Determine the area of the region between the curves:
f(x) = -(3)x^3 + (6.25)x^2 + (27.1875)x - (9.328125) and g(x) = (1)x^2 + (3)x -(3)
To determine the area of the region between the curves, you need to find the points of intersection of the two functions and then integrate the absolute difference between the functions over that interval.
1. Start by finding the points of intersection:
Set f(x) equal to g(x) and solve for x:
-(3)x^3 + (6.25)x^2 + (27.1875)x - (9.328125) = (1)x^2 + (3)x -(3)
2. Simplify the equation:
-3x^3 + 6.25x^2 + 27.1875x - 9.328125 = x^2 + 3x - 3
Rearrange the equation:
-3x^3 + 5.25x^2 + 24.1875x - 6.328125 = 0
3. Solve the equation for x using a numerical method, such as graphing or using a calculator. The solutions will give you the x-coordinates of the points of intersection.
4. Once you have the x-coordinates of the points of intersection, integrate the absolute difference between the two functions over the interval between those points to find the area. The integral can be set up as follows:
∫[a, b] |f(x) - g(x)| dx, where a and b are the x-coordinates of the points of intersection.
Evaluate this integral to find the area between the curves.
Note: The absolute value is used in the integral to ensure that the area is positive, regardless of the position of the curves with respect to each other.