Chapter 6

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

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 a grapher, you may want to graph these curves to see what they look like. $$y=x^{3 / 2} \quad from \quad x=0 to x=4$$

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

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

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 \(y=2-x^{2}\)

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

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 \(y=\sqrt{1-x^{2}}\)

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

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?

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 \(y=\sqrt{1-x^{2}}\)

In Exercises \(1-8 :\) $$\begin{array}{l}{\text { a. Set up an integral for the area of the surface generated by }} \\ {\text { revolving the given curve about the indicated axis. }} \\ {\text { b. Graph the curve to see what it looks like. If you can, graph }} \\ {\text { the surface too. }} \\ {\text { c. Use your utility's integral evaluator to find the surface's area }} \\ {\text { numerically. }}\end{array}$$ $$ x^{1 / 2}+y^{1 / 2}=3 \quad \text { from }(4,1) \text { to }(1,4) ; \quad x\text-axis $$

Subway car springs It takes a force of \(21,714\) lb 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 }} \\\ {\text { first half inch? the second half inch? Answer to the nearest }} \\\ {\quad \text { in.-lb. }}\end{array} \end{equation}

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 a grapher, you may want to graph these curves to see what they look like. $$x=\left(y^{3} / 6\right)+1 /(2 y) from y=2 to y=3$$

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

Bathroom scale A bathroom scale is compressed 1\(/ 16\) in. when a \(150-\) lb 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 lengths of the curves in Exercises \(1-12 .\) If you have a grapher, you may want to graph these curves to see what they look like. $$y=(3 / 4) x^{4 / 3}-(3 / 8) x^{2 / 3}+5, \quad 1 \leq x \leq 8$$

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

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 \(\mathrm{N} / \mathrm{m}\) ?

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 \begin{equation} \begin{array}{l}{\text { a. rectangles of height } 10 \text { . }} \\ {\text { b. rectangles of perimeter } 20 \text { . }}\end{array} \end{equation}

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

In Exercises \(1-8 :\) $$\begin{array}{l}{\text { a. Set up an integral for the area of the surface generated by }} \\ {\text { revolving the given curve about the indicated axis. }} \\ {\text { b. Graph the curve to see what it looks like. If you can, graph }} \\ {\text { the surface too. }} \\ {\text { c. Use your utility's integral evaluator to find the surface's area }} \\ {\text { numerically. }}\end{array}$$ $$ y=\int_{1}^{x} \sqrt{t^{2}-1} d t, \quad 1 \leq x \leq \sqrt{5} ; \quad x\text-axis $$

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 \begin{equation} \begin{array}{l}{\text { a. isosceles triangles of height } 6 \text { . }} \\\ {\text { b. semicircles with diameters running across the base of the solid. }}\end{array} \end{equation}

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

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.

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?

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 lateral surface area of the cone generated by revolvingthe 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.

Force of attraction When a particle of mass \(m\) is at \((x, 0),\) it is attracted toward the origin with a force whose magnitude is \(k / x^{2}\) . If the particle starts from rest at \(x=b\) and is acted on by no other forces, find the work done on it by the time it reaches \(x=a\) , \(0< a < b .\)

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 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.

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 \(y\) -axis. Check your result with the geometry formula Frustum surface area \(=\pi\left(r_{1}+r_{2}\right) \times\) slant height.

Find the volume of the given pyramid, which has a square base of area 9 and height \(5 .\)

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$y=x^{2},-1 \leq x \leq 2$$

A twisted solid A square of side length \(s\) lies in a plane perpendicular to a line \(L .\) One vertex of the square lies on \(L .\) As this square moves a distance \(h\) along \(L\) , the square turns one revolution about \(L\) to generate a corkscrew like column with square cross-sections. \begin{equation} \begin{array}{l}{\text { a. Find the volume of the column. }} \\ {\text { b. What will the volume be if the square turns twice instead of }} \\ {\text { once? Give reasons for your answer. }}\end{array} \end{equation}

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$y=\tan x, \quad-\pi / 3 \leq x \leq 0$$

Find the areas of the surfaces generated by revolving the curves in Exercises \(13-23\) about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$y=\sqrt{x}, \quad 3 / 4 \leq x \leq 15 / 4 ; \quad x -axis$$

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$x=\sin y, \quad 0 \leq y \leq \pi$$

Find the areas of the surfaces generated by revolving the curves in Exercises \(13-23\) about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$y=\sqrt{2 x-x^{2}}, \quad 0.5 \leq x \leq 1.5 ; \quad x -axis$$

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$x=\sqrt{1-y^{2}}, \quad-1 / 2 \leq y \leq 1 / 2$$

Find the areas of the surfaces generated by revolving the curves in Exercises \(13-23\) about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$y=\sqrt{x+1}, \quad 1 \leq x \leq 5 ; \quad x -axis$$

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$y^{2}+2 y=2 x+1 \quad from \quad(-1,-1) to (7,3)$$

Emptying a tank A vertical right-circular cylindrical tank measures 30 ft high and 20 ft in diameter. It is full of kerosene weighing 51.2 \(\mathrm{lb} / \mathrm{ft}^{3} .\) How much work does it take to pump the kerosene to the level of the top of the tank?

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$y=\sin x-x \cos x, \quad 0 \leq x \leq \pi$$

Find the areas of the surfaces generated by revolving the curves in Exercises \(13-23\) about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$x=(1 / 3) y^{3 / 2}-y^{1 / 2}, \quad 1 \leq y \leq 3 ; \quad y -axis$$

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$y=\int_{0}^{x} \tan t d t, \quad 0 \leq x \leq \pi / 6$$

Find the areas of the surfaces generated by revolving the curves in Exercises \(13-23\) about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$x=2 \sqrt{4-y}, \quad 0 \leq y \leq 15 / 4 ; \quad y -axis$$