The area of the region bounded by the curves y = |x^2 - 2x| and y = x is

Step-by-step solution

To find the area bounded by y = |x^2 - 2x| and y = x , we first determine their points of intersection by setting |x^2 - 2x| = x . Since y = x and y 0 , we must have x 0 . Case 1: x^2 - 2x 0 x 2 (since x 0 ). x^2 - 2x = x x^2 - 3x = 0 x = 0, 3 . The valid intersection is x = 3 . Case 2: x^2 - 2x -(x^2 - 2x) = x 2x - x^2 = x x^2 - x = 0 x = 0, 1 . The valid intersection is x = 1 . The intersection points are x = 0, 1, 3 . The modulus expression changes sign at x = 2 . This divides the region into three intervals: [0