The Question
$$x_n(c)=\frac{\int_{1}^{c} x_{n-1}(b)y_{n-1}(b)db}{\int_{1}^{c} y_{n-1}(b)x_{n-1}(b)'db}, x_1(c)=\frac{\int_{1}^{c} xf(x)dx}{\int_{1}^{c} f(x)dx} $$
$$y_n(c)=\frac{\int_{1}^{c} y_{n-1}(b)^2db}{\int_{1}^{c} y_{n-1}(b)x_{n-1}(b)'db}, y_1(c)=\frac{\int_{1}^{c} f(x)^2dx}{2\int_{1}^{c} f(x)dx} $$
My Understanding
So say we wanted to find the centroid of the region bounded by the curve $y=f(x)$, $y=0$, $x=1$, and $x=b$. The parametric function obtained would give the path that the centroid "travels" as $b$ goes from 1 to infinity. We could then find the centroid of the region bounded by that curve, $y=0$, $x=1$, and $x = b$. From this, you get a sequence of functions and I was wondering if/when does that sequence converge. I found a recursive sequence for the nth set of parametric equations but I don't know how to get the limit as n goes to infinity from there. I think if we start from functions like $y=\frac{1}{x}$, the sequence will converge but I'm not sure.