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Find the area of the region bounded by the curves f(x) = x³ - 3x and g(x) = 2x² that lies between their intersection points in the second quadrant (x in [-1, 0]).

1. 12/13
2. 7/12
3. 2/3
4. 3/5

Correct answer: 7/12

Solution

Intersection: x³-3x = 2x² => x³-2x²-3x = 0 => x(x²-2x-3) = 0 => x(x-3)(x+1) = 0. Roots: x = -1, 0, 3. On (-1,0): f(-0.5) - g(-0.5) = 1.375 - 0.5 = 0.875 > 0. Area = integral₋₁⁰(x³ - 3x - 2x²)dx. Antiderivative: x⁴/4 - 3x²/2 - 2x³/3. At x=0: 0. At x=-1: 1/4 - 3/2 + 2/3 = 3/12 - 18/12 + 8/12 = -7/12. Area = 0-(-7/12) = 7/12.

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