Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 9 / v / Solution

Rearrange the first integral from part (iv) and use the half-angle identities:Eliminating $\lambda$ givesSquaring removes the sign of the slope; the rising and falling sides of the same profile have opposite tangent angles but obey the same height relation. Heights and angles must remain on the regular branch with $\cos \psi >0$.