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 \(1-8 :\) 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 grapher's or computer's integral evaluator to find the surface's area numerically. \(y=\tan x, \quad 0 \leq x \leq \pi / 4 ; \quad x\) -axis

An \(80-\) lb child and a \(100-\) lb child are balancing on a seesaw. The \(80-\) lb child is 5 \(\mathrm{ft}\) from the fulcrum. How far from the fulcrum is the \(100-\) lb child?

Find the lengths of the curves in Exercises \(1-6\) $$ x=1-t, \quad y=2+3 t, \quad-2 / 3 \leq t \leq 1 $$

In Exercises 1 and \(2,\) find a formula for the area \(A(x)\) of the cross- sections of the solid perpendicular to the \(x\) -axis. The solid lies between planes perpendicular to the \(x\) -axis at \(x=-1\) and \(x=1 .\) In each case, the cross-sections perpendicular to the \(x\) -axis between these planes run from the semicircle \(y=-\sqrt{1-x^{2}}\) to the semicircle \(y=\sqrt{1-x^{2}}\) a. The cross-sections are circular disks with diameters in the \(x y\) -plane. b. The cross-sections are squares with bases in the \(x y\) -plane. c. The cross-sections are squares with diagonals in the \(x y\) -plane. (The length of a square's diagonal is \(\sqrt{2}\) times the length of its sides.) d. The cross-sections are equilateral triangles with bases in the \(x y\) -plane.

Stretching a spring A spring has a natural length of 10 in. An 800 -lb force stretches the spring to 14 in. a. Find the force constant. b. How much work is done in stretching the spring from 10 in. to 12 in.? c. How far beyond its natural length will a 1600 -lb force stretch the spring?

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 grapher's or computer's integral evaluator to find the surface's area numerically. \(y=x^{2}, \quad 0 \leq x \leq 2 ; \quad x\) -axis

Find the lengths of the curves in Exercises \(1-6\) $$ x=\cos t, \quad y=t+\sin t, \quad 0 \leq t \leq \pi $$

The ends of a log are placed on two scales. One scale reads 100 \(\mathrm{kg}\) and the other 200 \(\mathrm{kg}\) . Where is the log's center of mass?

In Exercises 1 and \(2,\) find a formula for the area \(A(x)\) of the cross- sections of the solid perpendicular to the \(x\) -axis. The solid lies between planes perpendicular to the \(x\) -axis at \(x=0\) and \(x=4 .\) The cross-sections perpendicular to the \(x\) -axis between these planes run from the parabola \(y=-\sqrt{x}\) to the parabola \(y=\sqrt{x}\) . a. The cross-sections are circular disks with diameters in the \(x y\) -plane. b. The cross-sections are squares with bases in the \(x y\) -plane. c. The cross-sections are squares with diagonals in the \(x y\) -plane. d. The cross-sections are equilateral triangles with bases in the \(x y\) -plane.

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 grapher's or computer's integral evaluator to find the surface's area numerically. \(x y=1, \quad 1 \leq y \leq 2 ; \quad y\) -axis

Find the lengths of the curves in Exercises \(1-6\) $$ x=t^{3}, \quad y=3 t^{2} / 2, \quad 0 \leq t \leq \sqrt{3} $$

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 grapher's or computer's integral evaluator to find the surface's area numerically. \(x=\sin y, \quad 0 \leq y \leq \pi ; \quad y\) -axis

You weld the ends of two steel rods into a right-angled frame. One rod is twice the length of the other. Where is the frame's center of mass? (Hint: Where is the center of mass of each rod?)

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 grapher's or computer's integral evaluator to find the surface's area numerically. \(x^{1 / 2}+y^{1 / 2}=3\) from \((4,1)\) to \((1,4) ; \quad x\) -axis

Find the lengths of the curves in Exercises \(1-6\) $$ x=(2 t+3)^{3 / 2} / 3, \quad y=t+t^{2} / 2, \quad 0 \leq t \leq 3 $$

Bathroom scale \(A\) bathroom scale is compressed 1\(/ 16\) in. when a 150 -lb person stands on it. Assuming that the scale be- haves 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.?

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 grapher's or computer's integral evaluator to find the surface's area numerically. \(y+2 \sqrt{y}=x, \quad 1 \leq y \leq 2 ; \quad y\) -axis

Exercises \(5-12\) give density functions of thin rods lying along various intervals of the \(x\) -axis. Use Equations \((3 a)\) through \((3 c)\) to find each rod's moment about the origin, mass, and center of mass. $$ \delta(x)=4, \quad 1 \leq x \leq 3 $$

Find the lengths of the curves in Exercises \(1-6\) $$ x=8 \cos t+8 t \sin t, \quad y=8 \sin t-8 t \cos t, \quad 0 \leq t \leq \pi / 2 $$

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

Use the shell method to find the volumes of the solids generated by re- volving the regions bounded by the curves and lines about the \(y\)-axis. \(y=x, \quad y=-x / 2, \quad x=2\)

