Final answer:
Find the intersection points of the curves f(x) and g(x) by setting them equal to each other, solve for x to get the limits of integration, and then calculate the definite integral of the difference between the curves over that interval to determine the area.
Step-by-step explanation:
To find the area of the region bounded by the curves f(x) = x^2 + 11x − 54 and g(x) = − x^2 + 5x + 2, you need to calculate the integral of the difference between the two functions, over the interval where they intersect. First, set f(x) equal to g(x) to find the intersection points (the limits of integration). Solve the equation x^2 + 11x − 54 = −x^2 + 5x + 2 to find the x-values of these points. After solving, you'll obtain two x-values, which will be the limits of integration x1 and x2. Compute the integral of the difference (−x^2 + 5x + 2) - (x^2 + 11x − 54) from x1 to x2. This will give you the total area between the two curves. Graphical or numerical methods may be used for calculating the definite integral if the anti-derivative is difficult to find by hand.