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Let f(x) = x for 0 <= x < 1/2, f(1/2) = 1/2, and f(x) = 1 - x for 1/2 < x <= 1; and g(x) = (x - 1/2)² for all real x. Find the area (in square units) of the region bounded by y = f(x) and y = g(x) between the lines 2x = 1 and 2x = sqrt(3).

1. 1/3 + sqrt(3)/4
2. sqrt(3)/4 - 1/3
3. 1/2 + sqrt(3)/4
4. 1/2 - sqrt(3)/4

Correct answer: sqrt(3)/4 - 1/3

Solution

Between x = 1/2 and x = sqrt(3)/2 the line 1 - x lies above the parabola, and the integral evaluates to sqrt(3)/4 - 1/3.

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