Exam 13: A: Definite Integrals - Techniques

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 6 years (until the couple reach age 64). ​

Find the average value of the function ​

The production from a particular assembly line is considered a continuous income stream with annual rate of flow given by (in thousands of dollars per year). Use Simpson's Rule with n = 4 to approximate the total income to 2 decimal places over the first 2 years, given by . ​

Find the area of the region enclosed by and . ​​

-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. ​

The Carter Car Service franchise has a continuous income stream with a monthly rate of flow modeled by (dollars per month). Find the average flow of income over years 4 to 5. Round your answer to the nearest cent. ​

The demand function for a product is , where p is the number of dollars and x is the number of units. If the equilibrium quantity is $7, what is the consumer's surplus? Round your answer to two decimal places. ​

Suppose that the income from a slot machine in a casino flows continuously at a rate of , where t is the time in hours since the casino opened. The total income during the first 8 hours is given by . Find the average income over the first 8 hours. ​

Use the Trapezoidal Rule to approximate with n = 6. 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 9% per year. Then the amount of radioactive waste that will accumulate in b years is given by . Evaluate this integral. ​

True or false. For the function gives the area between the graph of and the x-axis from . ​

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

Assume that a store finds that its sales revenue changes at a rate given by dollars per day, where t is the number of days after an advertising campaign ends and . Find the total sales for the second week after the campaign ends to . ​

The average life for a particular brand of car battery is given by the following probability density function where t is in years. ​Find the probability that a battery chosen at random lasts between 2 and 7 years. Round your answer to two decimal places. ​

Assume that a store finds that its sales revenue changes at a rate given by dollars per day, where t is the number of days after an advertising campaign ends and . Find the total sales for the second week after the campaign ends to . ​

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

A transmission repair firm that wants to offer a lifetime warranty on its repairs has determined that the probability density function for transmission failure after repair is , where t is the number of months after repair. What is the probability that a transmission chosen at random will last more than 4 months? Round to 3 decimal places. ​

A drug manufacturer has developed a time-release capsule with the number of milligrams of the drug in the bloodstream given by where x is in hours and . Find the average number of milligrams of the drug in the bloodstream for the first 3 hours after a capsule is taken. Round your answer to the nearest milligram. ​

Consider the following supply and demand schedules, with p in dollars and x as the number of units. Use Simpson's Rule to approximate the producer's surplus at market equilibrium to 2 decimal places. Note that market equilibrium can be found from the tables. ​

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.​ ​