Exam 14: Definite Integrals: Techniques of Integration

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 demand for q units of a certain product at p per unit is given by . Use Simpson's Rule with n = 6 to approximate the average price as demand ranges from 3 to 8 items to the nearest cent. ​

Use integration by parts to evaluate the integral ​

A 58-year-old couple are considering opening a business of their own. They will either purchase an established Gift and Card Shoppe or open a new Video Rental Palace. The Gift Shoppe has a continuous income stream with an annual rate of flow at time t given by (dollars per year) and the Video Palace has a continuous income stream with a projected annual rate of flow at time t given by (dollars per year). The initial investment is the same for both businesses, and money is worth 10% compounded continuously. Find the present value of the Video Palace over the next 8 years (until the couple reach age 66). Round your answer to the nearest dollar. ​

Evaluate the integral by integration. Round your answer to two decimal places. ​

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. ​

Use an integral formula to evaluate . ​

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. ​

Suppose the marginal cost for x units of a good is (dollars per unit) and if the fixed cost is $100 find the total cost function of the good? ​

Suppose the following table gives the supply and demand schedules, with p in dollars and x as the number of units. Use Simpson's Rule to approximate the consumer's surplus at market equilibrium. Note that market equilibrium can be found from the tables. ​ ​

Use an integral formula to evaluate .

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

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

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

Suppose the marginal cost for x units of a good is (dollars per unit) and if the fixed cost is $200. What is the total cost of producing 4 units of this good? Round your answer to the nearest dollar. ​

Evaluate the definite integral . ​

Find the average value of the given function over the given interval. ​

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

A 58-year-old couple are considering opening a business of their own. They will either purchase an established Gift and Card Shoppe or open a new Video Rental Palace. The Gift Shoppe has a continuous income stream with an annual rate of flow at time t given by (dollars per year) and the Video Palace has a continuous income stream with a projected annual rate of flow at time t given by (dollars per year). The initial investment is the same for both businesses, and money is worth 10% compounded continuously. Determine which is the better buy by finding the present value of each business over the next 7 years (until the couple reach age 65). ​

Use Simpson's Rule to approximate with n = 4. Round your answer to 2 decimal places. ​