Let O be the origin. y = c intersects the curve y = 2x - 3x^{3} 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.

**Solution**

Answer: 4/9.

*Trivial.*

Suppose the line cuts the curve at x = a and x = b. Then ∫_{0}^{b}(2x - 3x^{3}) dx = bc if c has the correct value (draw a picture). So b^{2} - 3/4 b^{4} = b (2b - 3b^{3}) or b = 2/3 and hence c = 4/9.

© John Scholes

jscholes@kalva.demon.co.uk

12 Dec 1998