Chapter 12

Evaluate the definite integral. $$ \int_{1}^{3} x^{2} \ln x d x $$

Write the partial fraction decomposition for the rational expression. Check your result algebraically. Then assign a value to the constant \(a\) and use a graphing utility to check the result graphically. $$ \frac{1}{(x+1)(a-x)} $$

$$ y=\frac{\ln x}{x^{2}}, y=0, x=1, x=e $$

Capitalized Cost In Exercises 51 and 52, find the capitalized cost \(C\) of an asset \((a)\) for \(n=5\) years, \((b)\) for \(n=10\) years, and (c) forever. The capitalized cost is given by \(C=C_{0}+\int_{0}^{n} c(t) e^{-r t} d t\) where \(C_{0}\) is the original investment, \(t\) is the time in years, \(r\) is the annual interest rate compounded continuously, and \(c(t)\) is the annual cost of maintenance (measured in dollars). [Hint: For part (c), see Exercises \(35-38 .]\) $$ C_{0}=\$ 650,000, c(t)=25,000, r=10 \% $$

Lumber Use The table shows the amounts of lumber used for residential upkeep and improvements (in billions of board-feet per year) for the years 1997 through \(2005 .\) (Source: U.S. Forest Service) \begin{tabular}{|l|c|c|c|c|c|} \hline Year & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline Amount & \(15.1\) & \(14.7\) & \(15.1\) & \(16.4\) & \(17.0\) \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|c|} \hline Year & 2002 & 2003 & 2004 & 2005 \\ \hline Amount & \(17.8\) & \(18.3\) & \(20.0\) & \(20.6\) \\ \hline \end{tabular} (a) Use Simpson's Rule to estimate the average number of board-feet (in billions) used per year over the time period. (b) A model for the data is $$ L=6.613+0.93 t+2095.7 e^{-t}, \quad 7 \leq t \leq 15 $$ where \(L\) is the amount of lumber used and \(t\) is the year, with \(t=7\) corresponding to 1997 . Use integration to find the average number of board- feet (in billions) used per year over the time period. (c) Compare the results of parts (a) and (b).

Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \int x^{2} e^{x} d x \quad \text { Integration by parts } $$

Writing What is the first step when integrating \(\int \frac{x^{2}}{x-5} d x ?\) Explain. (Do not integrate.)

Capitalized Cost In Exercises 51 and 52, find the capitalized cost \(C\) of an asset \((a)\) for \(n=5\) years, \((b)\) for \(n=10\) years, and (c) forever. The capitalized cost is given by \(C=C_{0}+\int_{0}^{n} c(t) e^{-r t} d t\) where \(C_{0}\) is the original investment, \(t\) is the time in years, \(r\) is the annual interest rate compounded continuously, and \(c(t)\) is the annual cost of maintenance (measured in dollars). [Hint: For part (c), see Exercises \(35-38 .]\) $$ C_{0}=\$ 650,000, c(t)=25,000(1+0.08 t), r=12 \% $$

Median Age The table shows the median ages of the U.S. resident population for the years 1997 through \(2005 .\) (Source: U.S. Census Bureau) \begin{tabular}{|l|c|c|c|c|c|} \hline Year & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline Median age & \(34.7\) & \(34.9\) & \(35.2\) & \(35.3\) & \(35.6\) \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|c|} \hline Year & 2002 & 2003 & 2004 & 2005 \\ \hline Median age & \(35.7\) & \(35.9\) & \(36.0\) & \(36.2\) \\ \hline \end{tabular} (a) Use Simpson's Rule to estimate the average age over the time period. (b) A model for the data is \(A=31.5+1.21 \sqrt{t}\), \(7 \leq t \leq 15\), where \(A\) is the median age and \(t\) is the year, with \(t=7\) corresponding to 1997 . Use integration to find the average age over the time period. (c) Compare the results of parts (a) and (b).

Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \int x^{4} \ln x d x \quad \text { Integration by parts } $$

Writing State the method you would use to evaluate each integral. Explain why you chose that method. (Do not integrate.) (a) \(\int \frac{2 x+1}{x^{2}+x-8} d x\) (b) \(\int \frac{7 x+4}{x^{2}+2 x-8} d x\)

Medicine A body assimilates a 12 -hour cold tablet at a rate \(\quad\) modeled \(\quad\) by \(\quad d C / d t=8-\ln \left(t^{2}-2 t+4\right)\), \(0 \leq t \leq 12\), where \(d C / d t\) is measured in milligrams per hour and \(t\) is the time in hours. Use Simpson's Rule with \(n=8\) to estimate the total amount of the drug absorbed into the body during the 12 hours.

Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \begin{aligned} &\int \frac{1}{x^{2}(x+1)} d x\\\ &\text { Partial fractions } \end{aligned} $$

Biology A conservation organization releases 100 animals of an endangered species into a game preserve. During the first 2 years, the population increases to 134 animals. The organization believes that the preserve has a capacity of 1000 animals and that the herd will grow according to a logistic growth model. That is, the size \(y\) of the herd will follow the equation \(\int \frac{1}{y(1-y / 1000)} d y=\int k d t\) where \(t\) is measured in years. Find this logistic curve. (To solve for the constant of integration \(C\) and the proportionality constant \(k\), assume \(y=100\) when \(t=0\) and \(y=134\) when \(t=2 .\) Use a graphing utility to graph your solution.

Medicine The concentration \(M\) (in grams per liter) of a six-hour allergy medicine in a body is modeled by \(M=12-4 \ln \left(t^{2}-4 t+6\right), 0 \leq t \leq 6\), where \(t\) is the time in hours since the allergy medication was taken. Use Simpson's Rule with \(n=6\) to estimate the average level of concentration in the body over the six- hour period.

Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \begin{aligned} &\int \frac{1}{x^{2}-75} d x\\\ &\text { Partial fractions } \end{aligned} $$

Health: Epidemic A single infected individual enters a community of 500 individuals susceptible to the disease. The disease spreads at a rate proportional to the product of the total number infected and the number of susceptible individuals not yet infected. A model for the time it takes for the disease to spread to \(x\) individuals is \(t=5010 \int \frac{1}{(x+1)(500-x)} d x\) where \(t\) is the time in hours. (a) Find the time it takes for \(75 \%\) of the population to become infected (when \(t=0, x=1\) ). (b) Find the number of people infected after 100 hours.

Consumer Trends The rate of change \(S\) in the number of subscribers to a newly introduced magazine is modeled by \(d S / d t=1000 t^{2} e^{-t}, 0 \leq t \leq 6\), where \(t\) is the time in years. Use Simpson's Rule with \(n=12\) to estimate the total increase in the number of subscribers during the first 6 years.

Prove that Simpson's Rule is exact when used to approximate the integral of a cubic polynomial function, and demonstrate the result for \(\int_{0}^{1} x^{3} d x, n=2\).

Probability The probability of finding between \(a\) and \(b\) percent iron in ore samples is modeled by \(P(a \leq x \leq b)=\int_{a}^{b} 2 x^{3} e^{x^{2}} d x, \quad 0 \leq a \leq b \leq 1\) (see figure). Find the probabilities that a sample will contain between (a) \(0 \%\) and \(25 \%\) and (b) \(50 \%\) and \(100 \%\) iron.

Population Growth \(\ln\) Exercises 57 and 58, use a graphing utility to graph the growth function. Use the table of integrals to find the average value of the growth function over the interval, where \(N\) is the size of a population and \(t\) is the time in days. $$ N=\frac{5000}{1+e^{4.8-1.9 t}}, \quad[0,2] $$

Revenue The revenue \(R\) (in millions of dollars per year) for Symantec Corporation from 1997 through 2005 can be modeled by \(R=\frac{1340 t^{2}+24,044 t+22,704}{-6 t^{2}+94 t+100}\) where \(t=7\) corresponds to 1997 . Find the total revenue from 1997 through \(2005 .\) Then find the average revenue during this time period. (Source: Symantec Corporation)

Find the area of the region bounded by the graphs of the given equations. $$ \begin{aligned} &y=x e^{-x}\\\ &y=0, x=4 \end{aligned} $$

