Chapter 5

Find the value of the expression accurate to four decimal places. a. \(\sinh 2\) b. \(\cosh 4\) c. \(\operatorname{sech} 3\)

Find the center of mass of the system comprising masses \(m_{k}\) located at the points \(x_{k}\) on a coordinate line. Assume that mass is measured in kilograms and distance is measured in meters. $$ \begin{array}{l} m_{1}=2, \quad m_{2}=4, \quad m_{3}=6 ; \quad x_{1}=-3, \quad x_{2}=-1, \\ x_{3}=4 \end{array} $$

Find the work done in lifting a 50 -lb sack of potatoes to a height of \(4 \mathrm{ft}\) above the ground.

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line.

Find the value of the expression accurate to four decimal places. a. \(\operatorname{csch} 3\) b. \(\tanh (-2)\) c. \(\operatorname{coth} 5\)

Find the center of mass of the system comprising masses \(m_{k}\) located at the points \(x_{k}\) on a coordinate line. Assume that mass is measured in kilograms and distance is measured in meters. $$ \begin{array}{l} m_{1}=3, \quad m_{2}=1, \quad m_{3}=5, \quad m_{4}=6 ; \quad x_{1}=-4, \\ x_{2}=-1, \quad x_{3}=1, \quad x_{4}=3 \end{array} $$

A rectangular swimming pool is \(40 \mathrm{ft}\) long, \(15 \mathrm{ft}\) wide, and \(9 \mathrm{ft}\) deep. If the pool is filled with water to a depth of \(8 \mathrm{ft}\), find the force exerted by the water (a) on the bottom of the pool and (b) on one end of the pool.

How much work is done in lifting a \(4-\mathrm{kg}\) bag of rice to a height of \(1.5 \mathrm{~m}\) above the ground?

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line.

Find the value of the expression accurate to four decimal places. a. \(\cosh 0\) b. \(\operatorname{sech}(-1)\) c. \(\operatorname{csch}(\ln 2)\)

Find the center of mass of the system comprising masses \(m_{k}\) located at the points \(x_{k}\) on a coordinate line. Assume that mass is measured in kilograms and distance is measured in meters. $$ \begin{array}{l} m_{1}=4, \quad m_{2}=3, \quad m_{3}=2, \quad m_{4}=4, \quad m_{5}=8 \\ x_{1}=-5, \quad x_{2}=-3, \quad x_{3}=-2, \quad x_{4}=2, \quad x_{5}=4 \end{array} $$

A particle moves a distance of \(100 \mathrm{ft}\) along a straight line. As it moves, it is acted upon by a constant force of magnitude \(5 \mathrm{lb}\) in a direction opposite to that of the motion. What is the work done by the force?

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line.

Find the value of the expression accurate to four decimal places. a. \(\sinh ^{-1} 1\) b. \(\cosh ^{-1} 2\) c. \(\operatorname{sech}^{-1} \frac{1}{3}\)

Find the center of mass of the system comprising masses \(m_{k}\) located at the points \(x_{k}\) on a coordinate line. Assume that mass is measured in kilograms and distance is measured in meters. $$ \begin{array}{l} m_{1}=6, \quad m_{2}=4, \quad m_{3}=5, \quad m_{4}=8, \quad m_{5}=4 \\ x_{1}=-4, \quad x_{2}=-2, \quad x_{3}=0, \quad x_{4}=3, \quad x_{5}=6 \end{array} $$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line.

Find the value of the expression accurate to four decimal places. a. \(\operatorname{csch}^{-1} 2\) b. \(\operatorname{csch}^{-1}(-2)\) c. \(\operatorname{coth}^{-1} \frac{3}{2}\)

Find the center of mass of the system comprising masses \(m_{k}\) located at the points \(P_{k}\) in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. $$ \begin{array}{l} m_{1}=4, \quad m_{2}=3, \quad m_{3}=5 ; \quad P_{1}(-3,-2), \quad P_{2}(-1,2), \\\ P_{3}(2,4) \end{array} $$

Find the work done by the force \(F(x)=2 x-1\) (measured in pounds) in moving an object along the \(x\) -axis from \(x=-2\) to \(x=4(x\) is measured in feet \() .\)

In Exercises 5 and 6, find the length of the line segment joining the two given points by finding the equation of the line and using Equation (2). Then check your answer by using the distance formula. $$ (0,0) \text { and }(3,8) $$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line.

Find the center of mass of the system comprising masses \(m_{k}\) located at the points \(P_{k}\) in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. $$ \begin{array}{l} m_{1}=2, \quad m_{2}=4, \quad m_{3}=1 ; \quad P_{1}(-2,2), \quad P_{2}(2,1), \\\ P_{3}(3,-1) \end{array} $$

Find the work done by the force \(f(x)=4 / x^{2}\) (measured in pounds) in moving a particle along the \(x\) -axis from \(x=1\) to \(x=6(x\) is measured in feet \()\).

