The area of the region defined by S = (x, y) : 2x^2 y 4 - 2|x| is

Step-by-step solution

The region S is bounded below by the parabola y = 2x^2 and above by the curve y = 4 - 2|x| . Since both y = 2x^2 and y = 4 - 2|x| are even functions, the region is symmetric about the y -axis. We can find the area in the first quadrant ( x 0 ) and multiply it by 2 . In the first quadrant, the upper curve is y = 4 - 2x . To find the point of intersection of the two curves in the first quadrant, we solve: 2x^2 = 4 - 2x 2x^2 + 2x - 4 = 0 x^2 + x - 2 = 0 (x + 2)(x - 1) = 0 Since x 0 , the intersection point is at x = 1