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

f(x) = x-x²

Half the area of f(x) = (1/2)∫₀¹ x-x² dx = 1/12

A line from the origin to a point on f(x) is (a,f(a))

Area/2 = (1/2)∫₀^a f(x) - mx dx = (1/2)∫₀^a x-x²-mx dx = = a²/2 - a³/3 -mx²/2 |₀^a = 1/12

m is the slope, m= f(a)/a = (a-a²)/a = 1-a

Multiplying A/2 thru by 12,

6a²-4a³-6a²+6a³ = 1

2a³=1

a = (1/2)^1/3

m = 1-a = 1-(1/2)^1/3 ≈ .205299