Exam 8: Integration Techniques, L'Hôpital's Rule, and Improper Integrals

​The base of a certain solid is the region bounded by the coordinate axes,the line ,and the graph of .For this solid,each cross section perpendicular to the -axis is a square whose side is across the base.The volume of this solid is ​

Evaluate the limit using L'Hopital's Rule if necessary.

Evaluate .Round your answer to three decimal places. ​

Evaluate the integral using the integration limits.Round your answer to three decimal places. ​

​Find an expression for ,in terms of only,that does not involve an integral. ​

Evaluate the limit using L'Hopital's Rule if necessary. ​

Find the indefinite integral. ​ ​

Find the definite integral. ​ ​

Determine whether the improper integral diverges or converges.Evaluate the integral if it converges. ​

Find the indefinite integral. ​ ​

Evaluate . ​

Use partial fractions to find the integral . ​

​Is the graph of concave up or concave down at ? Explain your answer. ​

Use integration tables to evaluate the integral . ​

Find the indefinite integral by making the substitution . ​ ​

Find . ​

A population is growing according to the logistic model ,where t is the time in days.Find the average population over the interval .Round your answer to one decimal place. ​

​The area of the region bounded by the graphs of , , ,and the -axis is ​

Find the indefinite integral. ​ ​

Find the indefinite integral. ​ ​