The area of the region bounded by the parabola y = -x^2 + 2x + 1 and the curve y = |x - 1| is :

Step-by-step solution

To find the points of intersection of the curves y = -x^2 + 2x + 1 and y = |x - 1| , we consider two cases based on the definition of the modulus function. Case 1: x 1 The equation becomes -x^2 + 2x + 1 = x - 1 x^2 - x - 2 = 0 (x - 2)(x + 1) = 0 Since x 1 , we get x = 2 . Case 2: x The equation becomes -x^2 + 2x + 1 = -(x - 1) = 1 - x x^2 - 3x = 0 x(x - 3) = 0 Since x The required area A is the integral of the upper curve minus the lower curve from x = 0 to x = 2 . We split the integral at x = 1 : A = ₀¹ ( (-x^2 +