Exam 13: Definite Integrals - Techniques

In an effort to make the distribution of income more nearly equal, the government of a country passes a tax law that changes the Lorenz curve from for 2004 to for 2005. Find the Gini coefficient of income for both years. Round your answer to six decimal places. ​

Suppose that the profit from a machine's production can be considered as a continuous income stream with annual rate of flow at time t given by (dollars per year). If money is worth 14%, compounded continuously, find the present value of this stream over the next 5 years. Round your answer to the nearest dollar. ​

If the marginal cost function for x units of a product is dollars per unit, and if the fixed cost is $120, find the total cost function. ​

Evaluate the definite integral . ​

Find the area of the shaded region. ​ ​

Use an integral formula to evaluate . ​

Evaluate the integral . ​

Evaluate the integral . ​

Evaluate . ​

The probability density function for the life span of an electronics part is where t is the number of months in service. Find the probability that any given part of this type lasts longer than 18 months. Round your answer to three decimal places. ​

A continuous income stream has an annual rate of flow at time t given by (dollars per year). Find the present value of this income stream for the next 9 years, if the money is worth 4% compounded continuously. Round to the nearest dollar. ​

Find the area of the region bounded by ​

If the supply function for a commodity is where p is in millions of dollars and x is the number of thousands of units. What is the producer's surplus at ? Round your answer to the nearest million dollars. ​

Suppose that the rate at which a nuclear power plant produces radioactive waste is proportional to the number of years it has been operating, according to in pounds per year. Suppose also that the waste decays exponentially at a rate of 6% per year. Then the amount of radioactive waste that will accumulate in b years is given by and this integral evaluates to . How much waste will accumulate in the long run? Take the limit as in the integral evaluated. Round your answer to the nearest pound, if it exists. ​

Find the area of the shaded region for the graph given below. ​ ​

Use the Trapezoidal Rule to approximate with n = 6. Round your answer to two decimal places. ​

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

Assume that the tax burden per capita (in dollars) for selected years from 1950 to 2002 can be modeled by where represents 1950. If the model remains valid, find the predicted average federal tax burden per capita from 2000 to 2010. Round your answer to the nearest dollar. ​

Evaluate the improper integral if it converges, or state that it diverges. ​

Evaluate the integral . ​