Chapter 7

Constant Force In Exercises \(1-4,\) determine the work done by the constant force. A 1200 -pound steel beam is lifted 40 feet.

Finding Distance Using Two Methods In Exercises 1 and \(2,\) find the distance between the points using (a) the Distance Formula and (b) integration. $$ (0,0), \quad(8,15) $$

Find the center of mass of the point masses lying on the \(x\) -axis. \(m_{1}=7, m_{2}=3, m_{3}=5\) \(x_{1}=-5, x_{2}=0, x_{3}=3\)

Writing a Definite Integral In Exercises \(1-6,\) set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=x^{2}-6 x} \\ {y_{2}=0}\end{array} $$

Constant Force In Exercises \(1-4,\) determine the work done by the constant force. An electric hoist lifts a 2500 -pound car 6 feet.

Finding Distance Using Two Methods In Exercises 1 and \(2,\) find the distance between the points using (a) the Distance Formula and (b) integration. $$ (1,2), \quad(7,10) $$

Find the center of mass of the point masses lying on the \(x\) -axis. \(m_{1}=7, m_{2}=4, m_{3}=3, m_{4}=8\) \(x_{1}=-3, x_{2}=-2, x_{3}=5, x_{4}=4\)

Writing a Definite Integral In Exercises \(1-6,\) set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=x^{2}+2 x+1} \\ {y_{2}=2 x+5}\end{array} $$

Finding the Volume of a Solid In Exercises \(1-6,\) set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the \(x\) -axis. $$ y=4-x^{2} $$

Constant Force In Exercises \(1-4,\) determine the work done by the constant force. A force of 112 newtons is required to slide a cement block 8 meters in a construction project.

Find the center of mass of the point masses lying on the \(x\) -axis. \(m_{1}=1, m_{2}=3, m_{3}=2, m_{4}=9, m_{5}=5\) \(x_{1}=6, x_{2}=10, x_{3}=3, x_{4}=2, x_{5}=4\)

Writing a Definite Integral In Exercises \(1-6,\) set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=x^{2}-4 x+3} \\ {y_{2}=-x^{2}+2 x+3}\end{array} $$

Constant Force In Exercises \(1-4,\) determine the work done by the constant force. The locomotive of a freight train pulls its cars with a constant force of 9 tons a distance of one-half mile.

Finding the Volume of a Solid In Exercises \(1-14,\) use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. Graph cannot copy $$ y=\frac{1}{2} x^{2}+1 $$

Find the center of mass of the point masses lying on the \(x\) -axis. \(m_{1}=8, m_{2}=5, m_{3}=5, m_{4}=12, m_{5}=2\) \(x_{1}=-2, x_{2}=6, x_{3}=0, x_{4}=3, x_{5}=-5\)

Writing a Definite Integral In Exercises \(1-6,\) set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=x^{2}} \\ {y_{2}=x^{3}}\end{array} $$

Finding the Volume of a Solid In Exercises \(1-14,\) use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. Graph cannot copy $$ y=\frac{1}{4} x^{2}, \quad y=0, \quad x=4 $$

Writing a Definite Integral In Exercises \(1-6,\) set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=3\left(x^{3}-x\right)} \\ {y_{2}=0}\end{array} $$

Finding the Volume of a Solid In Exercises \(1-14,\) use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. Graph cannot copy $$ y=\frac{1}{2} x^{3}, \quad y=0, \quad x=3 $$

Finding the Volume of a Solid In Exercises \(1-6,\) set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the \(x\) -axis. $$ y=2, \quad y=4-\frac{x^{2}}{4} $$

Writing a Definite Integral In Exercises \(1-6,\) set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=(x-1)^{3}} \\ {y_{2}=x-1}\end{array} $$

Hooke's Law In Exercises \(5-10\) , use Hooke's Law to determine the variable force in the spring problem. A force of 20 pounds stretches a spring 9 inches in an exercise machine. Find the work done in stretching the spring 1 foot from its natural position.

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\frac{3}{2} x^{2 / 3}, \quad[1,8] $$

Finding the Volume of a Solid In Exercises \(1-14,\) use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. Graph cannot copy $$ y=x^{2}, \quad y=4 x-x^{2} $$

Consider a beam of length \(L\) with a fulcrum \(x\) feet from one end (see figure). There are objects with weights \(W_{1}\) and \(W_{2}\) placed on opposite ends of the beam. Find \(x\) such that the system is in equilibrium. Two children weighing 48 pounds and 72 pounds are going to play on a seesaw that is 10 feet long.

