Explanation Given: The Lorenz curve with equation y = x 2 e x − 1 . Formula used: Gini index of Income Concentration: If y = f ( x ) is the equation of a Lorentz curve, then Gini Index = 2 ∫ 0 1 [ x − f ( x ) ] d x . The Gini index of income concentration lies between 0 and 1. Calculation: Calculate the Gini index of income concentration for the Lorenz curve, 2 ∫ 0 1 ( x − x 2 e x − 1 ) d x (1) First integrate the indefinite integral ∫ x 2 e x − 1 d x . Consider u = x 2 and d v = e x − 1 d x in the integral ∫ x 2 e x − 1 d x . Obtain the value of d u by differentiating the equation u = x 2 . d u = d d x ( x 2 ) d u = 2 x d x Obtain the value of v by integrating the equation d v = e x − 1 d x . ∫ d v = ∫ e x − 1 d x v = e − 1 + x + C Use Integration by-Parts Formula and simplify the integral ∫ x 2 e x − 1 d x as follows. ∫ x 2 e x − 1 d x = e − 1 + x x 2 − ∫ 2 x e − 1 + x d x = e − 1 + x x 2 − 2 ∫ x e − 1 + x Consider u = x and d v = e 1 + x d x in the integral ∫ x e − 1 + x . Obtain the value of d u by differentiating the equation u = x . d u = d d x ( x ) d u = d x Obtain the value of v by integrating the equation d v = e − 1 + x d x . ∫ d v = ∫ e − 1 + x d x v = e − 1 + x + C Use Integration by-Parts Formula and simplify the integral ∫ x e − 1 + x as follows

Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

13th Edition

ISBN: 9780321869838

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Publisher: PEARSON

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Chapter 6.3, Problem 70E

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Chapter 6.3, Problem 69E

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Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

Ch. 6.1 - Matched Problem 1 Find the area bounded by f(x) =...Ch. 6.1 - Matched Problem 2 Find the area between the graph...Ch. 6.1 - Prob. 3MP Ch. 6.1 - Matched Problem 4Find the area bounded by f(x) =6 ...Ch. 6.1 - Matched Problem 5Find the area bounded by f(x)=2x2...Ch. 6.1 - Matched Problem 6Find the area (to three decimal...Ch. 6.1 - Prob. 7MP Ch. 6.1 - Prob. 1ED Ch. 6.1 - In Problems 18, use geometric formulas to find the...Ch. 6.1 - In Problems 18, use geometric formulas to find the...

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