Q:

Finding The Slope of a Line Intersecting a Region $R$ Bounded by a Curve $f(x)$

Let $R$ be the region in the first quadrant enclosed by $f(x) = x - x^2$ and the $x$-axis. A line, passing through the origin, runs through region $R$ and cuts it so that the area of the lower portion of $R$ is seven times the area of the upper portion of $R$. Find the slope of the line.

A:

Hints: Draw a picture.
It is easy to compute the total area $A$. 
Draw a line of slope $m$ through the origin, where $m$ feels roughly right. The line meets the curve at $x=0$ and $x=1-m$.
The area of the upper part is
$$\int_0^{1-m} \left((x-x^2)-mx\right)\,dx.$$
Set this equal to $\frac{7}{8}A$.  

