Chapter 7

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. $$ \int_{0}^{4}(2 \sqrt{y}-y) d y $$

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. $$ \begin{array}{ll}{y=2 x^{2}, y=0,} & {x=2} \\ {\begin{array}{ll}{\text { (a) the } y \text { -axis }} & {\text { (b) the } x \text { -axis }} \\ {\text { (c) the line } y=8} & {\text { (d) the line } x=2}\end{array}}\end{array} $$

Propulsion Neglecting air resistance and the weight of the propellant, determine the work done in propelling a 10 -ton satellite to a height of (a) \(11,000\) miles above Earth and (b) \(22,000\) miles above Earth.

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\frac{1}{2}\left(e^{x}+e^{-x}\right), \quad[0,2] $$

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 $$ y=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad y=0, \quad x=0, \quad x=1 $$

Think About It In Exercises 13 and \(14,\) determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g .\) (Make your selection on the basis of a sketch of the region and not by performing any calculations.) $$ \begin{array}{l}{f(x)=x+1, \quad g(x)=(x-1)^{2}} \\\ {\begin{array}{llll}{\text { (a) }-2} & {\text { (b) } 2} & {\text { (c) } 10} & {\text { (d) } 4} & {\text { (e) } 8}\end{array}}\end{array} $$

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. $$ \begin{array}{l}{y=x^{2}, y=4 x-x^{2}} \\ {\begin{array}{ll}{\text { (a) the } x \text { -axis }} & {\text { (b) the line } y=6}\end{array}}\end{array} $$

Propulsion \(A\) lunar module weighs 12 tons on the surface of Earth. How much work is done in propelling the module from the surface of the moon to a height of 50 miles? Consider the radius of the moon to be 1100 miles and its force of gravity to be one-sixth that of Earth.

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

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 $$ y=\left\\{\begin{array}{ll}{\frac{\sin x}{x},} & {x>0} \\ {1,} & {x=0}\end{array}, y=0, \quad x=0, \quad x=\pi\right. $$

Find \(M_{x}, M_{y},\) and \((\overline{x}, \overline{y})\) for the laminas of uniform density \(\boldsymbol{\rho}\) bounded by the graphs of the equations. \(y=6-x, y=0, x=0\)

Think About It In Exercises 13 and \(14,\) determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g .\) (Make your selection on the basis of a sketch of the region and not by performing any calculations.) $$ \begin{array}{l}{f(x)=2-\frac{1}{2} x, \quad g(x)=2-\sqrt{x}} \\\ {\begin{array}{llll}{\text { (a) } 1} & {\text { (b) } 6} & {\text { (c) }-3} & {\text { (d) } 3} & {\text { (e) } 4}\end{array}}\end{array} $$

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. $$ \begin{array}{l}{y=4+2 x-x^{2}, \quad y=4-x} \\ {\text { (a) the } x \text { -axis }} & {\text { (b) the line } y=1}\end{array} $$

Pumping Water A rectangular tank with a base 4 feet by 5 feet and a height of 4 feet is full of water (see figure). The water weighs 62.4 pounds per cubic foot. How much work is done in pumping water out over the top edge in order to empty (a) half of the tank and (b) all of the tank?

Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ x=\frac{1}{3}\left(y^{2}+2\right)^{3 / 2}, \quad 0 \leq y \leq 4 $$

Find \(M_{x}, M_{y},\) and \((\overline{x}, \overline{y})\) for the laminas of uniform density \(\boldsymbol{\rho}\) bounded by the graphs of the equations. \(y=\sqrt{x}, y=0, x=4\)

Comparing Methods In Exercises 15 and \(16,\) find the area of the region by integrating (a) with respect to \(x\) and (b) with respect to \(y .\) (c) Compare your results. Which method is simpler? In general, will this method always be simpler than the other one? Why or why not? $$ \begin{array}{l}{x=4-y^{2}} \\ {x=y-2}\end{array} $$

Finding the Volume of a Solid In Exercises \(15-18\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(y=4\) . $$ y=x, \quad y=3, \quad x=0 $$

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

Finding the Volume of a Solid In Exercises \(15-22,\) 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 \(x\) -axis. Graph cannot copy $$ y=1-x $$

Comparing Methods In Exercises 15 and \(16,\) find the area of the region by integrating (a) with respect to \(x\) and (b) with respect to \(y .\) (c) Compare your results. Which method is simpler? In general, will this method always be simpler than the other one? Why or why not? $$ \begin{array}{l}{y=x^{2}} \\ {y=6-x}\end{array} $$

Finding the Volume of a Solid In Exercises \(15-18\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(y=4\) . $$ y=\frac{1}{2} x^{3}, \quad y=4, \quad x=0 $$

Pumping Water \(A\) cylindrical water tank 4 meters high with a radius of 2 meters is buried so that the top of the tank is 1 meter below ground level (see figure). How much work is done in pumping a full tank of water up to ground level? (The water weighs 9800 newtons per cubic meter.)

