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Find the area (in square units) of the region defined by the set {(x, y) belonging to R x R: x >= 0, 2x² <= y <= 4 - 2x}.

1. 8/3
2. 17/3
3. 13/3
4. 7/3

Correct answer: 8/3

Solution

The region is bounded by y = 2x² (parabola) below and y = 4-2x (line) above, for x>=0. Setting equal: 2x² = 4-2x => x²+x-2=0 => (x-1)(x+2)=0 => x=1 (for x>=0). At x=0: y ranges from 0 to 4. Integrating from x=0 to x=1: integral of (4-2x-2x²)dx = [4x - x² - (2/3)x³] from 0 to 1 = 4 - 1 - 2/3 = 3 - 2/3 = 7/3. Wait, that gives 7/3. Let me recheck: 4(1)-1²-(2/3)(1)³ = 4 - 1 - 2/3 = 3 - 2/3 = 9/3 - 2/3 = 7/3. So answer is 7/3.

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