Population Growth \(\ln\) Exercises 57 and 58, use a graphing utility to graph the growth function. Use the table of integrals to find the average value of the growth function over the interval, where \(N\) is the size of a population and \(t\) is the time in days. $$ N=\frac{375}{1+e^{4.20-0.25 t}}, \quad[21,28] $$

Environment The predicted cost \(C\) (in hundreds of thousands of dollars) for a company to remove \(p \%\) of a chemical from its waste water is shown in the table. \begin{tabular}{|c|c|c|c|c|l|} \hline\(p\) & 0 & 10 & 20 & 30 & 40 \\ \hline\(C\) & 0 & \(0.7\) & \(1.0\) & \(1.3\) & \(1.7\) \\ \hline \end{tabular} \begin{tabular}{|l|l|l|l|l|l|} \hline\(p\) & 50 & 60 & 70 & 80 & 90 \\ \hline\(C\) & \(2.0\) & \(2.7\) & \(3.6\) & \(5.5\) & \(11.2\) \\ \hline \end{tabular} A model for the data is given by \(C=\frac{124 p}{(10+p)(100-p)}, \quad 0 \leq p<100\) Use the model to find the average cost for removing between \(75 \%\) and \(80 \%\) of the chemical.

Find the area of the region bounded by the graphs of the given equations. $$ \begin{aligned} &y=\frac{1}{9} x e^{-x / 3}\\\ &y=0, x=0, x=3 \end{aligned} $$

Revenue The revenue (in dollars per year) for a new product is modeled by \(R=10,000\left[1-\frac{1}{\left(1+0.1 t^{2}\right)^{1 / 2}}\right]\) where \(t\) is the time in years. Estimate the total revenue from sales of the product over its first 2 years on the market.

Find the area of the region bounded by the graphs of the given equations. $$ \begin{aligned} &y=x \ln x\\\ &y=0, x=e \end{aligned} $$

Consumer and Producer Surpluses Find the consumer surplus and the producer surplus for a product with the given demand and supply functions. Demand: \(p=\frac{60}{\sqrt{x^{2}+81}}\), Supply: \(p=\frac{x}{3}\)

Population Growth The population of the United States was 76 million people in 1900 and reached 300 million people in 2006. From 1900 through 2006 , assume the population of the United States can be modeled by logistic growth with a limit of \(839.1\) million people. (Source: U.S. Census Bureau) (a) Write a differential equation of the form \(\frac{d y}{d t}=k y\left(1-\frac{y}{L}\right)\) where \(y\) represents the population of the United States (in millions of people) and \(t\) represents the number of years since 1900 . (b) Find the logistic growth model \(y=\frac{L}{1+b e^{-k t}}\) for this population. (c) Use a graphing utility to graph the model from part (b). Then estimate the year in which the population of the United States will reach 400 million people.

Find the area of the region bounded by the graphs of the given equations. $$ \begin{aligned} &y=x^{-3} \ln x\\\ &y=0, x=e \end{aligned} $$

Profit The net profits \(P\) (in billions of dollars per year) for The Hershey Company from 2002 through 2005 can be modeled by \(P=\sqrt{0.00645 t^{2}+0.1673}, \quad 2 \leq t \leq 5\) where \(t\) is time in years, with \(t=2\) corresponding to 2002 . Find the average net profit over that time period. (Source: The Hershey Co.)

Use a symbolic integration utility to evaluate the integral. $$ \int_{0}^{2} t^{3} e^{-4 t} d t $$

Use a symbolic integration utility to evaluate the integral. $$ \int_{1}^{4} \ln x\left(x^{2}+4\right) d x $$

Use a symbolic integration utility to evaluate the integral. $$ \int_{0}^{5} x^{4}\left(25-x^{2}\right)^{3 / 2} d x $$

Use a symbolic integration utility to evaluate the integral. $$ \int_{1}^{e} x^{9} \ln x d x $$

Demand A manufacturing company forecasts that the demand \(x\) (in units per year) for its product over the next 10 years can be modeled by \(x=500\left(20+t e^{-0.1 t}\right)\) for \(0 \leq t \leq 10\), where \(t\) is the time in years. (T) (a) Use a graphing utility to decide whether the company is forecasting an increase or a decrease in demand over the decade. (b) According to the model, what is the total demand over the next 10 years? (c) Find the average annual demand during the 10 -year period.

