Consider the region S = (x, y) : |x^2 - 3| y 2 . The exact area of the region S is

Step-by-step solution

The region S is defined by the inequalities |x^2 - 3| y 2 . For the region to exist, the lower bound must be less than or equal to the upper bound, so |x^2 - 3| 2 . This implies -2 x^2 - 3 2 1 x^2 5 . Thus, the domain of x is [- 5 , -1] [1, 5 ] . The interval (-1, 1) is excluded because the curve y = |x^2 - 3| is above y = 2 in this region. By symmetry about the y -axis, the required area is: A = 2 ₁^ 5 (2 - |x^2 - 3|) dx We split the integral at x = 3 where x^2 - 3 changes sign: A = 2 ( ₁^ 3 (2 - (3 - x^2)) dx + _