91Ó°ÊÓ

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{0} 3 d x $$

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. $$ f(x)=2 x-x^{3} $$

Find the indefinite integral and check the result by differentiation. $$ \int \frac{-3}{\sqrt{2 t+3}} d t $$

Use a computer or programmable calculator to approximate the definite integral using the Midpoint Rule and the Trapezoidal Rule for \(n=4\), \(8,12,16\), and 20. $$ \int_{0}^{4} \sqrt{2+3 x^{2}} d x $$

The projected fuel cost \(C\) (in millions of dollars per year) for an airline company from 2007 through 2013 is \(C_{1}=568.5+7.15 t\), where \(t=7\) corresponds to \(2007 .\) If the company purchases more efficient airplane engines, fuel cost is expected to decrease and to follow the model \(C_{2}=525.6+6.43 t\). How much can the company save with the more efficient engines? Explain your reasoning.

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

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. $$ \frac{d I}{d t}=500 \sqrt{t+1} $$