Capital Campaign The board of trustees of a college is planning a five-year capital gifts campaign to raise money for the college. The goal is to have an annual gift income \(I\) that is modeled by \(I=2000\left(375+68 t e^{-0.2 t}\right)\) for \(0 \leq t \leq 5\), where \(t\) is the time in years. (T) (a) Use a graphing utility to decide whether the board of trustees expects the gift income to increase or decrease over the five-year period. (b) Find the expected total gift income over the five-year period. (c) Determine the average annual gift income over the five-year period. Compare the result with the income given when \(t=3\).

Memory Model 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, \quad 0

Find the present value of the income \(c\) (measured in dollars) over \(t_{1}\) years at the given annual inflation rate \(r\). $$ c=100,000+4000 t, r=5 \%, t_{1}=10 \text { years } $$

Find the present value of the income \(c\) (measured in dollars) over \(t_{1}\) years at the given annual inflation rate \(r\). $$ c=30,000+500 t, r=7 \%, t_{1}=6 \text { years } $$

Find the present value of the income \(c\) (measured in dollars) over \(t_{1}\) years at the given annual inflation rate \(r\). $$ c=1000+50 e^{t / 2}, r=6 \%, t_{1}=4 \text { years } $$

Find the present value of the income \(c\) (measured in dollars) over \(t_{1}\) years at the given annual inflation rate \(r\). $$ c=5000+25 t e^{t / 10}, r=6 \%, t_{1}=10 \text { years } $$

Present Value A company expects its income \(c\) during the next 4 years to be modeled by \(c=150,000+75,000 t\) (a) Find the actual income for the business over the 4 years. (b) Assuming an annual inflation rate of \(4 \%\), what is the present value of this income?

Present Value A professional athlete signs a three-year contract in which the earnings can be modeled by \(c=300,000+125,000 t .\) (a) Find the actual value of the athlete's contract. (b) Assuming an annual inflation rate of \(3 \%\), what is the present value of the contract?

Find the future value of the income (in dollars) given by \(f(t)\) over \(t_{1}\) years at the annual interest rate of \(r\). If the function \(f\) represents a continuous investment over a period of \(t_{1}\) years at an annual interest rate of \(r\) (compounded continuously), then the future value of the investment is given by Future value \(=\mathrm{e}^{r t_{1}}\left[{ }^{t_{1}} f(t) e^{-r t} d t\right.\). $$ f(t)=3000, r=8 \%, t_{1}=10 \text { years } $$

Find the future value of the income (in dollars) given by \(f(t)\) over \(t_{1}\) years at the annual interest rate of \(r\). If the function \(f\) represents a continuous investment over a period of \(t_{1}\) years at an annual interest rate of \(r\) (compounded continuously), then the future value of the investment is given by Future value \(=\mathrm{e}^{r t_{1}}\left[{ }^{t_{1}} f(t) e^{-r t} d t\right.\). $$ f(t)=3000 e^{0.05 t}, r=10 \%, t_{1}=5 \text { years } $$

Finance: Future Value Use the equation from Exercises 77 and 78 to calculate the following. (Source: Adapted from Garman/Forgue, Personal Finance, Eighth Edition) (a) The future value of \(\$ 1200\) saved each year for 10 years earning \(7 \%\) interest. (b) A person who wishes to invest \(\$ 1200\) each year finds one investment choice that is expected to pay \(9 \%\) interest per year and another, riskier choice that may pay \(10 \%\) interest per year. What is the difference in return (future value) if the investment is made for 15 vears?

Use a program similar to the Midpoint Rule program on page 856 with \(n=12\) to approximate the area of the region bounded by the graphs of \(y=\frac{10}{\sqrt{x} e^{x}}, y=0, x=1\), and \(x=4\).