How is the intersection area of two curves intersecting at more than two points calculated?
Due to the behavior of the cubic function and the quadratic function, when these types of polynomial functions intersect they do it in more than two points, therefore the intersection area results from two sub regions, so to calculate the total area you simply have to calculate two areas and add them.
To understand the proposition of the area of intersection at more than two points, we have the following exercise:
Find the area of the region between the graphs of:
The area of the region between the cubic function and the quadratic function is the region of intersection between the two curves, so the first step is to find the region of intersection, for this we equal the two curves as follows:
As a second step we rearrange the terms all on one side of the equality as follows:
As a third step, the terms with equal signs and having the same exponent are canceled, and those having the same exponent are added algebraically:
As a fourth step we get the solution for the third degree equation, we can do it with the factorization method (common factor) as follows:
The x-coordinate of the intersection points are
x1= 0
x2= 2
x3= -2
As a fifth step to get the Y-coordinate of the intersection points we substitute x1= 0 ; x2= 2 ; x3= -2 in any of the two functions, in this case I will choose to substitute in the quadratic function:
This implies that the Y coordinates of the intersection points are:
y1 = 0
y2 = 0
y3 = -8
Therefore the coordinates of the intersection points are:
(0 ; 0)
(2 ; 0)
(-2 ; -8)
As a sixth step we find the intersection graph by applying GeoGebra software and at the same time check the intersection points:
Area calculation approach between cubic and quadratic function
Bibliographic references consulted and recommended
Calculus with Analytic Geometry. Author: Larson and Hostetler. 7th edition. Volume I.