Convergence rates of approximate sums of the areas of surfaces of revolution (Q974344)

A surface of revolution \(\Phi \dots \{(x, f(x)\cos \theta, f(x) \sin\theta), a\leq x\leq b, 0 \leq\theta\leq 2\pi\}\) is approximated by replacing the meridian of \(\Phi\) by a polygon of length \(n\) which by rotation yields a series of frustums inscribed in \(\Phi\). The polygon is defined by an \(n\)-division \(\Delta_n: a = s_0 \leq s_1 \leq \dots \leq s_{n-1} \leq s_n=b\) of the interval \([a,b]\). The area \(S(\Delta_n)\) of the union of these frustums is then used as an approximate sum of the area \(S\) of \(\Phi\). The authors prove three estimations for the limits of the error terms \(n^2|S- S(\Delta_n)|\) and \(n^2(S-S(\Delta_n))\), one of them for the general case whereas the other two use special assumptions such as \(\Phi\) being convex.