Section 6.1: Area Between Curves

Area between Two Curves

If f and g are continuous and f(x) ≥ g(x) over the interval [a, b], then the area bounded by y = f(x) and y = g(x) for a ≤ x ≤ b is given exactly by:

A = ∫ab [f(x) - g(x)] dx

Income Distribution

A Lorenz curve displays the distribution of income among a given population.

• The variable x represents the cumulative percentage of families at or below a given income level.
• The variable y represents the cumulative percentage of total family income received.

Gini Index of Income Concentration

If y = f(x) is the equation of a Lorenz curve, then

Gini index = 2∫01 [x - f(x)] dx

The Gini index is always a number between 0 and 1.

• A Gini index of 0 indicates absolute equality - all people share equally in the income.
• A Gini index of 1 indicates absolute inequality - one person has all the income and the rest have none.

Income Distribution in United States

The graph to the right shows the Gini index for households in the US, 1967 - 2016.