The area (in sq. units) of the region R = (x, y) : |y| x 2 - y^2 is

Step-by-step solution

The given region is bounded by the curves x = |y| and x = 2 - y^2 . To find the points of intersection, we equate the two expressions for x : |y| = 2 - y^2 |y|^2 + |y| - 2 = 0 (|y| + 2)(|y| - 1) = 0 Since |y| 0 , we have |y| = 1 , which gives y = -1 and y = 1 . The area of the region is given by the integral with respect to y : Area = _ -1 ¹ (x_ right - x_ left ) dy Area = _ -1 ¹ (2 - y^2 - |y|) dy Since the integrand is an even function, we can simplify this to: Area = 2 ₀¹ (2 - y^2 - y) dy Area = 2 [ 2y - y^3 3 -