Exam 14: Definite Integrals: Techniques of Integration

Suppose that the output of the machinery in a factory can be considered as a continuous income stream with an annual rate of flow at time t given by in thousands of dollars per year. If the annual interest rate is 4% compounded continuously, find the capital value of the machinery. Round your answer to the nearest dollar. ​

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. ​

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

The graph in the following figure gives the times that it takes a vehicle to reach speeds from 0 mph to 25 mph, in increments of 5 mph, with a curve connecting them. Count the squares under the curve to estimate this distance. Estimate the distance traveled by the vehicle in 14 seconds, to a speed of 25 mph. (Be careful with time units.)

Use the function from to and n equal subintervals with the function evaluated at the right-hand endpoints of each subinterval. Let the sum of the areas of the rectangles be S. Find by using the formula for the sum of the areas of the n rectangles. ​

Find the average value of the function over ​

Find the mean of the probability distribution if the probability density function is . Round your answer to two decimal places. ​

Evaluate the integral .

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. ​

Evaluate the integral .

Use an integral formula to evaluate . ​

Suppose that a vending machine service company models its income by assuming that money flows continuously into the machines, with the annual rate of flow given by in thousands of dollars per year. Find the total income from the machines predicted by the model over the first 5 years. Round your answer to the nearest thousand dollars. ​

Suppose that the demand function for an appliance is where q is the number of units and p is in dollars. What is the consumer's surplus if the equilibrium price is $17.6 and the equilibrium quantity is 21? Round your answer to the nearest dollar. ​

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

If the demand function for wheat is dollars, where x is the number of hundreds of bushels of wheat, estimate the consumer's surplus at Round answer to the nearest cent. ​

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. ​

When the area under from to is approximated, the formulas for the sum of n rectangles using left-hand endpoints and right-hand endpoints are: ​ Left-hand endpoints: ​ Right-hand endpoints: ​ Find ​

Evaluate . ​

The Lorenz curve for the income distribution in a certain country in 2005 is given by . Find the Gini coefficient of income for 2005 for this country. Round your answer to four decimal places. ​

Evaluate the integral . ​ ​