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The area of the region bounded by the curve y = sqrt(4 - x²) (upper semicircle of radius 2), the parabola x = sqrt(3) * y, and the x-axis is

1. (1/2)(1 + 2*pi/3 - 2*sqrt(3)/3)
2. 1 + 2*pi/3 - 2*sqrt(3)/3
3. (1/2)(2*sqrt(3)/3 - 1 - 2*pi/3)
4. None of these

Correct answer: (1/2)(1 + 2*pi/3 - 2*sqrt(3)/3)

Solution

The curve y = sqrt((4-x)²) = |4-x| gives only a V-shape, not a closed region with a parabola and x-axis that matches the options. The intended curve is y = sqrt(4 - x²), the upper semicircle of radius 2 centred at origin. The parabola x = sqrt(3)*y rewrites as y = x/sqrt(3). Intersection with semicircle: x² + x²/3 = 4 -> x² = 3 -> x = sqrt(3), y = 1. The bounded region lies between the y-axis, the parabola, and the arc from (0,2) to (sqrt(3),1), plus the area between the parabola and x-axis from 0 to sqrt(3).

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