Question

There is a line through the origin that divides the region bounded by the parabola y=2x-4x^2 and the x-axis into two regions with equal area. What is the slope of that line?

Answer 1

let the line be y = mx
find the x where the line crosses the parabola
Solve[2 x - 4 x^2 == m x, x]
x -> (2 - m)/4

Find the area between parabola and line
Integrate[2 x - 4 x^2 - m*x, {x, 0, (2 - m)/4}]
1/12 - m/8 + m^2/16 - m^3/96

Set that equal to half the area of the parabola and solve for m
Solve[1/12 - m/8 + m^2/16 - m^3/96 == .5*
Integrate[2 x - 4 x^2, {x, 0, .5}], m]

m -> 0.412599