Chapter 11

Use the value \(\int_{0}^{2} x^{3} d x=4\) to evaluate each definite integral. Explain your reasoning. (a) \(\int_{-2}^{0} x^{3} d x\) (b) \(\int_{-2}^{2} x^{3} d x\) (c) \(\int_{0}^{2} 3 x^{3} d x\)

The rate of increase of the number of married couples \(M\) (in thousands) in the United States from 1970 to 2005 can be modeled by \(\frac{d M}{d t}=1.218 t^{2}-44.72 t+709.1\) where \(t\) is the time in years, with \(t=0\) corresponding to \(1970 .\) The number of married couples in 2005 was 59,513 thousand. (Source: U.S. Census Bureau) (a) Find the model for the number of married couples in the United States. (b) Use the model to predict the number of married couples in the United States in \(2012 .\) Does your answer seem reasonable? Explain your reasoning.

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d C}{d x}=2.25 \quad x=100 $$

The rate of growth of the number of Internet users \(I\) (in millions) in the world from 1991 to 2004 can be modeled by \(\frac{d I}{d t}=-0.25 t^{3}+5.319 t^{2}-19.34 t+21.03\) where \(t\) is the time in years, with \(t=1\) corresponding to 1991\. The number of Internet users in 2004 was 863 million. (Source: International Telecommunication Union) (a) Find the model for the number of Internet users in the world. (b) Use the model to predict the number of Internet users in the world in 2012 . Does your answer seem reasonable? Explain your reasoning.

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d C}{d x}=\frac{20,000}{x^{2}} \quad x=10 $$

The table gives the marginal benefit and marginal cost of producing \(x\) units of a product for a given company. Plot the points in each column and use the regression feature of a graphing utility to find a linear model for marginal benefit and a quadratic model for marginal cost. Then use integration to find the benefit \(B\) and cost \(C\) equations. Assume \(B(0)=0\) and \(C(0)=425 .\) Finally, find the intervals in which the benefit exceeds the cost of producing \(x\) units, and make a recommendation for how many units the company should produce based on your findings. (Source: Adapted from \mathrm{\\{} T a y l o r , ~ E c o n o m i c s , ~ F i f t h ~ E d i t i o n ) ~ $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Number of units } & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Marginal benefit } & 330 & 320 & 290 & 270 & 250 \\ \hline \text { Marginal cost } & 150 & 120 & 100 & 110 & 120 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Number of units } & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Marginal benefit } & 230 & 210 & 190 & 170 & 160 \\ \hline \text { Marginal cost } & 140 & 160 & 190 & 250 & 320 \\ \hline \end{array} $$

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d R}{d x}=48-3 x \quad x=12 $$

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d R}{d x}=75\left(20+\frac{900}{x}\right) \quad x=500 $$

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d P}{d x}=\frac{400-x}{150} \quad x=200 $$

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d P}{d x}=12.5(40-3 \sqrt{x}) \quad x=125 $$

Find the amount of an annuity with income function \(c(t)\), interest rate \(r\), and term \(T\). $$ c(t)=\$ 250, \quad r=8 \%, \quad T=6 \text { years } $$

Find the amount of an annuity with income function \(c(t)\), interest rate \(r\), and term \(T\). $$ c(t)=\$ 500, \quad r=7 \%, \quad T=4 \text { years } $$

Find the amount of an annuity with income function \(c(t)\), interest rate \(r\), and term \(T\). $$ c(t)=\$ 1500, \quad r=2 \%, \quad T=10 \text { years } $$

Find the amount of an annuity with income function \(c(t)\), interest rate \(r\), and term \(T\). $$ c(t)=\$ 2000, \quad r=3 \%, \quad T=15 \text { years } $$

You are given the rate of investment \(d l / d t\). Find the capital accumulation over a five-year period by evaluating the definite integral Capital accumulation \(=\int_{0}^{5} \frac{d l}{d t} d t\) where \(t\) is the time in years. $$ \frac{d I}{d t}=500 $$

You are given the rate of investment \(d l / d t\). Find the capital accumulation over a five-year period by evaluating the definite integral Capital accumulation \(=\int_{0}^{5} \frac{d l}{d t} d t\) where \(t\) is the time in years. $$ \frac{d I}{d t}=100 t $$

You are given the rate of investment \(d l / d t\). Find the capital accumulation over a five-year period by evaluating the definite integral Capital accumulation \(=\int_{0}^{5} \frac{d l}{d t} d t\) where \(t\) is the time in years. $$ \frac{d I}{d t}=500 \sqrt{t+1} $$

You are given the rate of investment \(d l / d t\). Find the capital accumulation over a five-year period by evaluating the definite integral Capital accumulation \(=\int_{0}^{5} \frac{d l}{d t} d t\) where \(t\) is the time in years. $$ \frac{d I}{d t}=\frac{12,000 t}{\left(t^{2}+2\right)^{2}} $$

The total cost of purchasing and maintaining a piece of equipment for \(x\) years can be modeled by \(C=5000\left(25+3 \int_{0}^{x} t^{1 / 4} d t\right)\) Find the total cost after (a) 1 year, (b) 5 years, and (c) 10 years.

A company purchases a new machine for which the rate of depreciation can be modeled by \(\frac{d V}{d t}=10,000(t-6), \quad 0 \leq t \leq 5\) where \(V\) is the value of the machine after \(t\) years. Set up and evaluate the definite integral that yields the total loss of value of the machine over the first 3 years.

A deposit of \(\$ 2250\) is made in a savings account at an annual interest rate of \(6 \%\), compounded continuously. Find the average balance in the account during the first 5 years.

The rate of change of mortgage debt outstanding for one- to four-family homes in the United States from 1998 through 2005 can be modeled by \(\frac{d M}{d t}=5.142 t^{2}-283,426.2 e^{-x}\) where \(M\) is the mortgage debt outstanding (in billions of dollars) and \(t\) is the year, with \(t=8\) corresponding to \(1998 .\) In 1998 , the mortgage debt outstanding in the United States was \(\$ 4259\) billion. (Source: Board of Governors of the Federal Reserve System) (a) Write a model for the debt as a function of \(t\). (b) What was the average mortgage debt outstanding for 1998 through \(2005 ?\)

The velocity \(v\) of the flow of blood at a distance \(r\) from the center of an artery of radius \(R\) can be modeled by \(v=k\left(R^{2}-r^{2}\right), \quad k>0\) where \(k\) is a constant. Find the average velocity along a radius of the artery. (Use 0 and \(R\) as the limits of integration.)

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{3}^{6} \frac{x}{3 \sqrt{x^{2}-8}} d x $$

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{1 / 2}^{1}(x+1) \sqrt{1-x} d x $$

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{2}^{5}\left(\frac{1}{x^{2}}-\frac{1}{x^{3}}\right) d x $$

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{0}^{1} x^{3}\left(x^{3}+1\right)^{3} d x $$