Exam 13: A: Definite Integrals - Techniques

Suppose the rate of production of a new line of products is given by where x is the number of items produced and t is the number of weeks the products have been in production. How many units were produced in the first 3 weeks? Round your answer to the nearest unit produced. ​

Evaluate the definite integral . ​

Find the mean of the probability distribution if the probability density function is . ​

Evaluate the given integral with the Fundamental Theorem of Calculus . ​

For what value of c is the function a probability density function? ​

Find the shaded area between the given function and the x-axis. ​ ​

Suppose the number of daily sales of a product was found to be given by , x days after the start of an advertising campaign for this product. Find the average daily sales during the first 25 days of the campaign-that is, from to . Round your answer to the nearest dollar. ​

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

For what value of c does ? ​

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. ​

Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the right-hand endpoints of the subintervals. Compute the approximate area using up to four decimal places as needed. ​ from to subintervals. ​

Use an integral formula to evaluate . ​

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 function from to and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find by using the formula for the sum of the areas of the n rectangles (call this S). ​

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. ​

Approximate the area under the curve defined by the function over the interval x = 0 to x = 3 using the left-hand endpoints of three subintervals (rectangles). ​

Evaluate . ​

Suppose that when a new oil well is opened, its production is viewed as a continuous income stream with monthly rate of flow where t is time in months and f is in thousands of dollars per month. Find the total income over the next 40 years (480 months). Round your answer to one decimal place. ​

Use the table values and apply the Simpson's Rule to approximate to one decimal place.

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.