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Find the total area enclosed between g(x) = min{x, sin x, 1/2}, the x-axis, and the vertical lines x = -pi and x = pi.

1. pi² + 2*sqrt(3)
2. 2 + pi/3 + 2 - sqrt(3)
3. pi²/2 + pi/3 + 2 - sqrt(3)
4. pi/3 + 3 - sqrt(3)

Correct answer: pi²/2 + pi/3 + 2 - sqrt(3)

Solution

The function equals x on [-pi,0], sin x on [0,pi/6] union [5pi/6,pi], and 1/2 on [pi/6,5pi/6]. Integrating absolute values gives pi²/2 + (1-sqrt(3)/2) + pi/3 + (1-sqrt(3)/2) = pi²/2 + pi/3 + 2 - sqrt(3).

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