Exam 13: Definite Integrals - Techniques

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). ​

Find the average value of the function over ​

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

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. ​

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

Evaluate the integral . ​

Evaluate the integral .

Use integration by parts to evaluate . ​

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 . ​

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.​ ​

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). ​

Evaluate the integral . ​

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

Use Simpson's Rule to approximate with n = 4. Round your answer to 3 decimal places. ​

Use Simpson's Rule to approximate with n = 6. Round your answer to two decimal places. ​

For the interval [3,15] and n = 4, find the values of ​

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

With data for 1990 to 2002, the total receipts for world tourism (in billions of dollars per year) can be modeled by the function where represents 1985. Find the predicted total receipts for world tourism for the decade from 2003 to 2013. Round your answer to one decimal place. ​

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

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