- #1

Gary Smart

- 33

- 1

## Homework Statement

The problem consists of investigating the area between two functions of the forms (Parabolic segment):

: y = mx + c and y = ax^2 + bx + c

The investigation involves finding a combination that has one of each of the above functions and finding an area of one. The area between the functions has to be in the first quarter (positive x and y).

**2. The attempt at a solution**

My investigation up to now has found the following:

Function 1: 6x^2 - 2x + 8

Function 2: 4x + 8

These give points of intersection of: (0,8) and (1,12). These are the lower and upper bounds when integrating.

Integration produces:

Curve: 3x^2 - x^2 + 8x [Boundaries of 0 and 1]: Area under curve = 9 (between 0 and 1).

Line: 2x^2 + 8x [Boundaries of 0 and 1]; Area under line = 10 (Between 0 and 1).

=> Area between line and curve is: 10 - 9 = 1

**3. Further investigations**

I have already found that the coefficient of A (Parabola) has to be a multiple of 3, otherwise there will be a recurring decimal when integrated.

My next steps involve:

: Producing any generalisations that affect finding this specific area with these two functions.

: Proving/generalising using Algebra.