The motivating problem comes from a review of introductory calculus:
We want to derive the formula for the Lorenz curve $\theta(p)$ using integrals $p=F(x)$ (the proportion of the population with income less than x) and $G(x)$ (the average income earned by those making less than x).
I've determined that $F(x) = \int_0^x f(t)dt$, $G(x)= \int_0^x tf(t)dt$, and the mean $\mu = \int_0^{\infty} tf(t)dt$ (since all values of x are possible). The formula for $\theta$ can be found by taking the total amount of money earned by those making less than x over the total amount of money. If we consider that $\mu$ = T (total income) over N (total size of population), $G(x)$ equals I (total income earned by those making less than x) over A (total number of people making less than x), and $F(x)$ = A/N, then $\theta$ should be I/T = (N*$F(x)G(x)$)/(N*$\mu$), or $\theta(x) = \mu^{-1}F(x)G(x)$.
However this is apparently not the correct answer. $\theta$ should be $\mu^{-1} \int_0^p F^{-1}(p)$. I have no idea to get to this equation from the one derived above. Researching I've found that this $F^{-1}(p)$ function is something called the quantile function and is defined using an infimum, but this was not covered in the material this problem is supposed to test. Additionally to get the second derivative of $\theta(p)$ we need to take the derivative of $F^{-1}(p)$, which the problem states must be $1/f(F^{-1}(p))$. How can one derive these results using only elementary calculus?