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c Alculus in Context #5 Math 113, Chapters 5 & 6 How does our understanding of integration and the Fundamental Theorem of Calculus allow us to compare income inequality? The Gini index is a statistical measure of income distribution. It ranges from 0 (or 0%) for perfect equality to 1 for 100%) for perfect inequality. Source: https://data.worldbank.rs/indicator/SIPOV GINI?viewemap A country where every 2. Unequal 4. Total neque resident has the same income would have a Gini index of o. GI A country in which one resident earned all the income, while everyone else earned nothing would have a Gini index of 1 1. Perfectly 1. More Unequal G10 மே G1 w The Gini index is defined to be two times the area between the line of perfect equality ( x) and the Lorenz curve. (The Lorenz curve shows the percentage of income held by each percentile of the country's population. As an example, the Lorenz curve would plot a point at (50,20) if the bottom 50% of the population held 20% of the country's income.) Lorenz Curve - Ukraine 2020 y - 1.5625 -2.57812 +1.34382 +0.2406x +0.4313x 1 0.0 30.6 0.4 02 0 0.2 0.8 1 Talks were underway for Ukraine to join the European Union an alliance which currently includes the countries of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czechia, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Ireland, Italy, 0.4 0.6 Latvia, Lithuania, Luxembourg, Malta, Netherlands, Poland, Cumulative Percentage of the Population Portugal, Romania, Slovakia, Slovenia, Spain and Sweden. We are going to compare the income distribution of Ukraine with that of Poland, a country in the EU which is geographically close to Ukraine. 1. Above is a plot of the line of equality and the Lorenz curve for Ukraine in 2020. Shade the area we need t find first so that we can calculate the Gini index. Then, write an expression for THIS AREA that includes a definite integral. 2. Use the integral expression you wrote in #1 and the Fundamental Theorem of Calculus to find the Gini index for Ukraine in the year 2020. Show the substitutions necessary, but use fnint to do the calculation
Lex in p Lorenz Curve - Poland 2018 y = 2.0833x5 - 3.75x4 + 2.2708x3 + 0.1125x2 + 0.2833x 3. To the right is a plot of the line of equality and the Lorenz curve for Poland in the year 2018. Shade the area we need to find first so that we can calculate the Gini index. Then, write an expression for THIS AREA that includes a definite integral. 1 Poland -Line of Equality 0.8 Cumulative Wealth 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 Cumulative Percentage of the population 4. Use the integral expression you wrote in #3 and the Fundamental Theorem of Calculus to find the Gini index for Poland in the year 2018. Show the substitutions necessary, but use fnint to do the calculation.