Find the length of the line segment joining the two given points by finding the equation of the line and using Equation (2). Then check your answer by using the distance formula. $$ (-1,-2) \text { and }(3,6) $$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line.

Prove the identity. \(\cosh (-x)=\cosh x\)

Find the center of mass of the system comprising masses \(m_{k}\) located at the points \(P_{k}\) in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. $$ \begin{array}{l} m_{1}=3, \quad m_{2}=4, \quad m_{3}=6, \quad m_{4}=5 ; \quad P_{1}(-3,-2), \\ P_{2}(-2,3), \quad P_{3}(2,3), \quad P_{4}(4,-2) \end{array} $$

When a particle is at the point \(x\) on the \(x\) -axis, it is acted upon by a force of \(x^{2}+2 x\) newtons. Find the work done by the force in moving the particle from the origin to the point \(x=3(x\) is measured in meters \()\).

In Exencises \(7-18\), find the arc length of the graph of the given equation from \(P\) to \(Q\) or on the specified interval. $$ y=-2 x+3 $$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(y=x^{2}, \quad y=0, \quad x=2 ; \quad\) the \(y\) -axis

Oil Production Shortfall Energy experts disagree about when global oil production will begin to decline. In the following figure, the function \(f\) gives the annual world oil production in billions of barrels from 1980 to 2050 according to the U.S. Department of Energy projection. The function \(g\) gives the world oil production in billions of barrels per year over the same period according to longtime petroleum geologist Colin Campbell. Find an expression in terms of definite integrals involving \(f\) and \(g\) giving the shortfall in the total oil production over the period in question heeding Campbell's dire warnings.

Prove the identity. \(\tanh (-x)=-\tanh x\)

Find the center of mass of the system comprising masses \(m_{k}\) located at the points \(P_{k}\) in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. $$ \begin{array}{l} m_{1}=4, \quad m_{2}=1, \quad m_{3}=2, \quad m_{4}=5 ; \quad P_{1}(-2,3), \\ P_{2}(-1,4), \quad P_{3}(1,4), \quad P_{4}(4,-3) \end{array} $$

A particle moves along the \(x\) -axis from \(x=1\) to \(x=3 .\) As it moves, it is acted upon by a force \(F(x)=-3 x^{2}+x\). If \(x\) is measured in meters and \(F(x)\) is measured in newtons, find the work done by the force.

Find the arc length of the graph of the given equation from \(P\) to \(Q\) or on the specified interval. $$ y=\frac{2}{3} x^{3 / 2}-1 ; \quad P\left(4, \frac{13}{3}\right), Q(9,17) $$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(y=x^{3}, \quad y=0, \quad x=1 ; \quad\) the \(y\) -axis

Prove the identity. \(\operatorname{sech}^{2} x+\tanh ^{2} x=1\)

Find the centroid of the region bounded by the graphs of the given equations. $$ y=-\frac{2}{3} x+2, \quad y=0, \quad x=0 $$

When a particle is at the point \(x\) on the \(x\) -axis, it is acted upon by a force of \(\sin \pi x\) newtons. Find the work done by the force in moving the particle from \(x=1\) to \(x=2(x\) is measured in meters).

Find the arc length of the graph of the given equation from \(P\) to \(Q\) or on the specified interval. $$ y=2(x-1)^{3 / 2} ; \quad P(1,0), Q(5,16) $$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(y=-x^{2}+2 x, \quad y=0 ; \quad\) the \(y\) -axis

In Exercises \(9-40\), sketch the region bounded by the graphs of the given equations and find the area of that region. $$ y=x^{2}+3, \quad y=x+1, \quad x=-1, \quad x=1 $$

Prove the identity. \(\sinh ^{2} x=\frac{\cosh 2 x-1}{2}\)

Find the centroid of the region bounded by the graphs of the given equations. $$ y=x^{2}, \quad y=0, \quad x=1, \quad x=2 $$

A force of \(8 \mathrm{lb}\) is required to stretch a spring 2 in. beyond its natural length. Find the work required to stretch the spring 3 in. beyond its natural length.

Find the arc length of the graph of the given equation from \(P\) to \(Q\) or on the specified interval. $$ x=\frac{1}{4} y^{4}+\frac{1}{8 v^{2}} ; \quad P\left(\frac{3}{8}, 1\right), Q\left(\frac{129}{32}, 2\right) $$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(y=\sqrt{x-1}, \quad y=0, \quad x=5 ;\) the \(y\) -axis

In Exercises \(9-40\), sketch the region bounded by the graphs of the given equations and find the area of that region. $$ y=x^{3}+1, \quad y=x-1, \quad x=-1, \quad x=1 $$

Prove the identity. \(\cosh ^{2} x=\frac{1+\cosh 2 x}{2}\)

Find the centroid of the region bounded by the graphs of the given equations. $$ y=4-x^{2}, \quad y=0 $$