Exam 14: Definite Integrals: Techniques of Integration

Suppose spending for a wireless communications services (in billions of dollars per year) can be modeled by the function where represents 1995. Use equal subdivisions with right-hand endpoints to approximate the area under the graph of S(t) between and . Round your answer to one decimal place. ​

Find the area of the shaded region. ​

The supply function for a good is , where p is the number of dollars and x is the number of units. If the equilibrium price is $27 what is the producer's surplus at the equilibrium price? Round to the nearest cent. ​

Evaluate the integral . ​

Use an integral formula to evaluate . ​

Find the numerical value of by using the sum formulas. ​

The demand function for a product is where p is in millions of dollars and x is the number of thousands of units. If the equilibrium price is $6 million, what is the consumer's surplus? Round your answer to the nearest million dollars. ​

Use integration by parts to evaluate the integral . Note that evaluation may require integration by parts more than once. ​

Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral. ​ I. Integration by parts ​

Suppose a land developer is planning to dig a small lake and build a group of homes around it. To estimate the cost of the project, the area of the lake must be calculated from the proposed measurements (in feet) given in the following figure and in the table. Use Simpson's Rule to approximate the area of the lake. Round your answer to two decimal places. ​ where a = 300, b = 200, c = 400 ​

The following income distribution data define points on Lorenz curves, where x represents the fraction of a certain country's population and L(x) the cumulative percent of income held by fraction x. Use this data and a numerical method to evaluate and hence to find the Gini coefficient of income for this country in 1995 and 2003. Round your answers to three decimal places.

For the interval [-5,2] and for n = 3 find h.

The demand function for a product is , and the supply function for it is , where p is the number of dollars and x is the number of units. If the equilibrium price is $248 what is the producer's surplus at the equilibrium price? Round to the nearest cent. ​

Find the value of the sum ​

Find the area of the shaded region. Round to the nearest hundredth if necessary.

A small brewery considers the output of its bottling machine as a continuous income stream with an annual rate of flow at time t given by in thousands of dollars per year. Find the income from this stream for the next 30 years. Round your answer to the nearest dollar. ​

Suppose that the rate of production of a product (in units per week) is measured at the end of each of the first 5 weeks after start-up, and the following data are obtained. Use the Trapezoidal Rule to approximate the total number of units produced in the first 5 weeks. Round your answer to two decimal places. ​

The United States' spending for military (in billions of dollars per year) can be modeled by , where represents 1993. Estimate the total spending for the United States military between 2000 and 2005 by using 10 equal subdivisions and right-hand endpoints to approximate the area under the graph of M(t) between and . Round to three decimal places. ​

Use integration by parts to evaluate . ​

Suppose the Lorenz curve for the distribution of income of a certain country is given by Find the Gini coefficient of income. Round your answer to three decimal places. ​