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Chapter 11: Problem 28
Find the indefinite integral and check your result by differentiation. $$ \int(5-x) d x $$
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Health An epidemic was spreading such that \(t\) weeks after its outbreak it had infected \(N_{1}(t)=0.1 t^{2}+0.5 t+150, \quad 0 \leq t \leq 50\) people. Twenty-five weeks after the outbreak, a vaccine was developed and administered to the public. At that point, the number of people infected was governed by the model \(N_{2}(t)=-0.2 t^{2}+6 t+200\)
Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=x^{2}-x^{3} \quad[-1,0] $$
You are given the rate of investment \(d l / d t\). Find the capital accumulation over a five-year period by evaluating the definite integral Capital accumulation \(=\int_{0}^{5} \frac{d l}{d t} d t\) where \(t\) is the time in years. $$ \frac{d I}{d t}=\frac{12,000 t}{\left(t^{2}+2\right)^{2}} $$
Lorenz Curve Economists use Lorenz curves to illustrate the distribution of income in a country. Letting \(x\) represent the percent of families in a country and \(y\) the percent of total income, the model \(y=x\) would represent a country in which each family had the same income. The Lorenz curve, \(y=f(x)\), represents the actual income distribution. The area between these two models, for
Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=x^{3}-2 x+1, y=-2 x, x=1 $$
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