Finding a Region In Exercises \(7-12\) , the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{x}{2}\right] d x $$

Hooke's Law In Exercises \(5-10\) , use Hooke's Law to determine the variable force in the spring problem. An overhead garage door has two springs, one on each side of the door. A force of 15 pounds is required to stretch each spring 1 foot. Because of the pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the work done by the pair of springs.

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\frac{x^{4}}{8}+\frac{1}{4 x^{2}}, \quad[1,3] $$

Finding the Volume of a Solid In Exercises \(1-14,\) use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. Graph cannot copy $$ y=9-x^{2}, \quad y=0 $$

Consider a beam of length \(L\) with a fulcrum \(x\) feet from one end (see figure). There are objects with weights \(W_{1}\) and \(W_{2}\) placed on opposite ends of the beam. Find \(x\) such that the system is in equilibrium. In order to move a 600 -pound rock, a person weighing 200 pounds wants to balance it on a beam that is 5 feet long.

Finding a Region In Exercises \(7-12\) , the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-1}^{1}\left[\left(2-x^{2}\right)-x^{2}\right] d x $$

Hooke's Law In Exercises \(5-10\) , use Hooke's Law to determine the variable force in the spring problem. Eighteen foot-pounds of work is required to stretch a spring 4 inches from its natural length. Find the work required to stretch the spring an additional 3 inches.

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\frac{x^{5}}{10}+\frac{1}{6 x^{3}}, \quad[2,5] $$

Find the center of mass of the given system of point masses. \(\begin{array}{|c|c|c|c|}\hline m_{i} & {5} & {1} & {3} \\ \hline\left(x_{i}, y_{i}\right) & {(2,2)} & {(-3,1)} & {(1,-4)} \\ \hline\end{array}\)

Finding a Region In Exercises \(7-12\) , the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x $$

Hooke's Law In Exercises \(5-10\) , use Hooke's Law to determine the variable force in the spring problem. Seven and one-half foot-pounds of work is required to compress a spring 2 inches from its natural length. Find the work required to compress the spring an additional one-half inch.

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\frac{3}{2} x^{2 / 3}+4, \quad[1,27] $$

Finding the Volume of a Solid In Exercises \(1-14,\) use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. Graph cannot copy $$ y=x^{3 / 2}, \quad y=8, \quad x=0 $$

Find the center of mass of the given system of point masses. \(\begin{array}{|c|c|c|c|}\hline m_{i} & {10} & {2} & {5} \\\ \hline\left(x_{i}, y_{i}\right) & {(1,-1)} & {(5,5)} & {(-4,0)} \\\ \hline\end{array}\)

Finding a Region In Exercises \(7-12\) , the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(\sec ^{2} x-\cos x\right) d x $$

Finding the Volume of a Solid In Exercises \(7-10,\) set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the \(y\) -axis. $$ x=-y^{2}+4 y $$

Propulsion Neglecting air resistance and the weight of the propellant, determine the work done in propelling a five-ton satellite to a height of (a) 100 miles above Earth and (b) 300 miles above Earth.

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\ln (\sin x), \quad\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right] $$

Finding the Volume of a Solid In Exercises \(1-14,\) use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. Graph cannot copy $$ y=\sqrt{x-2}, \quad y=0, \quad x=4 $$

Find the center of mass of the given system of point masses. \(\begin{array}{|c|c|c|c|c|}\hline m_{i} & {12} & {6} & {4.5} & {15} \\\ \hline\left(x_{i}, y_{i}\right) & {(2,3)} & {(-1,5)} & {(6,8)} & {(2,-2)} \\\ \hline\end{array}\)

Finding a Region In Exercises \(7-12\) , the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-2}^{1}\left[(2-y)-y^{2}\right] d y $$

Finding the Volume of a Solid In Exercises \(11-14\) , find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. $$ \begin{array}{l}{y=\sqrt{x}, y=0, \quad x=3} \\ {\begin{array}{ll}{\text { (a) the } x \text { -axis }} & {\text { (b) the } y \text { -axis }} \\\ {\text { (c) the line } x=3} & {\text { (d) the line } x=6}\end{array}}\end{array} $$

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\ln (\cos x), \quad\left[0, \frac{\pi}{3}\right] $$

Finding the Volume of a Solid In Exercises \(1-14,\) use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. Graph cannot copy $$ y=-x^{2}+1, \quad y=0 $$

Find the center of mass of the given system of point masses. \(\begin{array}{|c|c|c|c|c|c|}\hline m_{i} & {3} & {4} & {2} & {1} & {6} \\\ \hline\left(x_{i}, y_{i}\right) & {(-2,-3)} & {(5,5)} & {(7,1)} & {(0,0)} & {(-3,0)} \\ \hline\end{array}\)