Find the slope of the line through the origin that splits the region bounded by the parabola y = x-x^2 and the x-axis into two regions of equal area

Answer 1

y = mx; mx = x - x²; x = 0 or x = 1 - m;

∫₀^1-m(x - x² - mx)dx = 1/2∫₀¹(x - x²)dx; (x²/2 - x³/3 - mx²/2)₀^1-m = 1/2(x²/2 - x³/3)₀¹;

(1 - m)²(1/2 - (1- m)/3 - m/2) = 1/12; (1 - m)²(1/6 - m/6) = 1/12; (1 - m)³ = 1/2; 1 - m = 1/2^1/3; m = 1 - 1/2^1/3