Chapter 5

Lifting \(a\) Chain In Exercises 19-22, consider a 15-foot chain that weighs 3 pounds per foot hanging from a winch 15 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. Wind up the entire chain with a 500 -pound load attached to it.

Use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=x^{2}, \quad y=4 x-x^{2}, \text { about the line } x=2 $$

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

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=x^{2}+1, \quad y=-x^{2}+2 x+5, \quad x=0, \quad x=3 $$

Determine the quadrants in which the solution of the differential equation is an increasing function. Explain. (Do not solve the differential equation.) $$ \frac{d y}{d x}=\frac{1}{2} x y $$

Approximation In Exercises 23 and \(24,\) determine which value best approximates the length of the arc represented by the integral. (Make your selection on the basis of a sketch of the arc and not by performing any calculations.) \(\int_{0}^{2} \sqrt{1+\left[\frac{d}{d x}\left(\frac{5}{x^{2}+1}\right)\right]^{2}} d x\) (a) 25 (b) 5 (c) 2 (d) -4 (e)

Boyle's Law A quantity of gas with an initial volume of 2 cubic feet and a pressure of 1000 pounds per square foot expands to a volume of 3 cubic feet. Find the work done by the gas using the integral \(W=\int_{V_{0}}^{v_{1}} p d V\) Assume that the pressure is inversely proportional to the volume.

Use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=4 x-x^{2}, \quad y=0, \text { about the line } x=5 $$

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(y)=\frac{y}{\sqrt{16-y^{2}}}, \quad g(y)=0, \quad y=3 $$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\sqrt{x}, \quad y=-\frac{1}{2} x+4, \quad x=0, \quad x=8 $$

Approximation In Exercises 23 and \(24,\) determine which value best approximates the length of the arc represented by the integral. (Make your selection on the basis of a sketch of the arc and not by performing any calculations.) \(\int_{0}^{\pi / 4} \sqrt{1+\left[\frac{d}{d x}(\tan x)\right]^{2}} d x\) (a) 3 (b) -2 (c) 4 (d) \(\frac{4 \pi}{3}\) (e) 1

Determine the quadrants in which the solution of the differential equation is an increasing function. Explain. (Do not solve the differential equation.) $$ \frac{d y}{d x}=\frac{1}{2} x^{2} y $$

Use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=\sqrt{x}, \quad y=0, \quad x=4, \text { about the line } x=6 $$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(y\) -axis. $$ y=3(2-x), \quad y=0, \quad x=0 $$

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ \({ }^{226} \mathrm{Ra} \quad 1599 \quad 10 \mathrm{~g}\)

Approximation In Exercises 25 and 26, approximate the arc length of the graph of the function over the interval [0,4] in four ways. (a) Use the Distance Formula to find the distance between the endpoints of the arc. (b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when \(x=0, x=1, x=2, x=3,\) and \(x=4 .\) Find the sum of the four lengths. (c) Use Simpson's Rule with \(n=10\) to approximate the integral yielding the indicated arc length. (d) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated are length. $$ f(x)=x^{3} $$

Hydraulic Press In Exercises 25 and \(26,\) use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force \(F\) (in pounds) and the distance \(x\) (in feet) the press moves is given. $$ \begin{array}{ll} \text { Force } & \text { Interval } \\ \hline F(x)=1000[1.8-\ln (x+1)] & 0 \leq x \leq 5 \end{array} $$

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ g(x)=\frac{4}{2-x}, \quad y=4, \quad x=0 $$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(y\) -axis. $$ y=9-x^{2}, \quad y=0, \quad x=2, \quad x=3 $$

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ { }^{226} \mathrm{Ra} \quad 1599 \quad \quad 1.5 \mathrm{~g} $$

Approximation In Exercises 25 and \(26,\) approximate the arc length of the graph of the function over the interval [0,4] in four ways. (a) Use the Distance Formula to find the distance between the endpoints of the arc. (b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when \(x=0, x=1, x=2, x=3,\) and \(x=4\). Find the sum of the four lengths. (c) Use Simpson's Rule with \(n=10\) to approximate the integral yielding the indicated arc length. (d) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated arc length. $$ f(x)=\left(x^{2}-4\right)^{2} $$

Hydraulic Press In Exercises 25 and \(26,\) use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force \(F\) (in pounds) and the distance \(x\) (in feet) the press moves is given. $$ F(x)=\frac{e^{x^{3}}-1}{100} \quad 0 \leq x \leq 4 $$

In Exercises \(27-34,\) (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. $$ f(x)=x\left(x^{2}-3 x+3\right), \quad g(x)=x^{2} $$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. Verify your results using the integration capabilities of a graphing utility. $$ y=\cos x, \quad y=0, \quad x=0, \quad x=\frac{\pi}{2} $$

(a) Use a graphing utility to graph the function \(f(x)=x^{2 / 3}\). (b) Can you integrate with respect to \(x\) to find the arc length of the graph of \(f\) on the interval [-1,8]\(?\) Explain. (c) Find the arc length of the graph of \(f\) on the interval [-1,8] .

