Question

Let R be the region in the first quadrant enclosed between the x-axis and the curve y = x - x^2. There is a straight line passing through the origin which cuts the region R into two regions so that the area of the lower region is 7 times the area of the upper region. Find the slope of this line.

Answer 1

always a good idea to have a picture in front of you when you do these problems

Answer 2

here is just the region R under y=x-x^2
I like to just write out the integral for R first just as a sort of starting point:\[R=\int_{0}^{1}x-x^2dx\]