Find the lengths of the curves in Exercises \(7-16\) . If you have a grapher, you may want to graph these curves to see what they look like. \(y=(1 / 3)\left(x^{2}+2\right)^{3 / 2} \quad\) from \(\quad x=0\) to \(x=3\)

Fish tank A horizontal rectangular freshwater fish tank with base \(2 \mathrm{ft} \times 4 \mathrm{ft}\) and height 2 \(\mathrm{ft}\) (interior dimensions) is filled to within 2 in. of the top. a. Find the fluid force against each side and end of the tank. b. If the tank is sealed and stood on end (without spilling), so that one of the square ends is the base, what does that do to the fluid forces on the rectangular sides?

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

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 grapher's or computer's integral evaluator to find the surface's area numerically. \(y=\int_{1}^{x} \sqrt{t^{2}-1} d t, \quad 1 \leq x \leq \sqrt{5} ; \quad x\) -axis

Use the shell method to find the volumes of the solids generated by re- volving the regions bounded by the curves and lines about the \(y\)-axis. \(y=2 x, \quad y=x / 2, \quad x=1\)

Find the lengths of the curves in Exercises \(7-16\) . If you have a grapher, you may want to graph these curves to see what they look like. \(y=x^{3 / 2}\) from \(x=0\) to \(x=4\)

Find the volumes of the solids in Exercises \(3-10\) . 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 \(y=\sec x\)

Semicircular plate \(A\) semicircular plate 2 ft in diameter sticks straight down into freshwater with the diameter along the surface. Find the force exerted by the water on one side of the plate.

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?

Use the shell method to find the volumes of the solids generated by re- volving the regions bounded by the curves and lines about the \(y\)-axis. \(y=x^{2}, \quad y=2-x, \quad x=0,\) for \(x \geq 0\)

Find the lengths of the curves in Exercises \(7-16\) . If you have a grapher, you may want to graph these curves to see what they look like. \(x=\left(y^{3} / 3\right)+1 /(4 y) \quad\) from \(\quad y=1\) to \(y=3\) (Hint: \(1+(d x / d y)^{2}\) is a perfect square.)

Find the volumes of the solids in Exercises \(3-10\) . 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 circular disks with diameters running from the \(y\) -axis to the parabola \(x=\sqrt{5} y^{2} .\)

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 \(x=a\) , \(0 < a < b\) .

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.

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

Find the lengths of the curves in Exercises \(7-16\) . If you have a grapher, you may want to graph these curves to see what they look like. \(x=\left(y^{3 / 2} / 3\right)-y^{1 / 2}\) from \(y=1\) to \(y=9\) (Hint: \(1+(d x / d y)^{2}\) is a perfect square.)

Find the volumes of the solids in Exercises \(3-10\) . 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.

Use the shell method to find the volumes of the solids generated by re- volving the regions bounded by the curves and lines about the \(y\)-axis. \(y=2 x-1, \quad y=\sqrt{x}, \quad x=0\)

Find the lengths of the curves in Exercises \(7-16\) . If you have a grapher, you may want to graph these curves to see what they look like. \(x=\left(y^{4} / 4\right)+1 /\left(8 y^{2}\right) \quad\) from \(y=1\) to \(y=2\) (Hint: \(1+(d x / d y)^{2}\) is a perfect square.)

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. a. Find the volume of the column. b. What will the volume be if the square turns twice instead of once? Give reasons for your answer.

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.

Use the shell method to find the volumes of the solids generated by re- volving the regions bounded by the curves and lines about the \(y\)-axis. \(y=3 /(2 \sqrt{x}), \quad y=0, \quad x=1, \quad x=4\)

Find the lengths of the curves in Exercises \(7-16\) . 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 \(y=3\) (Hint: \(1+(d x / d y)^{2}\) is a perfect square.)

Cavalieri's Principle A solid lies between planes perpendicular to the \(x\) -axis at \(x=0\) and \(x=12 .\) The cross-sections by planes perpendicular to the \(x\) -axis are circular disks whose diameters run from the line \(y=x / 2\) to the line \(y=x\) as shown in the accompanying figure. Explain why the solid has the same volume as a right circular cone with base radius 3 and height \(12 .\)

In Exercises \(13-24\) , 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 \(y=4\)

Use the shell method to find the volumes of the solids generated by re- volving the regions bounded by the curves and lines about the \(y\)-axis. Let \(g(x)=\left\\{\begin{array}{ll}{(\tan x)^{2} / x,} & {0< x \leq \pi / 4} \\\ {0,} & {x=0}\end{array}\right.\) a. Show that \(x g(x)=(\tan x)^{2}, 0 \leq x \leq \pi / 4\) b. Find the volume of the solid generated by revolving the shaded region about the \(y\)-axis.

Find the areas of the surfaces generated by revolving the curves in Exercises \(13-22\) 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