The area of the shaded region between the curves f(x) = x^4 - 4x^3 + 9x^2 and g(x) = -6x + 72 is approximately 431.33 square units.
To find the area between two curves, we need to determine the points of intersection between the curves and then integrate the difference between the curves over that interval. Let's find the points of intersection:
Setting f(x) equal to g(x), we have x^4 - 4x^3 + 9x^2 = -6x + 72. Rearranging the equation gives x^4 - 4x^3 + 9x^2 + 6x - 72 = 0.
Solving this equation may require numerical methods, such as graphing or using a calculator. The points of intersection are approximately x ≈ -2.12 and x ≈ 5.12.
To find the area between the curves, we integrate the difference between f(x) and g(x) over the interval [a, b], where a and b are the x-values of the points of intersection. Thus, the area is given by the integral of [f(x) - g(x)] dx from x = -2.12 to x = 5.12.
Evaluating the integral using numerical methods, we find that the area is approximately 431.33 square units.
Therefore, the area of the shaded region between the curves f(x) = x^4 - 4x^3 + 9x^2 and g(x) = -6x + 72 is approximately 431.33 square units. This represents the total enclosed area between the curves within the given interval.