Chapter 7

Pumping Gasoline In Exercises 23 and \(24,\) find the work done in pumping gasoline that weighs 42 pounds per cubic foot. (Hint: Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.) A cylindrical gasoline tank 3 feet in diameter and 4 feet long is carried on the back of a truck and is used to fuel tractors. The axis of the tank is horizontal. The opening on the tractor tank is 5 feet above the top of the tank in the truck. Find the work done in pumping the entire contents of the fuel tank into the tractor.

Finding Arc Length In Exercises \(17-26,\) (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ x=e^{-y}, \quad 0 \leq y \leq 2 $$

Finding the Volume of a Solid In Exercises \(23-26,\) use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=2 x-x^{2}, \quad y=0, \quad \text { about the line } x=4 $$

A circular plate of radius \(r\) feet is submerged vertically in a tank of fluid that weighs \(w\) pounds per cubic foot. The center of the circle is \(k\) feet below the surface of the fluid, where \(k>r .\) Show that the fluid force on the surface of the plate is \(F=w k\left(\pi r^{2}\right)\) (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.)

Finding the Volume of a Solid In Exercises \(23-30\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\frac{1}{\sqrt{x+1}}, \quad y=0, \quad x=0, \quad x=4 $$

Finding the Area of a Region In Exercises \(17-30,\) sketch the region bounded by the graphs of the equations and find the area of the region. $$ f(x)=\sqrt{x}+3, \quad g(x)=\frac{1}{2} x+3 $$

Pumping Gasoline In Exercises 23 and \(24,\) find the work done in pumping gasoline that weighs 42 pounds per cubic foot. (Hint: Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.) The top of a cylindrical gasoline storage tank at a service station is 4 feet below ground level. The axis of the tank is horizontal and its diameter and length are 5 feet and 12 feet, respectively. Find the work done in pumping the entire contents of the full tank to a height of 3 feet above ground level.

Finding Arc Length In Exercises \(17-26,\) (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ y=\ln x, \quad 1 \leq x \leq 5 $$

Finding the Volume of a Solid In Exercises \(23-26,\) use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=\sqrt{x}, \quad y=0, \quad x=4, \quad \text { about the line } x=6 $$

Finding the Volume of a Solid In Exercises \(23-30\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=x \sqrt{4-x^{2}}, \quad y=0 $$

Finding the Area of a Region In Exercises \(17-30,\) sketch the region bounded by the graphs of the equations and find the area of the region. $$ f(x)=\sqrt[3]{x-1}, \quad g(x)=x-1 $$

Lifting a Chain In Exercises \(25-28\) , consider a 20 -foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. Wind up the entire chain.

Finding Arc Length In Exercises \(17-26,\) (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ y=2 \arctan x, \quad 0 \leq x \leq 1 $$

Finding the Volume of a Solid In Exercises \(23-26,\) use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=x^{2}, \quad y=4 x-x^{2}, \quad \text { about the line } x=4 $$

A rectangular plate of height \(h\) feet and base \(b\) feet is submerged vertically in a tank of fluid that weighs \(w\) pounds per cubic foot. The center is \(k\) feet below the surface of the fluid, where \(k>h / 2\). Show that the fluid force on the surface of the plate is \(F=w k h b\).

Finding the Volume of a Solid In Exercises \(23-30\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\frac{1}{x}, \quad y=0, \quad x=1, \quad x=3 $$

Lifting a Chain In Exercises \(25-28\) , consider a 20 -foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. Wind up one- third of the chain.

Finding Arc Length In Exercises \(17-26,\) (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ x=\sqrt{36-y^{2}}, \quad 0 \leq y \leq 3 $$

Finding the Volume of a Solid In Exercises \(23-26,\) use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=\frac{1}{3} x^{3}, \quad y=6 x-x^{2}, \quad \text { about the line } x=3 $$

Finding the Volume of a Solid In Exercises \(23-30\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\frac{2}{x+1}, \quad y=0, \quad x=0, \quad x=6 $$

Finding the Area of a Region In Exercises \(17-30,\) sketch the region bounded by the graphs of the equations and find the area of the region. $$ f(y)=y(2-y), g(y)=-y $$

Lifting a Chain In Exercises \(25-28\) , consider a 20 -foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. Run the winch until the bottom of the chain is at the 10 -foot level.

Approximation In Exercises 27 and 28 , determine which value best approximates the length of the are represented by the integral. (Make your selection on the basis of a sketch of the arc, not by performing any calculations.) $$ \begin{array}{l}{\int_{0}^{2} \sqrt{1+\left[\frac{d}{d x}\left(\frac{5}{x^{2}+1}\right)\right]^{2}} d x} \\\ {\begin{array}{llll}{\text { (a) } 25} & {\text { (b) } 5} & {\text { (c) } 2} & {\text { (d) }-4} & {\text { (e) } 3}\end{array}}\end{array} $$

A square porthole on a vertical side of a submarine (submerged in seawater) has an area of 1 square foot. Find the fluid force on the porthole, assuming that the center of the square is 15 feet below the surface.

