Exam 13: Definite Integrals - Techniques

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. ​

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

The demand function for a certain product is and the supply function is where p is in millions of dollars and x is the number of thousands of units. Find the equilibrium point (x, p) and the consumer's surplus there. Round your answer to the nearest million dollars, where applicable. ​

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 . ​

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

Evaluate the integral . ​

Find the area of the region bounded by ​

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

Use formulas or numerical integration with a graphing calculator or computer to evaluate the given definite integral. Round answer to three 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 an integral formula to evaluate . ​

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. ​

Evaluate the definite integral . ​

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

Use the Trapezoidal Rule to approximate with n = 4. Round your answer to 3 decimal places. ​

Evaluate the definite integral . ​

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

Equations are given whose graphs enclose a region. Find the area of the region. ​ , , and ​

Evaluate . ​