Exam 7: Additional Topics in Integration

Evaluate the following improper integral whenever it is convergent. ​ Enter divergent if the integral is divergent.

The management of AstroWorld ("The Amusement Park of the Future") estimates that the number of visitors (in thousands) entering the amusement park t hours after opening time at 10 A.M. is given by ​ ​ Per hour. Determine the number of visitors admitted by noon. ​

Find the indefinite integral. ​ ​

Find the area of the region under the given curve over the indicated interval. ​ ​

Becky Wilkinson wants to establish a trust fund that will provide her children and heirs with a perpetual annuity in the amount of ​ ​ Thousand dollars per year beginning next year. If the trust fund will earn an interest rate of 10% per year compounded continuously, find the amount that she must place in the trust fund now. Hint: Use the formula

Find the indefinite integral. ​

Use the table of integrals to find the integral. ​ ​ Use C as the constant of integration.

Find the indefinite integral. ​ ​

Evaluate the definite integral by using the method of integration by parts. ​

Find the volume of the prolate spheroid (a solid of revolution in the shape of a football, i.e., elongated vertically) obtained by revolving the region under the graph of the function ​ ​ from to about the -axis.

Find the indefinite integral. ​

Use the table of integrals to find the integral. ​ ​

Evaluate the improper integral if it is convergent. ​

Find the volume of the solid of revolution obtained by revolving the region bounded above by the curve ​ ​ and below by the curve ​ ​ from to about the -axis.

Find a bound on the error in approximating the given definite integral using the trapezoidal rule with n intervals. ​ ​ If necessary, round the answer to seven decimal places. ​ Find a bound on the error in approximating the given definite integral using the Simpson's rule with n intervals. ​ ​ If necessary, round the answer to seven decimal places.

Use the table of integrals to find the integral. ​ ​ Use C as the constant of integration.

Sales of the latest recording by Brittania, a British rock group, are currently units/week (each unit representing 10,000 discs), where t denotes the number of weeks since the recording's release. Find an expression that gives the total number of discs sold as a function of t. ​

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. ​ ​

Use the formula ​ ​ to find the integral. ​

Thanks to smaller and more fuel-efficient models, American carmakers have doubled their average fuel economy over a 13-year period, from 1974 to 1987. The graph depicted in the figure gives the average fuel consumption in miles per gallon (mpg) of domestic-built cars over the period under consideration ( corresponds to the beginning of 1974). Use the trapezoidal rule to estimate the average fuel consumption of the domestic car built during this period. Round the result to the nearest hundredth. ​