Using data from a nation's census, an economist produced the following Lorenz curves for the distribution of that nation's income in $1962$ and $1972.$

$f(x)=\frac{3}{5}x+\frac{2}{5}{x}^2$ Lorenz curve for $1962$

$g(x)=\frac{2}{5}x+\frac{3}{5}{x}^2$ Lorenz curve for $1972$

Find the Gini index of income concentration for each Lorenz curve and interpret the results.

Identify the integrand for the computation of the Gini index for $1962$ and $1972.$ The Gini index for $1962$ is given by $2{\displaystyle \underset{0}{\overset{1}{}}}|,dx||$ and the Gini index for $1972$ is given by $2{\displaystyle \underset{0}{\overset{1}{}}}({\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}})dx.$