The total area enclosed between the curve y = |x^2 - 4x + 3| and the horizontal line y = 3 is :

Step-by-step solution

To find the points of intersection, we solve |x^2 - 4x + 3| = 3 . Case 1: x^2 - 4x + 3 = 3 x^2 - 4x = 0 x = 0, 4 . Case 2: x^2 - 4x + 3 = -3 x^2 - 4x + 6 = 0 , which has no real roots since its discriminant is negative ( 16 - 24 The region is bounded between x = 0 and x = 4 . The expression x^2 - 4x + 3 = (x-1)(x-3) changes sign at x = 1 and x = 3 . We split the required area A into three integrals: A = ₀⁴ (3 - |x^2 - 4x + 3|) dx A = ₀¹ (3 - (x^2 - 4x + 3)) dx + ₁³ (3 - (-x^2 + 4x - 3)) dx + ₃⁴ (3 - (x^2 - 4x + 3))