5.4 Applying Definite Integration: Area Between Curves and Average ValueMAC2233 (5.4)MDC5.4 Applying Definite Integration: Area Between Curves andAverage ValueArea Between Two Curves: If f(x) and g(x) are continuous with f(x) ≥ g(x) on the interval a ≤ x ≤ b, then the area A between the curves y = f(x) and y = g(x) over the interval is given by( ) ( )baA f x g x dx� �= -� ��. Average Value of a Function: If f(x) is a function that is continuous on the interval a ≤ x ≤ b, then the average value V of f(x) is ( )baV f x dxb a1=-�. Lorentz curves L(x) are a device used to measure the percentage of a society’s wealth that is possessed by a given percentage of its people. The line y = x represents complete equality; the closer L(x) is to y = x, the more equitable the distribution of wealth. The Gini Index GI is a measure of the inequality of wealth distribution (0 ≤ x ≤ 1); the closer to 0, the more equitable the distribution. It is calculated by GI = ( )x L x dx102� �-� ��.Sketch the given region R and then find its area:1. R is the region bounded by the curves 2. R is the region bounded by the curve yx21= and they = x2, y = -x2, and the line x = 1. lines bounded by y = x and xy8=.3. Find the average value of the given function f(x) over the interval [-1, 1]:( )212 6xf xx x+=+ +MAC2233 (5.4)MDC4. Find the average value V over the specified interval; graph the function along with the rectangle whose base is the given interval and whose height is the average value V:tg t e2( )-= over -1 ≤ t ≤ 25. Find the Gini index for the given Lorentz curve:( )L x x x2.7 0.3= +GI = ( )x L x dx102� �-� ��6. Tom invests $5,000 in an account that pays 6% compounded continuously. What is the average value of his investment over the next 10
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