To determine the area bounded by the functions f(x) = x² - 8x + 16, g(x) = 4x - 4, and the x-axis over the interval [1,4], we need to find the area between the curves and the x-axis within that interval.
1. First, we find the points of intersection between f(x) and g(x) by setting them equal to each other:
x² - 8x + 16 = 4x - 4
x² - 12x + 20 = 0
(x - 2)(x - 10) = 0
x = 2 or x = 10
2. The points of intersection within the interval [1,4] are 2 and 4.
3. Next, we integrate to find the area between the curves:
∫[1,2] (4x - 4) dx + ∫[2,4] (x² - 8x + 16 - (4x - 4)) dx
4. Calculate the integrals:
∫[1,2] (4x - 4) dx = 2x² - 4x |[1,2] = (2(2)² - 4(2)) - (2(1)² - 4(1)) = 4
∫[2,4] (x² - 8x + 16 - 4x + 4) dx = (1/3)x³ - 4x² + 16x |[2,4] = (1/3)(4)³ - 4(4)² + 16(4) - [(1/3)(2)³ - 4(2)² + 16(2)] = 20/3
5. Add the two areas together: 4 + 20/3 = 32/3 or approximately 10.67 square units.
Therefore, the area bounded by the functions f(x) = x² - 8x + 16, g(x) = 4x - 4, and the x-axis over the interval [1,4] is 32/3 square units.
