91Ó°ÊÓ

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. $$ f(x)=2 x^{2} \quad[1,3] $$

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. $$ f(x)=3 x^{2}+1 \quad[-1,3] $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. \(\int_{-1}^{4}|x-2| d x\)

Evaluate the definite integral. \(\int_{2}^{2}(x-3)^{4} d x\)

Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Multiple integrals may be necessary.) $$ f(x)=x\left(x^{2}-3 x+3\right), g(x)=x^{2} $$

Consumer and Producer Surpluses Factory orders for an air conditioner are about 6000 units per week when the price is 331 dollars and about 8000 units per week when the price is 303 dollars. The supply function is given by \(p=0.0275 x\). Find the consumer and producer surpluses. (Assume the demand function is linear.)

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 \(0 \leq x \leq 100,\) indicates the "income inequality" of a country. In \(2005,\) the Lorenz curve for the United States could be modeled by \(y=\left(0.00061 x^{2}+0.0218 x+1.723\right)^{2}, \quad 0 \leq x \leq 100\) where \(x\) is measured from the poorest to the wealthiest families. Find the income inequality for the United States in \(2005 .\)

Find the cost function for the marginal cost and fixed cost. $$ \begin{array}{ll}{\text {Marginal Cost}} & {\text {Fixed } \operatorname{Cost}(x=0)} \\ {\frac{d C}{d x}=\frac{\sqrt[4]{x}}{10}+10} & {\$ 2300}\end{array} $$

In Exercises 67 and \(68,\) find the revenue and demand functions for the given marginal revenue. (Use the fact that \(R=0\) when \(x=0\).) $$ \frac{d R}{d x}=225-3 x $$

Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all -values in the interval for which the function is equal to its average value. Function \(\quad\) Interval \(f(x)=e^{x / 4} \quad[0,4]\)