Use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. \(y=10 x \sqrt{125-x^{3}}, y=0\)

Finding the Volume of a Solid In Exercises \(23-30\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=e^{-x}, \quad y=0, \quad x=0, \quad y=1 $$

Finding the Area of a Region In Exercises \(17-30,\) sketch the region bounded by the graphs of the equations and find the area of the region. $$ f(y)=y^{2}+1, g(y)=0, \quad y=-1, \quad y=2 $$

Lifting a Chain In Exercises \(25-28\) , consider a 20 -foot chain that weighs 3 pounds per foot hanging from a winch 20 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.

Approximation In Exercises 27 and 28 , determine which value best approximates the length of the are represented by the integral. (Make your selection on the basis of a sketch of the arc, not by performing any calculations.) $$ \begin{array}{l}{\int_{0}^{\pi / 4} \sqrt{1+\left[\frac{d}{d x}(\tan x)\right]^{2}} d x} \\ {\begin{array}{llll}{\text { (a) } 3} & {\text { (b) }-2} & {\text { (c) } 4} & {\text { (d) } \frac{4 \pi}{3}} & {\text { (e) } 1}\end{array}}\end{array} $$

Use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. \(y=x e^{-x / 2}, y=0, x=0, x=4\)

Finding the Volume of a Solid In Exercises \(23-30\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=e^{x / 4}, \quad y=0, \quad x=0, \quad x=6 $$

Finding the Area of a Region In Exercises \(17-30,\) sketch the region bounded by the graphs of the equations and find the area of the region. $$ f(y)=\frac{y}{\sqrt{16-y^{2}}}, \quad g(y)=0, \quad y=3 $$

Approximation In Exercises 29 and \(30,\) 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 are. (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. $$ f(x)=x^{3} $$

Choosing a Method In Exercises \(29-32,\) use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. $$ \begin{array}{ll}{y=x^{3}, \quad y=0,} \quad {x=2} \\\ {\begin{array}{llll}{\text { (a) the } x \text { -axis }} & {\text { (b) the } y \text { -axis }} & {\text { (c) the line } x=4}\end{array}}\end{array} $$

Use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. \(y=5 \sqrt[3]{400-x^{2}}, y=0\)

Finding the Volume of a Solid In Exercises \(23-30\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=x^{2}+1, y=-x^{2}+2 x+5, \quad x=0, \quad x=3 $$

Approximation In Exercises 29 and \(30,\) 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 are. (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. $$ f(x)=\left(x^{2}-4\right)^{2} $$

Choosing a Method In Exercises \(29-32,\) use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. $$ \begin{array}{l}{y=\frac{10}{x^{2}}, \quad y=0, \quad x=1, \quad x=5} \\\ {\begin{array}{llll}{\text { (a) the } x \text { -axis }} & {\text { (b) the } y \text { -axis }} & {\text { (c) the line } y=10}\end{array}}\end{array} $$

Use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. \(y=\frac{8}{x^{2}+4}, y=0, x=-2, x=2\)

Finding the Volume of a Solid In Exercises \(23-30\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\sqrt{x}, \quad y=-\frac{1}{2} x+4, \quad x=0, \quad x=8 $$

Finding the Area of a Region In Exercises \(17-30,\) sketch the region bounded by the graphs of the equations and find the area of the region. $$ g(x)=\frac{4}{2-x}, \quad y=4, \quad x=0 $$

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.

Work by a Constant Force State the definition of work done by a constant force.

Choosing a Method In Exercises \(29-32,\) use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. $$ \begin{array}{l}{x^{1 / 2}+y^{1 / 2}=a^{1 / 2}, \quad x=0, \quad y=0} \\\ {\begin{array}{llll}{\text { (a) the } x \text { -axis }} & {\text { (b) the } y \text { -axis }} & {\text { (c) the line } x=a}\end{array}}\end{array} $$

Finding the Volume of a Solid In Exercises 31 and 32 ,find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(y\) -axis. $$ y=3(2-x), \quad y=0, \quad x=0 $$

Finding the Area of a Region In Exercises \(31-36,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region analytically, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=x\left(x^{2}-3 x+3\right), \quad g(x)=x^{2} $$

Work by a Variable Force State the definition of work done by a variable force.

(a) Define fluid pressure. (b) Define fluid force against a submerged vertical plane region.

Choosing a Method In Exercises \(29-32,\) use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. $$ \begin{array}{l}{x^{2 / 3}+y^{2 / 3}=a^{2 / 3}, \quad a>0 \text { (hypocycloid) }} \\ {\begin{array}{ll}{\text { (a) the } x \text { -axis }} & {\text { (b) the } y \text { -axis }}\end{array}}\end{array} $$

Finding the Area of a Region In Exercises \(31-36,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region analytically, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=x^{4}-2 x^{2}, \quad y=2 x^{2} $$