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

y = mx; mx = 2x - 4x²; x = 0 or x = (2 - m)/4; 2x - 4x² = 0; x = 0 or x = 1/2; Area S = ∫₀^1/2(2x - 4x²)dx =

(x² - 4x³/3)₀^1/2 = 1/4 - 1/6 = 1/12; ∫₀^{(2 - m)/4}(2x - 4x² - mx)dx = 1/24; (x² - 4x³/3 - mx²/2)₀^(m-2)/4 = 1/24;

(2 - m)²/16(1 - (2 - m)/3 - m/2) = 1/24; (2 - m)²/16(1/3 - m/6) = 1/24; (2 - m)³ = 4; m = 2 - 4^1/3