In Exercises 27 and \(28,\) find the center of mass of the point masses lying on the \(x\) -axis. $$ \begin{array}{l} m_{1}=6, m_{2}=3, m_{3}=5 \\ x_{1}=-5, x_{2}=1, x_{3}=3 \end{array} $$

In Exercises \(27-30\), use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line. \(y=x^{3}, \quad y=0, \quad x=2\) (a) the \(x\) -axis (b) the \(y\) -axis (c) the line \(x=4\)

(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. $$ y=x^{4}-2 x^{2}, \quad y=2 x^{2} $$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. Verify your results using the integration capabilities of a graphing utility. $$ y=e^{x-1}, \quad y=0, \quad x=1, \quad x=2 $$

Astroid Find the total length of the graph of the astroid \(x^{2 / 3}+y^{2 / 3}=4\)

In Exercises 27 and \(28,\) find the center of mass of the point masses lying on the \(x\) -axis. $$ \begin{array}{l} m_{1}=12, m_{2}=1, m_{3}=6, m_{4}=3, m_{5}=11 \\ x_{1}=-6, x_{2}=-4, x_{3}=-2, x_{4}=0, x_{5}=8 \end{array} $$

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line. \(y=\frac{10}{x^{2}}, \quad y=0, \quad x=1, \quad x=5\) (a) the \(x\) -axis (b) the \(y\) -axis (c) the line \(y=10\)

(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. $$ f(x)=x^{4}-4 x^{2}, \quad g(x)=x^{2}-4 $$

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=e^{-x^{2}}, \quad y=0, \quad x=0, \quad x=2 $$

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ \begin{array}{llll} { }^{14} \mathrm{C} & 5715 & 5 \mathrm{~g} \end{array} $$

In Exercises 29 and \(30,\) find the center of mass of the given system of point masses. $$ \begin{array}{|l|c|c|c|} \hline m_{i} & 5 & 1 & 3 \\ \hline\left(x_{1}, y_{1}\right) & (2,2) & (-3,1) & (1,-4) \\ \hline \end{array} $$

(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. $$ f(x)=x^{4}-4 x^{2}, g(x)=x^{3}-4 x $$

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\ln x, \quad y=0, \quad x=1, \quad x=3 $$

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ { }^{14} \mathrm{C} \quad 5715\quad \quad 3.2 \mathrm{~g} $$

Think About It Explain why the two integrals are equal. \(\int_{1}^{e} \sqrt{1+\frac{1}{x^{2}}} d x=\int_{0}^{1} \sqrt{1+e^{2 x}} d x\) Use the integration capabilities of a graphing utility to verify that the integrals are equal.

In Exercises 29 and \(30,\) find the center of mass of the given system of point masses. $$ \begin{array}{|l|c|c|c|c|} \hline m_{i} & 12 & 6 & \frac{15}{2} & 15 \\ \hline\left(x_{1}, y_{1}\right) & (2,3) & (-1,5) & (6,8) & (2,-2) \\ \hline \end{array} $$

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line. \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}, \quad a>0\) (hypocycloid) (a) the \(x\) -axis (b) the \(y\) -axis

(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. $$ f(x)=1 /\left(1+x^{2}\right), \quad g(x)=\frac{1}{2} x^{2} $$

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=2 \arctan (0.2 x), \quad y=0, \quad x=0, \quad x=5 $$

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ { }^{239} \mathrm{Pu} \quad 24.100 \quad \quad 2.1 \mathrm{~g} $$

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. $$ y=\sqrt{x}, y=0, x=4 $$

Consider a solid that is generated by revolving a plane region about the \(y\) -axis. Describe the position of a representative rectangle when using (a) the shell method and (b) the disk method to find the volume of the solid.

(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. $$ f(x)=6 x /\left(x^{2}+1\right), \quad y=0, \quad 0 \leq x \leq 3 $$

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\sqrt{2 x}, \quad y=x^{2} $$

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ { }^{239} \mathrm{Pu} \quad 24,100 \quad \quad \quad \quad \quad 0.4 \mathrm{~g} $$