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. $$ y=4-x^{2}, \quad 0 \leq x \leq 2 $$

Finding the Volume of a Solid In Exercises \(15-22,\) 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 \(x\) -axis. Graph cannot copy $$ y=\frac{1}{x} $$

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. $$ y=x^{2}-1, \quad y=-x+2, \quad x=0, \quad x=1 $$

Finding the Volume of a Solid In Exercises \(15-18\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(y=4\) . $$ y=\frac{3}{1+x}, \quad y=0, \quad x=0, \quad x=3 $$

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. $$ y=x^{2}+x-2, \quad-2 \leq x \leq 1 $$

Find \(M_{x}, M_{y},\) and \((\overline{x}, \overline{y})\) for the laminas of uniform density \(\boldsymbol{\rho}\) bounded by the graphs of the equations. \(y=\sqrt{x}, y=\frac{1}{2} x\)

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. $$ y=-x^{3}+2, \quad y=x-3, \quad x=-1, \quad x=1 $$

Finding the Volume of a Solid In Exercises \(15-18\) , find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(y=4\) . $$ y=\sec x, \quad y=0, \quad 0 \leq x \leq \frac{\pi}{3} $$

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. $$ y=\frac{1}{x}, \quad 1 \leq x \leq 3 $$

Finding the Volume of a Solid In Exercises \(15-22,\) 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 \(x\) -axis. Graph cannot copy $$ y=x^{3}, \quad x=0, \quad y=8 $$

Finding the Volume of a Solid In Exercises \(19-22,\) find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(x=5\) . $$ y=x, \quad y=0, \quad y=4, \quad x=5 $$

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

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. $$ y=\frac{1}{x+1}, \quad 0 \leq x \leq 1 $$

Finding the Volume of a Solid In Exercises \(15-22,\) 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 \(x\) -axis. Graph cannot copy $$ y=4 x^{2}, \quad x=0, \quad y=4 $$

Finding the Volume of a Solid In Exercises \(19-22,\) find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(x=5\) . $$ y=3-x, \quad y=0, \quad y=2, \quad x=0 $$

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. $$ y=-x^{2}+3 x+1, \quad y=-x+1 $$

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. $$ y=\sin x, \quad 0 \leq x \leq \pi $$

Finding the Volume of a Solid In Exercises \(15-22,\) 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 \(x\) -axis. Graph cannot copy $$ x+y=4, \quad y=x, \quad y=0 $$

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank when the tank is half full, where the diameter is 3 feet and the gasoline weighs 42 pounds per cubic foot.

Find \(M_{x}, M_{y},\) and \((\overline{x}, \overline{y})\) for the laminas of uniform density \(\boldsymbol{\rho}\) bounded by the graphs of the equations. \(y=x^{2 / 3}, y=0, x=8\)

Finding the Volume of a Solid In Exercises \(19-22,\) find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(x=5\) . $$ x=y^{2}, \quad x=4 $$

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. $$ y=x, \quad y=2-x, \quad y=0 $$

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. $$ y=\cos x, \quad-\frac{\pi}{2} \leq x \leq \frac{\pi}{2} $$

Finding the Volume of a Solid In Exercises \(15-22,\) 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 \(x\) -axis. Graph cannot copy $$ y=\sqrt{x+2}, \quad y=x, \quad y=0 $$

Find \(M_{x}, M_{y},\) and \((\overline{x}, \overline{y})\) for the laminas of uniform density \(\boldsymbol{\rho}\) bounded by the graphs of the equations. \(y=x^{2 / 3}, y=4\)

Finding the Volume of a Solid In Exercises \(19-22,\) find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(x=5\) . $$ x y=3, \quad y=1, \quad y=4, \quad x=5 $$

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. $$ y=\frac{4}{x^{3}}, \quad y=0, \quad x=1, \quad x=4 $$