1. ## Integration Question

I'm having a bit of trouble with this question:

In what ratio does te X-axis divide the area of the region bounded by the parabolas $\displaystyle y=4x-x^2$ and $\displaystyle y=x^2-x$

2. Hello, acevipa!

I assume you made a sketch . . .

In what ratio does the x-axis divide the area of the region
bounded by the parabolas: .$\displaystyle y\:=\:4x-x^2\,\text{ and }\,y\:=\:x^2-x$
Code:
            |
*    |          *
|
|
*   |    *    *
| *:::::*
*  |*:::::::*
* |:::::::*
- - - - *:-:-:* - * - - - -
|  *
|
*|          *
|

The parabola intersect when: .$\displaystyle x^2-x \:=\:4x-x^2 \quad\Rightarrow\quad x \:=\:0,\:\tfrac{5}{2}$

Find the total area between the parabolas: .$\displaystyle A_{\text{total}} \;=\;\int^{\frac{5}{2}}_0\bigg[(4x-x^2) - (x^2-x)\bigg]\,dx$

Find the area below the x-axis: .$\displaystyle A_{\text{below}} \;=\;-\int^1_0\bigg[x^2-x\bigg]\,dx$

Find the area above the x-axis: .$\displaystyle A_{\text{above}} \;=\;A_{\text{total}} - A_{\text{below}}$

Finally, form the ratio: .$\displaystyle \frac{A_{\text{above}}}{A_{\text{below}}}$