Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 112 / 2 / i

Normalize the surface area of the spherical cap by writingTwo elementary descriptions of a spherical cap are useful. The spherical polar coordinates formula givesProjecting orthogonally onto the last $n-1$ coordinates givesIndeed, the upper hemisphere is the graph $x_1=\sqrt{1-\Vert x^{\prime }{\Vert }^2}$, whose surface area factor is $(1-\Vert x^{\prime }{\Vert }^2{)}^{-1/2}\le 1/\cos \rho$ on $\Vert x^{\prime }\Vert \le \sin \rho$. This establishes the upper projection bound directly.

Set $F(\rho )={\sin }^n\rho -q(\rho )$. The supplied ratio bound implies $c_n<\sqrt{(n+1)/(2\pi )}/n\le 1/2$ for $n\ge 2$. Consequently, at $\rho =\pi /4$,Also,On $[\pi /4,\pi /2]$, the expression in parentheses is strictly decreasing; it starts positive and ends negative. Thus $F$ first increases and then decreases, so its minimum on this interval is at an endpoint. Since $F(\pi /2)=1/2$, both endpoint values are nonnegative. The spherical cap area upper bound follows: