Chapter 5

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=\sqrt{1+x^{3}}, \quad y=\frac{1}{2} x+2, \quad x=0 $$

Length of a Catenary Electrical wires suspended between two towers form a catenary (see figure) modeled by the equation \(y=20 \cosh \frac{x}{20}, \quad-20 \leq x \leq 20\) where \(x\) and \(y\) are measured in meters. The towers are 40 meters apart. Find the length of the suspended cable.

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=x \sqrt{\frac{4-x}{4+x}}, \quad y=0, \quad x=4 $$

Carbon-14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree growing today. A piece of ancient charcoal contains only \(15 \%\) as much of the radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of \({ }^{14} \mathrm{C}\) is 5715 years.)

Find the arc length from (-3,4) clockwise to (4,3) along the circle \(x^{2}+y^{2}=25\). Show that the result is one-fourth the circumference of the circle.

In Exercises \(35-40,\) sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=2 \sin x, \quad g(x)=\tan x, \quad-\frac{\pi}{3} \leq x \leq \frac{\pi}{3} $$

Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(y=e^{-x^{2} / 2}, \quad y=0, \quad x=0, \quad x=2\) (a) 3 (b) -5 (c) 10 (d) 7 (e) 20

Find the principal \(P\) that must be invested at rate \(r\), compounded monthly, so that \(\$ 500,000\) will be available for retirement in \(t\) years. $$ r=7 \frac{1}{2} \%, \quad t=20 $$

In Exercises \(35-38\), set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the \(x\) -axis. $$ y=\frac{1}{3} x^{3}, \quad 0 \leq x \leq 3 $$

In Exercises \(31-40,\) find \(M_{x}, M_{y},\) and \((\bar{x}, \bar{y})\) for the laminas of uniform density \(\rho\) bounded by the graphs of the equations. $$ y=x^{2 / 3}, y=0, x=8 $$

In Exercises \(35-38,\) (a) use a graphing utility to graph the plane region bounded by the graphs of the equations, and (b) use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the \(y\) -axis. $$ x^{4 / 3}+y^{4 / 3}=1, x=0, y=0, \text { first quadrant } $$

Sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=\sin x, g(x)=\cos 2 x,-\frac{\pi}{2} \leq x \leq \frac{\pi}{6} $$

Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(y=\arctan x, \quad y=0, \quad x=0, \quad x=1\) (a) 10 (b) \(\frac{3}{4}\) (c) 5 (d) -6 (e) 15

In Exercises \(35-38\), set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the \(x\) -axis. $$ y=2 \sqrt{x}, \quad 4 \leq x \leq 9 $$

In Exercises \(31-40,\) find \(M_{x}, M_{y},\) and \((\bar{x}, \bar{y})\) for the laminas of uniform density \(\rho\) bounded by the graphs of the equations. $$ y=x^{2 / 3}, y=4 $$

(a) use a graphing utility to graph the plane region bounded by the graphs of the equations, and (b) use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the \(y\) -axis. $$ y=\sqrt{1-x^{3}}, y=0, x=0 $$

Sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=\cos x, \mathrm{~g}(x)=2-\cos x, 0 \leq x \leq 2 \pi $$

A region bounded by the parabola \(y=4 x-x^{2}\) and the \(x\) -axis is revolved about the \(x\) -axis. A second region bounded by the parabola \(y=4-x^{2}\) and the \(x\) -axis is revolved about the \(x\) -axis. Without integrating, how do the volumes of the two solids compare? Explain.

Find the time necessary for \(\$ 1000\) to double if it is invested at a rate of \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. $$ r=7 \% $$

In Exercises \(35-38\), set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the \(x\) -axis. $$ y=\frac{x^{3}}{6}+\frac{1}{2 x}, \quad 1 \leq x \leq 2 $$

(a) use a graphing utility to graph the plane region bounded by the graphs of the equations, and (b) use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the \(y\) -axis. $$ y=\sqrt[3]{(x-2)^{2}(x-6)^{2}}, y=0, x=2, x=6 $$

Sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=\sec \frac{\pi x}{4} \tan \frac{\pi x}{4}, g(x)=(\sqrt{2}-4) x+4, \quad x=0 $$

Find the time necessary for \(\$ 1000\) to double if it is invested at a rate of \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. $$ r=6 \% $$

In Exercises \(35-38\), set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the \(x\) -axis. $$ y=\frac{x}{2}, \quad 0 \leq x \leq 6 $$

(a) use a graphing utility to graph the plane region bounded by the graphs of the equations, and (b) use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the \(y\) -axis. $$ y=\frac{2}{1+e^{1 / x}}, y=0, x=1, x=3 $$

Sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=x e^{-x^{2}}, \quad y=0, \quad 0 \leq x \leq 1 $$

If the portion of the line \(y=\frac{1}{2} x\) lying in the first quadrant is revolved about the \(x\) -axis, a cone is generated. Find the volume of the cone extending from \(x=0\) to \(x=6\).

The population (in millions) of a country in 2001 and the expected continuous annual rate of change \(k\) of the population for the years 2000 through 2010 are given. (Source: U.S. Census Bureau, International Data Base) (a) Find the exponential growth model \(P=C e^{k t}\) for the population by letting \(t=0\) correspond to 2000 . (b) Use the model to predict the population of the country in \(2015 .\) (c) Discuss the relationship between the sign of \(k\) and the change in population for the country. $$ \begin{array}{ll} \text { Country } & 2001 \text { Population } & k \\ \end{array} $$ $$ \begin{array}{lll} \text { Bulgaria } & 7.7 & -0.009 \end{array} $$

In Exercises 39 and 40 , set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the \(y\) -axis. $$ y=\sqrt[3]{x}+2, \quad 1 \leq x \leq 8 $$

Think About It In Exercises 39 and \(40,\) determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(y\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(y=2 e^{-x}, y=0, x=0, x=2\) (a) \(\frac{3}{2}\) (b) -2 (c) 4 (d) 7.5 (e) 15

Sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=3^{x}, \quad g(x)=2 x+1 $$

Use the disk method to verify that the volume of a right circular cone is \(\frac{1}{3} \pi r^{2} h,\) where \(r\) is the radius of the base and \(h\) is the height.

The population (in millions) of a country in 2001 and the expected continuous annual rate of change \(k\) of the population for the years 2000 through 2010 are given. (Source: U.S. Census Bureau, International Data Base) (a) Find the exponential growth model \(P=C e^{k t}\) for the population by letting \(t=0\) correspond to 2000 . (b) Use the model to predict the population of the country in \(2015 .\) (c) Discuss the relationship between the sign of \(k\) and the change in population for the country. $$ \begin{array}{ll} \text { Country } & 2001 \text { Population } & k \\ \end{array} $$ $$ \text { Cambodia } \quad 12.7 \quad 0.018 $$

In Exercises 39 and 40 , set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the \(y\) -axis. $$ y=9-x^{2}, \quad 0 \leq x \leq 3 $$

Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(y\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.)\(y=\tan x, y=0, x=0, x=\frac{\pi}{4}\) (a) 3.5 (b) \(-\frac{9}{4}\) (c) 8 (d) 10 (e) 1

In Exercises 41-44, (a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=2 \sin x+\sin 2 x, \quad y=0, \quad 0 \leq x \leq \pi $$

Use the disk method to verify that the volume of a sphere is \(\frac{4}{3} \pi r^{3}\).

One hundred bacteria are started in a culture and the number \(N\) of bacteria is counted each hour for 5 hours. The results are shown in the table, where \(t\) is the time in hours. $$ \begin{array}{|l|c|c|c|c|c|c|}\hline t & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline N & 100 & 126 & 151 & 198 & 243 & 297 \\\\\hline\end{array}$$ (a) Use the regression capabilities of a graphing utility to find an exponential model for the data. (b) Use the model to estimate the time required for the population to quadruple in size.

In Exercises \(41-44\), set up and evaluate the integrals for finding the area and moments about the \(x\) - and y-axes for the region bounded by the graphs of the equations. (Assume \(\rho=1\).) $$ y=x^{2}, y=x $$

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=2 \sin x+\cos 2 x, \quad y=0, \quad 0 < x \leq \pi $$

A sphere of radius \(r\) is cut by a plane \(h(h

The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours. (a) Find the initial population. (b) Write an exponential growth model for the bacteria population. Let \(t\) represent time in hours. (c) Use the model to determine the number of bacteria after 8 hours. (d) After how many hours will the bacteria count be \(25,000 ?\)

In Exercises 41 and \(42,\) use the integration capabilities of a graphing utility to approximate the surface area of the solid of revolution. $$ \begin{aligned} &y=\ln x\\\ &\text { revolved about the } y \text { -axis } \end{aligned} $$

In Exercises \(41-44\), set up and evaluate the integrals for finding the area and moments about the \(x\) - and y-axes for the region bounded by the graphs of the equations. (Assume \(\rho=1\).) $$ y=\frac{1}{x}, y=0,1 \leq x \leq 4 $$

Machine Part A solid is generated by revolving the region bounded by \(y=\sqrt{9-x^{2}}\) and \(y=0\) about the \(y\) -axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-third of the volume is removed. Find the diameter of the hole.

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=\frac{1}{x^{2}} e^{1 / x}, \quad y=0, \quad 1 \leq x \leq 3 $$

A cone of height \(H\) with a base of radius \(r\) is cut by a plane parallel to and \(h\) units above the base. Find the volume of the solid (frustum of a cone) below the plane.

The management at a certain factory has found that a worker can produce at most 30 units in a day. The learning curve for the number of units \(N\) produced per day after a new employee has worked \(t\) days is \(N=30\left(1-e^{k t}\right)\). After 20 days on the job, a particular worker produces 19 units. (a) Find the learning curve for this worker. (b) How many days should pass before this worker is producing 25 units per day?

In Exercises \(41-44\), set up and evaluate the integrals for finding the area and moments about the \(x\) - and y-axes for the region bounded by the graphs of the equations. (Assume \(\rho=1\).) $$ y=2 x+4, y=0,0 \leq x \leq 3 $$

{Volume of a Torus } A torus is formed by revolving the region bounded by the circle \(x^{2}+y^{2}=1\) about the line \(x=2\) (see figure). Find the volume of this "doughnut-shaped" solid. (Hint: The integral \(\int_{-1}^{1} \sqrt{1-x^{2}} d x\) represents the area of a semicircle.)