Let O be the origin. y = c intersects the curve y = 2x - 3x3 at P and Q in the first quadrant and cuts the y-axis at R. Find c so that the region OPR bounded by the y-axis, the line y = c and the curve has the same area as the region between P and Q under the curve and above the line y = c.
Suppose the line cuts the curve at x = a and x = b. Then ∫0b(2x - 3x3) dx = bc if c has the correct value (draw a picture). So b2 - 3/4 b4 = b (2b - 3b3) or b = 2/3 and hence c = 4/9.
© John Scholes
12 Dec 1998