Exam 13: Techniques of Integration

A model for the ability M of a child to memorize, measured on a scale from 0 to 10, is $M = 1 + 1.6 t \ln t , 0 < t \leq 3$ where t is the child's age in years. Find the average value predicted by the model for a child's ability to memorize between second and third birthdays. Round your answer to three decimal places.

Use integration by parts to evaluate $\int 4 x e ^ { - 5 x } d x$

Determine the amount of money required to set up a charitable endowment that pays the amount $P$ each year indefinitely for the annual interest rate compounded continuously. $P = \$ 12,000 , r = 6 \%$

A body assimilates a 12-hour cold tablet at a rate modeled by $\frac { d C } { d t } = 6 - \ln \left( t ^ { 2 } - 2 t + 4 \right)$ , $0 \leq t \leq 12$ where t is measured in milligrams per hour and t is the time in hours. Use Simpson's Rule with n = 12 to estimate the total amount of the drug absorbed into the body during the 12 hours.

Use integration by parts to evaluate $\int x ^ { 2 } e ^ { - 3 x } d x$ . Note that evaluation may require integration by parts more than once.

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $\int _ { 0 } ^ { 1 } \left( \frac { x ^ { 2 } } { 2 } + 1 \right) d x , n = 4$

Use the error formulas to find n such that the error in the approximation of the definite integral $\int _ { 3 } ^ { 6 } \frac { 1 } { x } d x$ is less than 0.0001 using the Trapezoidal Rule.

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $\int _ { - \infty } ^ { 0 } e ^ { 6 x } d x$

Find the definite integral. $\int _ { 1 } ^ { 4 } x ^ { 3 } \ln x d x$

A business is expected to yield a continuous flow of profit at the rate of $600,000 per year. If money will earn interest at the nominal rate of 8% per year compounded continuously, what is the present value of the business forever?