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Find the area of the region bounded by the curves f(x) = 9*x² - 9*x + 2, g(x) = 9*x² - 18*x + 8, and the vertical line x = 1.

1. 1/3
2. 1/2
3. 2/3
4. 3/4

Correct answer: 1/2

Solution

The difference f(x)-g(x) = 9x-6 = 0 at x=2/3. The two parabolas intersect at x=2/3. We need the left boundary: find where each curve hits x-axis or use the natural region. Integrate |f-g| = |9x-6| from 2/3 to 1 (since the region is bounded by x=1 on right and intersection at x=2/3 on left). Area = integral from 2/3 to 1 of (9x-6)dx = [9x²/2 - 6x] from 2/3 to 1 = (9/2-6) - (9*(4/9)/2 - 4) = (-3/2) - (2-4) = -3/2+2 = 1/2.

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