Final answer:
To determine the area of the shaded region between the curves y = -x² + 6x and y = x² - 4x, calculate the points of intersection, set up the integral of the difference of the functions over the interval defined by the intersections, then perform the integration.
Step-by-step explanation:
The problem involves finding the area of the shaded region between two parabolas, where the curves are defined by y = -x² + 6x and y = x² - 4x. To solve this, we first find the points of intersection of the two curves which are the solutions to the equation -x² + 6x = x² - 4x. These points will define the limits of integration for calculating the area.
To find the shaded region, we integrate the top function minus the bottom function across the interval defined by the intersection points. The integral's result will give us the total area of the shaded region between the two curves.
It is worth noting that finding areas between curves is a standard application of integration often covered in calculus courses in high school, which may also be relevant to certain Advanced Placement (AP) courses.