Suppose we give an equation of the form $f(x_1,x_2,..., x_n)=C$, with $f$ a smooth function, and assume this is such that defines a closed surface in $\mathbb{R}^{n+1}$. Assume also that the equation is such that one cannot in general isolate one variable in terms of the others. My question is: how to compute the enclosed volume V(C) as a function of C, the only "external" parameter in the equation? Is there any systematic algorithm to do that?
If there exists no general method of computing the exact volume, then:
- Is there some way of "estimating" the dependence of the volume with the constant, say $V\sim C^n$? (neglecting multiplicative constants)
- Could one compute $dV=\frac{\partial V}{\partial C}dC$, the volume of the infinitesimal shell that lies between $f=C$ and $f=C+dC$?
As a specific example, consider, for $C\geq 1$ \begin{align}\label{} \sqrt{1+(x_1-0)^2}+\sqrt{1+(x_2-x_1)^2}+...+\sqrt{1+(x_{n}-x_{n-1})^2}+\sqrt{1+(0-x_n)^2}=(n+1)C \end{align} which is related to computing the length of a path from $0$ to $0$. I have made an attempt of solution, but it's not general, and only works for some limiting cases of $C$.
Any suggestions or references would be greatly appreciated.