) where these 2 curves[f(x) and g(x)] meet. Then, find integral of |f(x) - g(x)| from A to B to calculate area. f(x) = g(x) ---> 10 + 3x - x^2 = 4x + 4, so x^2 + x - 6 = 0, so x^2 + 3x - 2x - 6 = 0, so (x-2)(x-3) = 0, so x = 2, 3. Now, area = integral[f(x) - g(x)] from 2 to 3 ---> integral [x^2 + x - 6] from 2 to 3 ---> [0.33x^3 + 0.5x^2 - 6x]from 2 to 3 ---> 0.333(3^3 - 2^3) + 0.5(3^2 - 2^2) - 6(3-2) = 6.33 + 2.5 - 6 = 2.83 units Ans

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