Chapter 6

In Exercises \(1-6,\) find the mass \(M\) and center of mass \(\overline{x}\) of the linear wire covering the given interval and having the given density \(\delta(x)\) . $$1 \leq x \leq 4, \quad \delta(x)=\sqrt{x}$$

Find the volumes of the solids. The solid lies between planes perpendicular to the \(x\) -axis at \(x=0\) and \(x=4 .\) The cross-sections perpendicular to the axis on the interval \(0 \leq x \leq 4\) are squares whose diagonals run from the parabola \(y=-\sqrt{x}\) to the parabola \(y=\sqrt{x}\) .

Find the lengths of the curves in Exercises \(1-12 .\) If you have graphing software, you may want to graph these curves to see what they look like. $$y=(1 / 3)\left(x^{2}+2\right)^{3 / 2} \quad \text { from } \quad x=0 \text { to } x=3$$

In Exercises \(1-8\) a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $$ y=\tan x, \quad 0 \leq x \leq \pi / 4 ; \quad x- axis $$

In Exercises \(1 - 8 :\) a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $$y = x ^ { 2 } , \quad 0 \leq x \leq 2 ; \quad x -axis$$

In Exercises \(1-6,\) find the mass \(M\) and center of mass \(\overline{x}\) of the linear wire covering the given interval and having the given density \(\delta(x)\) . $$-3 \leq x \leq 3, \quad \delta(x)=1+3 x^{2}$$

Find the volumes of the solids. The solid lies between planes perpendicular to the \(x\) -axis at \(x=-1\) and \(x=1 .\) The cross-sections perpendicular to the \(x\) -axis are circular disks whose diameters run from the parabola \(y=x^{2}\) to the parabola \(y=2-x^{2}.\)

Find the lengths of the curves in Exercises \(1-12 .\) If you have graphing software, you may want to graph these curves to see what they look like. $$y=x^{3 / 2} \text { from } x=0 \text { to } x=4$$

In Exercises \(1 - 8 :\) a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $$x y = 1 , \quad 1 \leq y \leq 2 ; \quad y -axis$$

In Exercises \(1-6,\) find the mass \(M\) and center of mass \(\overline{x}\) of the linear wire covering the given interval and having the given density \(\delta(x)\) . $$0 \leq x \leq 3, \quad \delta(x)=\frac{1}{x+1}$$

Find the volumes of the solids. The solid lies between planes perpendicular to the \(x\) -axis at \(x=-1\) and \(x=1 .\) The cross-sections perpendicular to the \(x\) -axis between these planes are squares whose bases run from the semicircle \(y=-\sqrt{1-x^{2}}\) to the semicircle \(y=\sqrt{1-x^{2}}.\)

Spring constant It took 1800 \(\mathrm{J}\) of work to stretch a spring from its natural length of 2 \(\mathrm{m}\) to a length of 5 \(\mathrm{m} .\) Find the spring's force constant.

In Exercises \(1 - 8 :\) a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $$x = \sin y , \quad 0 \leq y \leq \pi ; \quad y -axis$$

In Exercises \(1-6,\) find the mass \(M\) and center of mass \(\overline{x}\) of the linear wire covering the given interval and having the given density \(\delta(x)\) . $$1 \leq x \leq 2, \quad \delta(x)=\frac{8}{x^{3}}$$

Find the volumes of the solids. The solid lies between planes perpendicular to the \(x\) -axis at \(x=-1\) and \(x=1 .\) The cross-sections perpendicular to the \(x\) -axis between these planes are squares whose diagonals run from the semicircle \(y=-\sqrt{1-x^{2}}\) to the semicircle \(y=\sqrt{1-x^{2}}.\)

Find the lengths of the curves in Exercises \(1-12 .\) If you have graphing software, you may want to graph these curves to see what they look like. $$x=\left(y^{3 / 2} / 3\right)-y^{1 / 2} \text { from } y=1 \text { to } y=9$$

Stretching a spring A spring has a natural length of 10 in. An 800 -lb force stretches the spring to 14 in. \begin{equation} \begin{array}{l}{\text { a. Find the force constant. }} \\ {\text { b. How much work is done in stretching the spring from } 10 \text { in. }} \\ \quad {\text { to } 12 \text { in.? }} \\ {\text { c. How far beyond its natural length will a } 1600 \text { -lb force stretch }} \\ \quad {\text { the spring? }}\end{array} \end{equation}

