Question Detail

First, find the roots of the curve to determine the integration intervals:y = x(x² - 6x + 8) = x(x-2)(x-4).The roots are x=0, x=2, and x=4.We need to integrate the function in each interval and take the absolute value for areas below the x-axis.1. Find the indefinite integral: ∫(x³ - 6x² + 8x) dx = x⁴/4 - 2x³ + 4x².2. Evaluate for the interval [0, 2]: [x⁴/4 - 2x³ + 4x²] from 0 to 2 = (16/4 - 2(8) + 4(4)) - 0 = (4 - 16 + 16) = 4.3. Evaluate for the interval [2, 4]: [x⁴/4 - 2x³ + 4x²] from 2 to 4 = (4⁴/4 - 2(4³) + 4(4²)) - (16/4 - 2(8) + 4(4)) = (64 - 128 + 64) - (4 - 16 + 16) = 0 - 4 = -4.4. Total area = |Area[0,2]| + |Area[2,4]| = |4| + |-4| = 4 + 4 = 8 square units.