In Exercises \(1-6,\) find the mass \(M\) and center of mass \(\overline{x}\) of the linear wire covering the given interval and having the given density \(\delta(x)\) . $$\delta(x)=\left\\{\begin{array}{ll}{4,} & {0 \leq x \leq 2} \\ {5,} & {2 < x \leq 3}\end{array}\right.$$

Find the volumes of the solids. The base of a solid is the region between the curve \(y=2 \sqrt{\sin x}\) and the interval \([0, \pi]\) on the \(x\) -axis. The cross-sections perpendicular to the \(x\) -axis are a. equilateral triangles with bases running from the \(x\) -axis to the curve as shown in the accompanying figure. b. squares with bases running from the \(x\) -axis to the curve.

Find the lengths of the curves in Exercises \(1-12 .\) If you have graphing software, you may want to graph these curves to see what they look like. $$x=\left(y^{4} / 4\right)+1 /\left(8 y^{2}\right) \quad \text { from } \quad y=1 \text { to } y=2$$

Stretching a rubber band A force of 2 \(\mathrm{N}\) will stretch a rubber band 2 \(\mathrm{cm}(0.02 \mathrm{m}) .\) Assuming that Hooke's Law applies, how far will a \(4 . \mathrm{N}\) force stretch the rubber band? How much work does it take to stretch the rubber band this far?

In Exercises \(1 - 8 :\) a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $$y + 2 \sqrt { y } = x , \quad 1 \leq y \leq 2 ; \quad y -axis$$

In Exercises \(1-6,\) find the mass \(M\) and center of mass \(\overline{x}\) of the linear wire covering the given interval and having the given density \(\delta(x)\) . $$\delta(x)=\left\\{\begin{array}{cc}{2-x,} & {0 \leq x<1} \\ {x,} & {1 \leq x \leq 2}\end{array}\right.$$

Find the volumes of the solids. The solid lies between planes perpendicular to the \(x\)-axis at \(x=-\pi / 3\) and \(x=\pi / 3 .\) The cross-sections perpendicular to the \(x\)-axis are a. circular disks with diameters running from the curve \(y=\tan x\) to the curve \(y=\sec x .\) b. squares whose bases run from the curve \(y=\tan x\) to the curve \(y=\sec x .\)

Find the lengths of the curves in Exercises \(1-12 .\) If you have graphing software, you may want to graph these curves to see what they look like. $$x=\left(y^{3} / 6\right)+1 /(2 y) \quad \text { from } \quad y=2 \text { to } y=3$$

Stretching a spring If a force of 90 \(\mathrm{N}\) stretches a spring 1 \(\mathrm{m}\) beyond its natural length, how much work does it take to stretch the spring 5 \(\mathrm{m}\) beyond its natural length?

In Exercises \(1 - 8 :\) a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $$x = \int _ { 0 } ^ { y } \tan t d t , \quad 0 \leq y \leq \pi / 3 ; \quad y -axis$$

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region bounded by the parabola \(y=x^{2}\) and the line \(y=4\)

Find the volumes of the solids. The base of a solid is the region bounded by the graphs of \(y=3 x\) , \(y=6,\) and \(x=0 .\) The cross-sections perpendicular to the \(x\)-axis are a. rectangles of height \(10 .\) b. rectangles of perimeter \(20 .\)

Subway car springs It takes a force of \(21,714\) ib to compress a coil spring assembly on a New York City Transit Authority subway car from its free height of 8 in. to its fully compressed height of 5 in. \begin{equation}\begin{array}{l}{\text { a. What is the assembly's force constant? }} \\ {\text { b. How much work does it take to compress the assembly the }} \\ \quad {\text { first half inch? the second half inch? Answer to the nearest }} \\ {\quad \text { in.-lb. }}\end{array}\end{equation}

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region bounded by the parabola \(y=25-x^{2}\) and the \(x\) -axis

Find the volumes of the solids. The base of a solid is the region bounded by the graphs of \(y=\sqrt{x}\) and \(y=x / 2 .\) The cross-sections perpendicular to the \(x\)-axis are a. isosceles triangles of height 6 . b. semicircles with diameters running across the base of the solid.

Find the lengths of the curves in Exercises \(1-12 .\) If you have graphing software, you may want to graph these curves to see what they look like. $$y=\left(x^{3} / 3\right)+x^{2}+x+1 /(4 x+4), \quad 0 \leq x \leq 2$$

Bathroom scale A bathroom scale is compressed 1\(/ 16\) in. when a \(150-1 b\) person stands on it. Assuming that the scale behaves like a spring that obeys Hooke's Law, how much does someone who compresses the scale 1\(/ 8\) in. weigh? How much work is done compressing the scale 1\(/ 8\) in.?

Find the lateral (side) surface area of the cone generated by revolving the line segment \(y = x / 2,0 \leq x \leq 4 ,\) about the \(x\) -axis. Check your answer with the geometry formula Lateral surface area \(= \frac { 1 } { 2 } \times\) base circumference \(\times\) slant height.

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region bounded by the parabola \(y=x-x^{2}\) and the line \(y=-x\)

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines in Exercises \(7-12\) about the \(y\) -axis. $$ y=x^{2}, \quad y=2-x, \quad x=0, \quad \text { for } x \geq 0 $$

Find the volumes of the solids. The solid lies between planes perpendicular to the \(y\) -axis at \(y=0\) and \(y=2 .\) The cross-sections perpendicular to the \(y\) -axis are cir- cular disks with diameters running from the \(y\) -axis to the parabola \(x=\sqrt{5} y^{2}.\)

Find the lengths of the curves in Exercises \(1-12 .\) If you have graphing software, you may want to graph these curves to see what they look like. $$y=\frac{x^{3}}{3}+\frac{1}{4 x}, \quad 1 \leq x \leq 3$$

Lifting a rope A mountain climber is about to haul up a \(50-\mathrm{m}\) length of hanging rope. How much work will it take if the rope weighs 0.624 \(\mathrm{N} / \mathrm{m} ?\)

Find the lateral surface area of the cone generated by revolving the line segment \(y = x / 2,0 \leq x \leq 4 ,\) about the \(y\) -axis. Check your answer with the geometry formula Lateral surface area \(= \frac { 1 } { 2 } \times\) base circumference \(\times\) slant height.

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region enclosed by the parabolas \(y=x^{2}-3\) and \(y=-2 x^{2}\)

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines in Exercises \(7-12\) about the \(y\) -axis. $$ y=2-x^{2}, \quad y=x^{2}, \quad x=0 $$

Find the volumes of the solids. The base of the solid is the disk \(x^{2}+y^{2} \leq 1 .\) The cross-sections by planes perpendicular to the \(y\) -axis between \(y=-1\) and \(y=1\) are isosceles right triangles with one leg in the disk.

Find the lengths of the curves in Exercises \(1-12 .\) If you have graphing software, you may want to graph these curves to see what they look like. $$y=\frac{x^{5}}{5}+\frac{1}{12 x^{3}}, \quad \frac{1}{2} \leq x \leq 1$$

Leaky sandbag A bag of sand originally weighing 144 lb was lifted at a constant rate. As it ruse, sand also leaked out at a constant rate. The sand was half gone by the time the bag had been lifted to 18 ft. How much work was done lifting the sand this far? (Neglect the weight of the bag and lifting equipment.)

Find the surface area of the cone frustum generated by revolving the line segment \(y = ( x / 2 ) + ( 1 / 2 ) , 1 \leq x \leq 3 ,\) about the \(x\) -axis. Check your result with the geometry formula Frustum surface area \(= \pi \left( r _ { 1 } + r _ { 2 } \right) \times\) slant height.

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region bounded by the \(y\) -axis and the curve \(x=y-y^{3}\) , \(0 \leq y \leq 1\)

Lifting an elevator cable An electric elevator with a motor at the top has a multistrand cable weighing 4.5 \(\mathrm{lb} / \mathrm{ft}\) . When the car is at the first floor, 180 \(\mathrm{ft}\) of cable are paid out, and effectively 0 \(\mathrm{ft}\) are out when the car is at the top floor. How much work does the motor do just lifting the cable when it takes the car from the first floor to the top?

Find the surface area of the cone frustum generated by revolving the line segment \(y = ( x / 2 ) + ( 1 / 2 ) , 1 \leq x \leq 3 ,\) about the \(x\) -axis. Check your result with the geometry formula Frustum surface area \(= \pi \left( r _ { 1 } + r _ { 2 } \right) \times\) slant height.